impressed altogether and at once, or gradually and successively. And this
motion (being always directed the same way with the generating force),
if the body moved before, is added to or subducted from the former
motion, according as they directly conspire with or are directly contrary
to each other; or obliquely joined, when they are oblique, so as to pro-
duce a new motion compounded from the determination of both.
T H E I L L U S T R A T E D O N T H E S H O U L D E R S O F G I A N T S
L A W I I I . T O E V E R Y A C T I O N T H E R E I S A L W A Y S O P P O S E D A N E Q U A L R E A C T I O N : O R T H E M U T U A L A C T I O N S O F T W O B O D I E S U P O N E A C H O T H E R A R E A L W A Y S E Q U A L , A N D D I R E C T E D T O C O N T R A R Y P A R T S .
Whatever draws or presses another is as m u c h drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavor to relax or u n b e n d itself, will draw the horse as m u c h towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as m u c h as it advances that of the other. If a body impinge u p o n another, and by its force change the m o t i o n of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. T h e changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.
C O R O L L A R Y I . A B O D Y B Y T W O F O R C E S C O N J O I N E D W I L L D E S C R I B E T H E D I A G O N A L O F A P A R A L L E L O G R A M , I N T H E S A M E T I M E T H A T I T W O U L D D E S C R I B E T H E S I D E S , B Y T T H O S E F O R C E S A P A R T .
If a body in a given time, by the force M impressed apart in the place A, should with a u n i f o r m m o t i o n be carried from A to B; and by the force N impressed apart in the same place, should be carried from A to C; complete the parallelogram ABCD, and, by b o t h forces acting togeth- er, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M , by which the body is carried towards the line B D . T h e body therefore will arrive at the line B D in the same time, w h e t h e r the force N be impressed or not; and therefore at the end of that time it will be f o u n d somewhere in the line BD. By the same argument, at the end
I S A A C N E W T O N
of the same time it will be found somewhere in the line CD. Therefore it will be f o u n d in the point D, where b o t h lines meet. But it will move in a right line f r o m A to D, by Law I.
C O R O L L A R Y I I . A N D H E N C E I S E X P L A I N E D T H E C O M P O S T I O N O F A N Y O N E D I R E C T F O R C E A D , O U T O F A N Y T W O O B L I Q U E F O R C E S A C A N D C D ; A N D , O N T H E C O N T R A R Y , T H E R E S O L U T I O N O F A N Y O N E D I R E C T F O R C E A N D I N T O T W O O B L I Q U E F O R C E S A C A N D C D : W H I C H C O M P O S I T I O N A N D R E S O L U T I O N A R E A B U N D A N T L Y C O N F I R M E D F R O M M E C H A N I C S .
As if the unequal radii O M and O N drawn from the center O of any wheel, should sustain the weights A and P by the cords M A and N P ; and the forces of those weights to move the wheel were required. T h r o u g h the center O draw the right line K O L , meeting the cords p e r p e n d i c u - larly in K and L; and from the center O, with O L the greater of the distances O K and O L , describe a circle, meeting the cord M A in D : and drawing O D , make A C parallel and D C perpendicular thereto. Now, it being indifferent w h e t h e r the points K, L, D, of the cords be fixed to the plane of the wheel or not, the weights will have the same effect w h e t h e r they are suspended from the points K and L, or from D and L. Let the whole force of the weight A be represented by the line AD, and let it be resolved into the forces A C and C D ; of which the force AC, drawing the radius O D directly from the center, will have n o effect to move the wheel: but the other force D C , drawing the radius D O perpendicularly, will have the same effect as if it drew perpendicularly the radius O L equal to O D ; that is, it will have the same effect as the weight P, if that weight is to the weight A as the force D C is to the force DA; that is (because of the similar triangles A D C , D O K ) , as O K to O D or OL. Therefore the weights A and P, which are reciprocally as the radii O K and O L that lie in the same right line, will be equipollent, and so remain in equilibrio; which is the well k n o w n property of the balance, the lever, and the wheel. If either weight is greater than in this ratio, its force to move the wheel will be so much greater.
