Principia Mathematica (Motte Translation, 1848) — part 5 of 45

Since the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural things ; and the modernSj laying aside substantial forms and occult qualities, have endeav- oured to subject the phsenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect ; as rational, which proceeds accurately by demonstration ; and practical. To practical me- chanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accu- rate is called geometrical ; what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic ; and if any could work with perfect accuracy, he would be the most perfect mechanic of all ; for the description of right lines and circles, upon which geometry is founded, belongs to me- chanics. Geometry does not teach us to draw these lines, but requires them to be drawn ; for it reqiiires that the learner should first be taught to describe these accurately, before he enters upon geometry ; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics ; and by geometry the use of them, when so solved, is shown ; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any mo- tions, accurately proposed and demonstrated. This part of mechanics was

Ixviii THE author's preface.

cultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power), iio otherwise than as it moved weights by those powers. Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly? those things which relate to gravity, levity, elastic force, the resist- ance of fluids, and the like forces, whether attractive or impulsive ; and therefore we offer this work as the mathematical principles of philosophy ; for all the difficulty of philosophy seems to consist in this — from the phaenom- ena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phasnomena ; and to this end the general propositions in the first and second book are directed. In the third book we give an example of this in the explication of the System of the World ; for by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phaenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the mo- tions of the planets, the comets, the moon, and the sea. I wish we could de- rive the rest of the phaenomena of nature by the same kind of reasoning from mechanical principles ; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other ; which forces being unknown, philosophers have hitherto at- tempted the search of nature in vain ; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy. In the publication of this work the most acute and universally learned Mr. Edmund Halley not only assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing ; for when he had obtained of me my demonstra- tions of the figure of the celestial orbits, he continually pressed me to com- municate the same to the Royal Society, who afterwards, by their kind en- couragement and entreaties, engaged me to think of publishing them. But after I had begun to consider the inequalities of the lunar motions, and had entered upon some other things relating to the laws and measures of gravity, and other forces ; and the figures that would be described by bodies attracted according to given laws ; and the motion of several bodies moving among themselves ; the motion of bodies in resisting mediums ; the forces, densities, and motions, of mediums ; the orbits of the comets, and such like ;

Ixix

I deferred that publication till I had made a search into those matters^ and could put forth the whole together. What relates to the lunar motions (be- ing imperfect), I have put all together in the corollaries of Prop. 66, to avoid being obliged to propose and distinctly demonstrate the several things there contained in a method more prolix than the subject deserved, and in- terrupt the series of the several propositions. Some things, found out after the rest, I chose to insert in places less suitable, rather than change the number of the propositions and the citations. I heartily beg that what I have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and in- vestigated by new endeavours of my readers.

ISAAC NEWTON.

Cambridge, Trinity College, May 8, 1686.

In the second edition the second section of the first book was enlarged. In the seventh section of the second book the theory of the resistances of fluids was more accurately investigated, and confirmed by new experiments. In the third book the moon's theory and the praecession of the equinoxes were more fully deduced from their principles ; and the theory of the comets was confirmed by more examples of the calculation of their orbits, done also with greater accuracy.

In this third edition the resistance of mediums is somewhat more largely handled than before; and new experiments of the resistance of heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on ; and there are added new observations of Mr. Pound's of the proportion of the diameters of Jupiter to each other : there are, besides, added Mr. Kirk's observations of the comet in 1680 ; the orbit of that comet com- puted in an ellipsis by Dr. Halley ; and the orbit of the comet in 1723, computed by Mr. Bradley.

BOOK I.

THE

MATHEMATICAL PRMCIPLES

OF

NATURAL PHILOSOPHY

DEFINITIONS.

DEFINITION L

The quantity of ^natter is the measure of the same, arising from its density and bulk conjunctly. Thus air of a double density, in a double space, is quadruple in quan- tity ; in a triple space, sextuple in quantity. The same thing is to be un- derstood of snow, and line dust or powders, that are condensed by compres- sion or liquefaction ; and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body ; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.

DEFINITION II.

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly. The motion of the whole is the sum of the motions of all the parts ; and therefore in a body double in quantity, with equal velocity, the motion is double ; with twice the velocity, it is quadruple.

DEFINITION III.

The vis insita, or innate force of Qnatter, is a power of resisting, hy which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line. This force is ever proportional to the body whose force it is ; and differs

nothing from the inactivity of the mass, but in our manner of conceiving

74 THE MATHEMATICAL PRINCIPLES

it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertice, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition ; and the exercise of this force may be considered both as resistance and impulse ; .it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed ; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distin- guished ; nor are those bodies always truly at rest, which commonly are taken to be so.

