Movement and Weight 

'Force with material movement and weight 

with percussion are the jour accidental powers 

in which all the worhj of mortals have their 

being and their end! 

Speak first of the movement then of the weight because it is produced 
by the movement, then of the force which proceeds from the weight 
and the movement, then of the percussion which springs from the 
weight the movement and often from the force. c.a. 155 v. b 

The action of a pole drawn through still water resembles that of 
running water against a stationary pole. c.a. 79 r. c 

Nothing that can be moved is more powerful in its simple move- 
ment than its mover. c.a. 91 v. b 



The science of weights is led into error by its practice, and in many 
instances this is not in harmony with this science nor is it possible to 
bring it into harmony; and this is caused by the poles of the balances 
by means of which the science of such weights is formed, which poles 
according to the ancient philosophers were placed by nature as poles of 
a mathematical line and in some cases in mathematical points, and 
these points and lines are devoid of substance whereas practice makes 
them possessed of substance, since necessity so constrains as needful to 
support the weight of these balances together with the weights which 
are reckoned upon them. 

I have found that the ancients were in error in their reckoning of 
weights, and that this error has arisen because in a considerable part of 




their science they have made use of poles which had substance and in a 
considerable part of mathematical poles, that is such as exist in the 
mind or are without substance; which errors I set down here below. 

c.a. 93 v. b 


In equal movements made in equal time the mover will always have 
more power than the thing which is moved. And the mover will be so 
much the more powerful than the thing moved in proportion as the 
movement of this thing moved exceeds the length of movement of its 
mover; and the difference of the power of the mover over that of the 
thing moved will be so much less in proportion as the length of 
the movement made by the thing moved is less than the movement 
of this mover. 

But observe, O reader, that in this case you must take count of the 
air which becomes so much the more condensed in front of the thing 
moved as this thing moved is of greater speed; for this air is capable of 
being condensed in an infinite degree. This however could not happen 
with the movements made by things which are moved within water, 
for this is not capable of being condensed, as may be proved by placing 
it in a vessel with a narrow mouth, since for lack of the knowledge of 
some motive power you will not be able to place within it more than 
the natural capacity the vessel will contain. And it is just the contrary 
with the air, for if it is forced into vessels with very narrow mouths 
which contain a quantity of water, and the vessel is tilted at such an 
angle that the water shut up in it is between the mouth of the vessel 
and the air which has been condensed, the power of the condensed air 
drives the water of the vessel with such fury as to penetrate through 
the air for some distance, until the air that remains in the vessel can 
return to its first natural state. 

But to return to our proposition, we may say that among movable 
things of the same gravity that one will have the slower movement of 
which the front that cleaves the air takes up more space; and so con- 
versely as it occupies less air, not however extending itself in such a 
degree of thinness as to cause its weight to fail, for where there is no 
weight there is no local movement through the air. 



There can be no local movement through the air unless it proceeds 
from greater or less density than the density of this air. 

And if my opponent should maintain that the density which the 
condensed air acquires in front of the thing moved is the same in 
front of the mover, and is so much the more in the case of the mover 
in proportion as it comes in contact with a greater quantity of air in 
front of itself when struck and condensed than does the thing moved 
by it, as we see with a hand when it throws a stone through the air: 
the answer to this is that it is impossible for the movement of the 
mover added to the movement of the thing moved to be either swifter 
or less swift than the movement made by the thing moved, nor can it 
ever be that in any part of its accidental movement its speed equals that 
of its mover; and this is proved in the accidental movement, where the 
thing moved lessens its speed at every stage of its movement, although 
the percussion of the thing moved is greater at some distance from the 
mover than it is when close at hand. 

And this we see with an arrow from a bow when its point is resting 
against wood, for though the cord drive it with all the force of the 
bow it only penetrates the wood a very little distance, but does the con- 
trary if it has some movement. Some say that the arrow in moving 
propels a wave of air in front of itself, and that this wave by means of 
its movement prevents the course of the arrow from being impeded. 
This is incorrect however because everything which is moved exhausts 
and impedes its mover. The air therefore which passes in waves in front 
of the arrow does so because of the movement of this arrow, and it 
lends little or no help of movement to its mover, which has to be 
moved by the same mover, but rather checks and shortens the move- 
ment of the thing moved. 