If the weight p, equal to the weight P, is partly suspended by the cord Np, partly sustained by the oblique plane pG; draw p H , N H , the f o r m e r
T H E I L L U S T R A T E D O N T H E S H O U L D E R S O F G I A N T S
perpendicular Co the horizon, the latter to the plane pG; and if the force of the weight p tending downwards is represented by the line p H , it may be resolved into the forces p N , H N . If there was any plane p Q , p e r p e n - dicular to the cord p N , cutting the other plane p G in a line parallel to the horizon, and the weight p was supported only by those planes p Q , pG, it would press those planes perpendicularly with the forces p N , H N ; to wit, the plane p Q with the force p N , and the plane p G with the force
I S A A C N E W T O N
H N . And therefore if the plane p Q was taken away, so that the weight might stretch the cord, because the cord, n o w sustaining the weight, sup- plies the place of the plane that was removed, it will be strained by the same force N w h i c h pressed u p o n the plane before. Therefore, the t e n - sion of this oblique cord p N will be to that of the other perpendicular cord P N as p N to p H . A n d therefore if the weight p is to the weight A in a ratio c o m p o u n d e d of the reciprocal ratio of the least distances of the cords P N , A M , from the center of the wheel, and of the direct ratio of p H to p N , the weights will have the same effect towards moving the wheel and will therefore sustain each other; as any one may find by experiment.
But the weight p pressing u p o n those two oblique planes may be considered as a wedge between the two internal surfaces of a body split by it; and hence the forces of the wedge and the mallet may be deter- mined; for because the force with which the weight p presses the plane p Q is to the force with w h i c h the same, w h e t h e r by its o w n gravity, or by the blow of a mallet, is impelled in the direction of the line p H towards b o t h the planes, as p H to p H ; and to the force with which it presses the other plane pG, as p N to N H . A n d thus the force of the screw may be deduced from a like resolution of forces; it being n o other than a wedge impelled with the force of a lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is farther confirmed. For on what has been said depends the whole doctrine of mechanics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are c o m p o u n d e d of wheels, pullies, levers, cords, and weights, ascending directly or obliquely, and other mechanical powers; as also the force of the tendons to move the bones of animals.
o p p o s i t e p a g e
Newton's diagram of the fist reflecting telescope he built in 1668. C O R O L L A R Y I I I . T H E Q U A N T I T Y O F M O T I O N , W H I C H I S C O L L E C T E D B Y T A K I N G T H E S U M O F T H E M O T I O N S D I R E C T E D T O W A R D S T H E S A M E P A R T S , A N D T H E D I F F E R E N C E O F T H O S E T H A T A R E D I R E C T E D T O C O N T R A R Y P A R T S , S U F F E R S N O C H A N G E F R O M T H E A C T I O N O F B O D I E S A M O N G T H E M S E L V E S .
T H E I L L U S T R A T E D O N T H E S H O U L D E R S O F G I A N T S
For action and its opposite reaction are equal, by Law III, and there- fore, by Law II, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be sub- ducted from the motion of that which follows; so that the sum win be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both; and therefore the difference of the motions directed towards opposite parts will remain the same.
Thus if a spherical body A with two parts of velocity is triple of a spherical body B w h i c h follows in the same right line with ten parts of velocity, the m o t i o n of A will be to that of B as 6 to 10. Suppose, then, their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, u p o n the meeting of the bodies, if A acquire 3, 4, or 5 parts of motion, B will lose as many; and therefore after reflection A will proceed with 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9 , 1 0 , 1 1 , or 12 parts of m o t i o n , and therefore after meeting proceed with 15, 16, 17, or 18 parts, the body B, losing so many parts as A has got, will either p r o - ceed with 1 part, having lost 9, or stop and remain at rest, as having lost its whole progressive m o t i o n of 10 parts; or it will go back with 1 part, having not only lost its whole motion, but (if I may so say) one part more; or it will go back with 2 parts, because a progressive m o t i o n of 12 parts is taken off. And so the sums of the conspiring motions 15+1 or 16+0, and the differences of the contrary motions 17-1 and 18-2, will always be equal to 16 parts, as they were before the meeting and reflec- tion of the bodies. But, the motions being k n o w n with which the bodies proceed after reflection, the velocity of either will be also k n o w n , by taking the velocity after to the velocity before reflection, as the m o t i o n after is to the m o t i o n before. As in the last case, where the m o t i o n of the body A was of 6 parts before reflection and of 18 parts after, and the velocity was of 2 parts before reflection, the velocity thereof after reflection will be f o u n d to be of 6 parts; by saying, as the 6 parts of motion, before to 18 parts after, so are 2 parts of velocity before reflection to 6 parts after.