DEFINITION IV.

An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

This force consists in the action only ; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertice only. Impressed forces are of different origins \ as from percussion, from pressure, from centripetal force.

DEFINITION y.

A centripetal force is that by which bodies are drawn or impelled^ or any

way tend, toiuards a point as to a centre.

Of this sort is gravity, by which bodies tend to the centre of the earth ; magnetism, by which iron tends to the loadstone ; and that force, what- ever it is, by which the planets are perpetually drawn aside from the rec- tilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to re- cede from the hand that turns it ; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and„as soon as ever it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits ; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right lines, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would

OF NATURAL PHILOSOPHY. 75

go oif from it in a right line, and that with an uniform motion, if the re- sistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the velo- city of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it devi- ate from a rectilinear course, and the farther it will go. If a leaden ball, projected from the top of a moimtain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground ; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velo- city, we may at pleasure increase the distance to which it might be pro- jected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls ; or lastly, so that it might never fall to the earth, but go forward into the celestial spaces, and pro- ceed in its motion in infinitum. And after the same manner that a pro- jectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the recti- linear way, which by its innate force it would pursue ; and would be made to revolve in the orbit which it now describes ; nor could the moon with- out some such force, be retained in its orbit. If this force was too small, it would not sufficiently turn the moon out of a rectilinear course : if it was too great, it would turn it too much, and draw down the moon from its orbit towards the earth. It is necessary, that the force be of a just quantity, and it belongs to the mathematicians to find the force, that may serve exactly to retain a body in a given orbit, with a given velocity ; and vice versa, to determine the curvilinear way, into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force.

The quantity of any centripetal force may be considered as of three kinds; absolute, accelerative, and motive.

DEFINITION VI.

The absolute quantity of a centripetal force is the measure of the same^ proportional to the efficacy of the cause that propagates it from the cen- tre, through tlie spaces round about.

Thus the magnetic force is greater in one load-stone and less in another^, according to their sizes and strength of intensity.,

76 THE MATHEMATICAL PRINCIPLES

DEFINITION YII.

The accelerafive quantity of a centripetal force is the 'measure of the same, proportional to the velocity which it generates in a given ti^ne.

Thus the force of the same load-stone is greater at a less distance, and less at a greater : also the force of gravity is greater in valleys, less on tops of exceeding high mountains ; and yet less (as shall hereafter be shown), at greater distances from the body of the earth ; but at equal distan- ces, it is the same everywhere ; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small.

DEFINITION VIII.

The motive quantity of a centripetal force, is the Tneasiire of the same,

proportional to the motion which it generates in a given time.

Thus the weight is greater in a greater body, less in a less body ; and, in the same body, it is greater near to the earth, and less at remoter dis- tances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight ; and it is al- ways known by the quantity of an equal and contrary force just sufficient to hinder the descent of the body.

These quantities of forces, we may, for brevity's sake, call by the names of motive, accelerative, and absolute forces ; and, for distinction's sake, con- sider them, with respect to the bodies that tend to the centre ; to the places of those bodies ; and to the centre of force towards which they tend ; that is to say, I refer the motive force to the body as an endeavour and propen- sity of the whole towards a centre, arising from the propensities of the several parts taken together ; the accelerative force to the place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them ; and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about ; whether that cause be some central body (such as is the load-stone, in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical .causes and seats.

Wherefore the accelerative force will stand in the same relation to the ■matlve, as celerity does to motion. For the quantity of motion arises from the celerity drawn into the quantity of matter ; and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum. of the actions of the accelerative force, upon the several particles ,.-of the W^J:) is the motive force of the whole. Hence it is, that near the

OP NATURAL PHILOSOPHY. 77

surface of the earth; where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body : but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those re- gions, where the accelerative gravity is diminished into one half, the weight of a body two or three times less, will be four or six times less.

I likewise call attractions and impulses, m the same sense, accelerative, and motive ; and use the words attraction, impulse or propensity of any sort towards a centre, promiscuously, and indiiferently, one for another ; considering those forces not physically, but mathematically : wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points) ; when at any time I happen to speak of centres as attracting, or as endued with attractive powers.

SCHOLIUM.

Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be under- stood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the re- moving of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.

I. Absolute, true, and mathematical time, of itself, and from its own na- ture flows equably without regard to anything external, and by another name is called duration : relative, apparent, and common time, is some sen- sible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ; such as an hour, a day, a month, a year.