The impetus generated in still water has a different efTect from that 
generated in still air. This is proved from the fact that water in itself is 
never compressed by means of any movement made below its surface, 
as the air is within itself when struck by a moving thing. And this we 
may readily learn from the bubbles with which the water is encum- 
bered from its surface to its bed, which cluster round about as the 
water fills up the vacuum of itself that the fish leaves behind it as it 
penetrates; and the movements of this water strike and drive this fish, 
because water only has weight within water when it has movement, 



and this is the primary cause of the increase of movement for its mover. 

c.a. 108 r. a 

I find that force is infinite together with time; and that weight is 
finite together with the weight of the whole globe of the terrestrial 

I find that the stroke of indivisible time is movement, and that this 
movement is of many varieties, namely natural, accidental and partici- 
pating; and this participating movement ends its greatest power when 
it changes from the accidental to the natural, that is in the middle of 
its course; and the natural is more powerful at the end than in any 
other place; the accidental is strongest in the third and weakest at the 
close. c.a. 117 r. c 

Weight, force, a blow and impetus are the children of movement 
because they are born from it. 

Weight and force always desire their death, and each is maintained 
by violence. 

Impetus is frequently the cause why movement prolongs the desire 
of the thing moved. c.a. 123 r. a 

Of water of uniform weight, depth, breadth and declivity that por- 
tion is swifter which is nearest the surface; and this occurs because the 
water that is uppermost is contiguous to the air, which offers but little 
resistance through its being lighter than the water; and the water that 
is below is contiguous to the earth, which offers great resistance through 
being immovable and heavier than water. It follows that the part which 
is more distant from this base has less resistance than that above which 
is contiguous to the air, for this is light and mobile. c.a. 124 r. a 

Gravity and levity are accidental powers which are produced by one 
element being drawn through or driven into another. 
No element has gravity or levity within its own element. 

c.a. 131 r. b 

If all the bed of the sea were covered with men lying down these men 
would sustain the whole of the element of water, consequently each 
man would find that he had a column of water a mile long upon his 
back. For if the whole sea is all supported upon its bed each part of the 
bed sustains its part of the water. c.a. 153 r. a 



Impetus at every stage of time becomes less by degrees, and the pro- 
longation of its essence is caused by the air or the water, which closes 
up behind the movable thing, filling up the vacuum which the movable 
thing that penetrates it leaves of itself. And this air is more powerful to 
strike and compress the movable thing by direct percussion, than is the 
air which is so olaced as to resist the penetration of this movable thing 
by becoming compressed; and it is this compression of the air which 
diminishes the fury of the aforesaid impetus in the movable thing. 

Impetus is the impression of local movement transmuted from the 
mover to the movable thing and maintained by the air or by the water 
as they move in order to prevent the vacuum. 

The impetus of the movable thing within the water is different from 
the impetus of the movable thing within the air, and these differences 
result from the varieties of the aforesaid liquids, because air is con- 
densable to infinity and water is not. 

The impetus of the water is divided into two parts through its being 
of two natures, one of which is simple and the other complex. The 
simple is entirely beneath the surface of the water, the other is complex, 
that is it is between the air and the water, as is seen with boats. 

The simple impetus does not condense the water in front of the 
movement of the fish, but moves the water behind the movement of 
the fish with the same speed that the mover has; and the wave of the 
water that is over against it will never be swifter than its mover. 

But the movement of the boat, called complex movement because it 
shares with the water and the air, is divided into three chief parts 
because this movement is carried on in three directions, namely against 
the course of the river, in the direction of its current, and crosswise, 
that is along the breadth of the river. c.a. 168 v. b 

If the movement of the oar or of the wing be swifter than the water 
or the air driven by them, that amount of movement which is left in 
the water or the air is completed by the oar or the wing in an opposite 

But if the movement of this water or air be in itself swifter than that 
of the oar or wing this oar and wing will not move against this water 
or air. 



And if the movement of the water or of the air be in itself of the 
same swiftness as that of the oar or the wing that moves in it then the 
oar and wing will follow the movement of the water and the air. 

c.a. 175 v. b 

The compression which the flame produces of itself, which increases 
within the resisting wall of the mortar, is that which produces the im- 
petuous movement of its ball; and this impetus cannot be created with 
less density of flame or less swiftness in its rate of increase. Such swift- 
ness of increase cannot take place within a wall of less resistance than 
that of this mortar. It follows therefore that the expansion which the 
flame makes as it rushes out of the mortar into the air, losing this 
density and directness of course, causes a loss of as much density as it 
acquires in its expansion and it ceases to follow the flight of the ball to 
the extent to which it bends to the . . . c.a. 176 v. a 

The movement of water within water acts as the movement of air 
within air. c.a. 184 v. a 

Anything which descends freely acquires fresh momentum at every 
stage of its movement. 