I S A A C N E W T O N
But if the bodies are either n o t spherical, or, moving in different right lines, impinge obliquely one u p o n the other, and their motions after reflection are required, in those cases we are first to determine the posi- tion of the plane that touches the concurring bodies in the point of c o n - course; then the m o t i o n of each body (by Corol. II) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, because the bodies act u p o n each other in the direction of a line perpendicular to this plane, the parallel motions are to be retained the same after reflection as before; and to the perpendicular motions we are to assign equal changes towards the contrary parts; in such manner that the sum of the conspiring and the difference of the contrary motions may remain the same as before. From such kind of reflections also some- times arise the circular motions of bodies about their o w n centers. But these are cases w h i c h I do n o t consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.
C O R O L L A R Y I V . T H E C O M M O N C E N T E R O F G R A V I T Y OF T W O O R M O R E B O D I E S D O E S N O T A L T E R I T S S T A T E O F M O T I O N O R R E S T B Y T H E A C T I O N S O F T H E B O D I E S A M O N G T H E M S E L V E S : A N D T H E R E F O R E T H E C O M M O N C E N T E R O F G R A V I T Y O F A L L B O D I E S A C T I N G U P O N E A C H O T H E R ( E X C L U D I N G O U T W A R D A C T I O N S A N D I M P E D I M E N T S ) I S E I T H E R A T R E S T , O R M O V E S U N I F O R M L Y I N A R I G H T L I N E .
For if two points proceed with an u n i f o r m m o t i o n in right lines, and their distance be divided in a given ratio, the dividing point will be either at rest, or proceed uniformly in a right line. This is demonstrated here- after in Lem. X X I I I and its Corol., w h e n the points are moved in the same plane; and by a like way of arguing, it may be demonstrated w h e n the points are not moved in the same plane. Therefore if any n u m b e r of bodies move uniformly in right lines, the c o m m o n center of gravity of any two of t h e m is either at rest, or proceeds uniformly in a right line; because the line which connects the centers of those two bodies so m o v - ing is divided at that c o m m o n center in a given ratio. In like manner the c o m m o n center of those two and that of a third body will be either at rest or moving uniformly in a right line because at that center the
T H E I L L U S T R A T E D O N T H E S H O U L D E R S O F G I A N T S
m
o p p o s i t e p a g e
The universe according to Newton. Here the major principles are determined from
gravitational forces acting on bodies of various masses. The principles behind the model of the solar system arc true for other star systems or even galaxies.
distance between the c o m m o n center of the two bodies, and the center of this last, is divided in a given ratio. In like manner the c o m m o n cen- ter ot these three, and ot a fourth body, is either at rest, or moves uni- formly in a right line; because the distance between the c o m m o n center ot the three bodies, and the center of the fourth is there also divided in a given ratio, and so on ad infinitum. Therefore, in a system of bodies where there is neither any mutual action a m o n g themselves, nor any for- eign force impressed u p o n t h e m from without, and w h i c h consequently move uniformly in right lines, the c o m m o n center of gravity of them all is either at rest or moves uniformly forward in a right line.
Moreover, in a system of two bodies mutually acting u p o n each other, since the distances between their centers and the c o m m o n center ot gravity ot both are reciprocally as the bodies, the relative motions of those bodies, w h e t h e r of approaching to or of receding from that center, will be equal a m o n g themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the c o m m o n center of those bodies, by their mutual action between themselves, is neither promoted nor retarded, nor suffers any change as to its state of m o t i o n or rest. But in a system ot several bodies, because the c o m m o n center of gravity of any two acting mutually u p o n each other suffers no change in its state by that action; and m u c h less the c o m m o n center of gravity of the others with which that action does not intervene: but the distance between those two centers is divided by the c o m m o n center of gravity of all the bodies into parts reciprocally proportional to the total sums of those bodies whose centers they are: and therefore while those two centers retain their state ot m o t i o n or rest, the c o m m o n center of all does also retain its state: it is manifest that the c o m m o n center of all never suffers any change in the state of its m o t i o n or rest from the actions of any two bodies between themselves. But in such a system all the actions ot the bodies a m o n g themselves either happen between two bodies, or are composed of actions interchanged between some two bodies; and therefore they do never produce any alteration in the c o m m o n center of all as to its state of m o t i o n or rest. W h e r e f o r e since that center, w h e n the bodies do not act mutually one u p o n another, either is at rest or moves
I S A A C N E W T O N
!
uniformly forward in some right line, it will, notwithstanding the mutual actions of the bodies among themselves, always persevere in its state, either of rest, or of proceeding uniformly in a right line, unless it is forced out of this state by the action of some power impressed from w i t h o u t u p o n the whole system. And therefore the same law takes place in a system consisting of many bodies as in one single body, with regard to their per- severing in their state of motion or of rest. For the progressive motion, w h e t h e r of one single body, or of a whole system of bodies, is always to be estimated from the m o t i o n of the center of gravity.