II. Absolute space, in its own nature, without regard to anything exter- nal, remains always similar and immovable. Relative space is some mo- vable dimension or measure of the absolute spaces ; which our senses de- termine by its position to bodies ; and which is vulgarly taken for immo- vable space ; such is the dimension of a subterraneous, an cereal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude ; but they do not re- main always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains al- ways the same, will at one time be one part of the absolute space into which

78 THE MATHEMATICAL PRINCIPLES

the air passes ; at another time it will be another part of the same, and so, absolutely understood, it will be perpetually mutable.

III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space ; not the situation, nor the external surface of the body. For the places of equal solids are always equal ; but their superfices, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts ; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places ; and therefore the place of the whole is the same thing with the sum of the places of the parts, and for that reason, it is internal, and in the whole body.

IV. Absolute motion is the translation of a body from one absolute place into another ; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses ; or that part of its cavity which the body fills, and which therefore moves together with the ship : and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space ; partly from the relative motion of the ship on the earth ; and if the body moves also relatively in the ship ; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative mo- tions as well of the ship on the earth, as of the body in the ship ; and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved toward the east, with a velocity of 10010 parts ; while the ship itself, with a fresh gale, and full sails, is carried towards the west, with a velocity ex- pressed by 10 of those parts ; but a sailor walks in the ship towards the cast, with 1 part of the said velocity ; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10001 parts, and relatively on the earth towards the west, with a velocity of 9 of those parts.

Absolute time, in astronomy, is distinguished from relative, by the equa- tion or correction of the vulgar time. For the natural days are truly un- equal, though they are commonly considered as equal, and used for a meas- ure of time; astronomers correct this inequality for their more accurate . deducing of the .celestial motions. It may be, that there is no such thing ;as an equable mod:ion, whereby time may be accurately measured. All mo-

OF NATURAL PHILOSOPHY. 79

tions may be accelerated and retarded, but the true, or equable, progress of absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all : and therefore it ought to be distinguished from what are only sensible measures thereof ; and out of which we collect it, by means of the astronomical equation. The necessity of which equation, for deter- mining the times of a phsenomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter.

As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession ; and in space as to order of situation. It is from their essence or nature that they are places ; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places ; and trans- lations out of those places, are the only absolute motions.

But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body con- sidered as immovable, we define all places ; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones ; and that without any inconvenience in common af- fairs ; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.

But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest ; but impossi- ble to know, from the position of bodies to one another in our regions, whether any of these do keep the same position to that remote body ; it follows that absolute rest cannot be determined from the position of bodies in our regions.

It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion ; and the impetus of bodies moving forward, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be deter-

so THE MATHEMATICAL PRINCIPLES

mined by the translation of it from those which only seem to rest ; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise; all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions ; and though that translation were not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel ; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell.

A property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it ; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions ; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place ; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Where- fore, entire and absolute motions can be no otherwise determined than by immovable places ; and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another : and upon this account must ever remain unmoved ; and do thereby constitute immovable space.

The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved ; but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies mth which the former is compared, that by their giving way, that relation may be changed, in which the re- lative rest or motion of this other body did consist. Again, true motion suiFers always some change from any force impressed upon the moving body ; but relative motion does not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, Avith which the comparison is made, that the relative position may be pre- served, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Upon which accounts, true motion does by no means consist in such relations.

The effects which distinguish absolute from relative motion are, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute cir- cular motion, they are greater or less, according to the quantity of the

OF NATURAL PHILOSOPHY. SI

motion. If a vessol, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at- rest together with the water ; after, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion ; the surface of the water will at first be plain, as before the vessel began to move : but the vessel, by grad- ually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I hiive experi- enced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. Tliis ascent of the water shows its en- deavour to recede from the axis of its motion ; and the true and absolute circular motion of the water, which is here directly contrary to the rela- tive, discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest, it pro- duced no endeavour to recede from the axis ; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its en- deavour to recede from the axis ; and this endeavour showed the real cir- cular motion of the water perpetually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect ; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps par- take of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them ; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes,. endeavour to recede from the axis of their motions.

Wherefore relative quantities are not the quantities themselves, whose- names they bear, but those sensible measures of them (either accurate or inaccurate), which are commonly used instead of the measured quantities themselves. And if the meaning of words is to be determined by their

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82

THE MATHEMATICAL PRINCIPLES

use, then by the names time, space, place and motion, their measures are properly to be understood ; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the sacred writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quan- tities themselves with their relations and vulgar measures.