If a power can move a body through a certain space in a certain time 
it does not necessarily follow that the half of this power will move the 
whole of the body over half the space in the whole of that time, or over 
the whole of the space in double the time. c.a. 202 v. b 

Movements are of [. . .] kinds of which the first is called temporal, 
because it is concerned solely with the movement of time, and this 
embraces within itself all the others; the second is concerned with the 
life of things; the third is termed mental, and this resides in animated 
bodies; the fourth is that of the images of things which are spread 
through the air in straight lines: this class does not appear to be subject 
to time, for it is made indivisible in time and that which cannot be 
divided in the mind is not found among us; the fifth is that of sounds 
which proceed through the air, and this will be treated of later, as also 
of odours and savours, and this we may call movement of the senses; 
the other is called material movement, concerning which we shall 
make our treatise. 



But we shall define movements merely as being of two kinds, of 
which one is material and the other incorporeal, because it is not per- 
ceptible to the visual sense, or we may say that the one is visible the 
other invisible, although among the invisible movements there are a 
considerable number of material movements, such as the movement 
of Saturn, and as there would be with a number of wheels revolving. 
Wherefore we may say that the two kinds of movement are such that 
the one is that which is united with bodies the other with the spirit. But 
among these movements that of the images of things amid the air is 
swiftest, because it covers a great space at the same time as it is very 
brief, and this loses [. . .] through distance, because the air thickens; 
the second is that of the mind. 

Of the movements of the senses we will only mention that of hearing 
because it operates in visible bodies, and works by means of time, as is 
shown in noises, peals of thunder, sounds and voices. Of smell taste and 
touch we will not speak, because they do not form part of our subject. 

Also one might speak of the influences of the planets and of God. 

c.a. 203 v. a 



If a power moves a weight a certain distance in a certain time the 
same power will move the half of this weight double the distance in 
the same time. 

Or this whole power [will move] all the weight half the distance in 
half the time, or the whole power in the same time will move double 
the weight half the distance, or the whole power in half the time [will 
move] the whole weight half the distance. c.a. 212 v. b 


Impetus is a power transmitted from the mover to the movable thing, 
and maintained by the wave of the air within the air which this mover 
produces; and this arises from the vacuum which would be produced 
contrary to the natural law if the air which is in front of it did not fill 
up the vacuum, so causing the air which is driven from its place by 



the aforesaid mover to flee away. And the air that goes before it would 
not fill up the place from which it is divided if it were not that another 
body of air filled up the place from whence this air was divided; and so 
of necessity it follows in succession. And this movement would con- 
tinue to infinity if the air were capable of being condensed to infinity. 

c.a. 219 v. a 



Force is spiritual essence which by fortuitous violence is united to 
weighty bodies, restrained from following their natural inclination, in 
which, although of brief duration, it nevertheless often shows itself of 
marvellous power. c.a. 253 r. c 

[A hymn to force] 

Force is all in the whole of itself and all in every part of itself. 

Force is a spiritual capacity, an invisible power which is implanted 
by accidental violence in all bodies that are withheld from their natural 

Force is nothing else than a spiritual capacity, an invisible power 
which is created and implanted by accidental violence by sensible bodies 
in insensible ones, giving to these a semblance of life; and this life is 
marvellous in its workings, constraining and transforming in place and 
shape all created things, running with fury to its own destruction, and 
producing different effects in its course as occasion requires. 

Tarrying makes it great and quickness makes it weak. 

It lives by violence and dies from liberty. 

It transforms and constrains every body with change of position and 

Great power gives it great desire of death. 

It drives away with fury whatever opposes its destruction. 

Transmuter of various forms. 

Lives always in hostility to whoever controls it. 

Always sets itself against natural desires. 

From small beginnings it slowly becomes larger, and makes itself a 
dreadful and marvellous power. 



And constraining itself it constrains everything. 

. . . dwells in bodies which are kept away from their natural course 
and use. 

. . . willingly consumes itself. 

. . . force is all in all and all through all the body where it is pro- 

. . . Power (. . . nza) [potenza?] is only a desire of flight. 

Always it desires to grow weak and to spend itself. 

Itself constrained it constrains every body. 

Without it nothing moves. 

Without it no sound or voice is heard. 

Its true seed is in sentient bodies. 