T H E I L L U S T R A T E D O N T H E S H O U L D E R S O F G I A N T S
C O R O L L A R Y V . T H E M O T I O N S O F B O D I E S I N C L U D E D I N A G I V E N S P A C E A R E T H E S A M E A M O N G T H E M S E L V E S , W H E T H E R T H A T S P A C E I S A T R E S T , O R M O V E U N I F O R M L Y F O R W A R D S I N A R I G H T L I N E W I T H O U T A N Y C I R C U L A R M O T I O N .
For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by supposition), in both cases the same; and it is from those sums and dif- ferences that the collisions and impulses do arise with w h i c h the bodies mutually impinge one u p o n another. Wherefore (by Law II), the effects of those collisions will be equal in b o t h cases; and therefore the mutual motions of the bodies a m o n g themselves in the one case will remain equal to the mutual motions of the bodies a m o n g themselves in the other. A clear proof of which we have from the experiment of a ship; where all motions happen after the same manner, w h e t h e r the ship is at rest, or is carried uniformly forwards in a right line.
C O R O L L A R Y V I . I F B O D I E S , A N Y H O W M O V E D A M O N G T H E M S E L V E S , A R E U R G E D I N T H E D I R E C T I O N O F P A R A L L E L L I N E S B Y E Q U A L A C C E L E R A T I V E F O R C E S , T H E Y W I L L A L L C O N T I N U E T O M O V E A M O N G T H E M S E L V E S , A F T E R T H E S A M E M A N N E R A S I F T H E Y H A D B E E N U R G E D B Y N O S U C H F O R C E S .
For these forces acting equally (with respect to the quantities of the bodies to be moved), and in the direction of parallel lines, will (by Law II) move all the bodies equally (as to velocity), and therefore will never produce any change in the positions or motions of the bodies a m o n g themselves.
S C H O L I U M .
Hitherto I have laid d o w n such principles as have been received by mathematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time, and that the motion of projectiles was in the curve of a parabola; experience agreeing with both, unless so far as these motions are a little retarded by the resist- ance of the air. W h e n a body is falling, the u n i f o r m force of its gravity
I S A A C N E W T O N
acting equally, impresses, in equal particles of time, equal forces u p o n that body, and therefore generates equal velocities; and in the whole time impresses a whole force, and generates a whole velocity proportional to the time. And the spaces described in proportional times are as the veloc- ities and the times conjunctly; that is, in a duplicate ratio of the times. And w h e n a body is thrown upwards, its u n i f o r m gravity impresses forces and takes off velocities proportional to the times; and the times of ascend- ing to the greatest heights are as the velocities to be taken off, and those heights are as the velocities and the times conjunctly, or in the duplicate ratio of the velocities. And if a body be projected in any direction, the motion arising from its projection is c o m p o u n d e d with the m o t i o n aris- ing from its gravity. As if the body A by its m o t i o n of projection alone could describe in a given time the right line AB, and with its m o t i o n of falling alone could describe in the same time the altitude AC; complete the paralellogram A B D C , and the body by that c o m p o u n d e d m o t i o n will at the end of the time be f o u n d in the place D ; and the curve line AED, which that body describes, will be a parabola, to which the right line AB will be a tangent in A; and whose ordinate B D will be as the square of the line AB. O n the same Laws and Corollaries depend those things which have been demonstrated concerning the times of the vibration of pendulums, and are confirmed by the daily experiments of p e n d u l u m clocks. By the same, together with the third Law, Sir Christ. W r e n , Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did sever- ally determine the rules of the congress and reflection of hard bodies, and m u c h about the same time communicated their discoveries to the Royal Society, exactly agreeing a m o n g themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher W r e n , and, lastly, Mr. Huygens. But Sir Christopher W r e n confirmed the truth of the thing before the Royal Society by the exper- iment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely u p o n that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well