Weight is all in all its vertical obstacle and all in every part of it. 

If the oblique obstacle opposed to the weight is loosened and free it 
will not make any resistance to this weight but will fall down with it in 

Weight naturally passes to its desired position. 

Every part of this force contains the whole opposite to weight. 

And often they are victors one over the other. 

They are in the grip of the same natural law, and the more powerful 
conquers the less. 

Weight changes [its position] unwillingly and force is always on the 
point of fleeing. 

Weight is corporeal and force is incorporeal. 

Weight is material and force is spiritual. 

If the one desires flight from itself and death, the other wishes for 
stability and permanence. 

They are often producers one of another: 

If weight brings forth force and force weight. 

If weight conquers force and force weight. 

And if they are of like nature they make long company. 

If the one is eternal the other is fleeting. c.a. 302 v. b 


I ask whether if an arrow is shot from a cross-bow [a distance of] 
four hundred braccia a cross-bow made in the same proportions but of 



four times the force and size will not send the arrow four times as far. 

I ask if you have cross-bows, of equal weight, and elevated in these 
various thicknesses, [diagram] of the same length, what effect will it 
make in the distances upon the same arrow. 

And if a cross-bow sends an arrow weighing two ounces a distance 
of four hundred braccia how many braccia will it send one of four 
ounces ? 


Force cannot exist in bodies without either force or weight together 
with movement. 

Force is caused by violent movement by means of weight or other 

If a thing which moves continuously is given fresh momentum by 
greater movement the thing moved redoubles the velocity of its move- 
ment: for example a revolving wheel, such as the potter's lathe which 
revolves; — by adding to it the movement of the foot this wheel becomes 
swifter; also if a ball moving in a certain direction be struck by the 
player along the line of its movement this movement will be accelerated. 

c.a. 314 v. b 


The movement made by a spherical heavy substance in the air. 

There are two movements which can be made by a spherical heavy 
substance in the air, one of which is called simple, the other compound. 
Simple is that movement in which the surface of the movable thing 
moves as much as its centre; compound is that in which the surface of 
the movable thing is in itself more swift than its centre. 

Simple movement 

Simple movement is that in which the movable thing moves equally 
in every part. 

In compound movement there is no part which moves with a move- 
ment equal to that of the whole, unless it is the diameter, which makes 
itself the seat of the revolving movement. 




The compound movement is transformed into as many different 
aspeets as there are different sides with which it strikes against the 
obstacles that are in its path. 

The simple movement is changed into compound movement, if its 
movement is impeded in any part of its sides. 

In a long course compound movement made in the air resolves itself 
into simple movement; and the fact of this happening makes it more 
certain that the cause of the simple movement is also the cause of the 
compound movement; and this is shown in every wheel to which the 
revolving impulse is imparted, for it endures but little and is constantly 
growing less. c.a. 315 r. a 

Every impetuous movement bends towards the less resistance as it 
flies from the greater. c.a. 315 r. b 

Force is caused by the movement of the lever in its counterlever, and 
by this it is infused into the bodies which it moves. c.a. 316 v. b 

Every heavy substance not held back out of its natural place desires 
to descend more by a direct line than by an arc. This is shown because 
every body whatever it may be, that is away from its natural place r 
which preserves it, desires to regain its first perfection in as brief a 
space of time as possible; and since the chord is described in a less time 
than the arc of the same chord it follows from this that every body 
which is away from its natural place desires to descend more speedily 
by a chord than by an arc. 

From this three things follow: — the first is that the movement of 
gravity in the balance is not entirely natural. This is evident from the 
fact that the arms of this balance as they descend describe an arc, and 
as a consequence curved lines. The second is that the heavy movement 
in the arm of the balance which descends is not entirely violent, since 
in this manner it acquires in its descent natural movement. 

The third is that the heavy movement in the balance is half-way 
between the natural and the violent. 

This is evident seeing that every natural movement is violent or 
indeed is beyond nature. ca. 335 v. f 

Among bodies of varying substance and of similar shape that which, 
has most weight descends most rapidly. 




That spherical and heavy body is of the slowest movement in which 
the contact that it makes with the plane where it touches is nearest to 
the perpendicular drawn from its centre. c.a. 338 r. b 

Weight. [With diagrams] 

The middle of each weight is in a perpendicular line with the centre 
of its support. 

When a man standing or sitting takes a weight in his arms it is 
necessary for the support on which he is resting to be in the middle 
between this weight and himself. 

It is impossible for the force exerted by a man's arm to be able with 
the weight supported by it to extend beyond the upright position with- 
out the counter action of the above named opposite weight. 

Suppose you were to say: — I wish to lean my whole back against a 
wall, and sit on the ground with legs extended in such a way as to 
touch the whole corner with legs and back, and I will take a weight 
in my hands and bring it near to me and move it away with the actual 
force of my arms, and I shall not be moving my back or my head or 
any part of me so as to create any counterpoise to the weight moved 
by my arms, and nevertheless this will be done effectively. 

To this I reply that the effect which the force exerts will not in this 
case extend to any other function than that of keeping the arms united 
with the trunk, as though without flexible joints and in one single 
piece, making this piece like a bar of iron bent in two right angles, 
the extremity of the upper part carrying itself in a perpendicular line 
as far as the middle of the base or the opposite lower part, and if there 
is the burden of a weight superimposed upon this upper extremity this 
weight will exert force below itself upon the perpendicular line of its 
base. c.a. 349 r. b 

A man about to give a great blow with his arms so places himself 
that all his power is on the opposite side to that of the place at which 
he intends to strike, for the thing which moves most exerts most power 
upon the thing that resists the movement. c.a. 352 v. c 

Every impression is preserved for a time in its sensitive object; 
and that which was of greater power will be preserved in its ob' 



ject for a longer time, and for a shorter time with the less powerful. 

In this connection I apply the term sensitive to such object as by any 
impression is changed from that which was at first an insensitive ob- 
ject; — that is one which, while changing from its first state preserves 
within itself no impression of the thing which has moved it. The sensi- 
ble impression is that of a blow received upon a resounding substance, 
such as bells and suchlike things, or like the note in the ear, which, 
indeed, unless it preserved the impression of the notes, could never 
derive pleasure from hearing a voice alone; for when it passes imme- 
diately from the first to the fifth note the effect is as though one heard 
these two notes at the same time, and thus perceived the true harmony 
which the first makes with the fifth; but if the impression of the first 
note did not remain in the ear for an appreciable space of time, the 
fifth, which follows immediately after the first, would seem alone, and 
one note cannot create any harmony, and consequently any song what- 
soever occurring alone would seem to be devoid of charm. 

So, too, the radiance of the sun or other luminous body remains in 
the eye for some time after it has been seen; and the motion of a single 
firebrand whirled rapidly in a circle causes this circle to seem one con- 
tinuous and uniform flame. 

The drops of rain water seem continuous threads descending from 
their clouds; and so herein one may see how the eye preserves the 
impressions of the moving things which it sees. 

The insensitive objects which do not preserve the impressions of the 
things which are opposite to them are mirrors, and any polished sub- 
stance, which, so soon as ever the thing of which it bears the impression 
is removed from before it, becomes at once entirely deprived of that 
impression. We may, therefore, conclude that it is the action of the 
mover pressing against the body moved by it which moves this body 
in the direction in which it moves. 

Amongst the cases of impressions being preserved in various bodies 
we may also instance the wave, the eddies of the water, the winds in 
the air, and a knife stuck into a table, which on being bent in one 
direction and then released, retains for a long time a quivering move- 
ment, all its movements being reciprocal one of another, and all may 
be said to be approaching towards the perpendicular of the surface 
where the knife is fixed by its point. 



The voice impresses itself through the air without displacement of 
air, and strikes upon the objects and returns back to its source. 

The concussion of liquid with solid bodies is of a different character 
from the above-mentioned cases of concussion; and the concussion of 
liquid with liquid also varies from the foregoing. 

Of the concussion of solid with liquid there is seen an example in 
the shores of the ocean, which receive the waters full on their rocks and 
hurl them against the steep crags; and oftentimes it happens that before 
the course of the wave is half completed, the stones carried by it return 
to the sea from whence they came; and their power of destruction is 
increased by the might of the wave which falls back from the lofty 
cliffs, c.a. 360 r. a 

Force never has weight, although it often performs the function of 
The force is always equal to the weight which produces it. 
This is proved by the . . . c.a. 382 r. a 

That body weighs less upon the air which rests upon a greater ex- 
panse of air. We may take as an example the gold from which money 
is made which is extremely heavy, but which when spread out in fine 
leaf for gilding maintains itself upon the air with each slightest move- 
ment of this air. 

[With drawing] 

The hollows of the wings underneath the shoulders receive the revo- 
lution of the air near the starting point of the wings, and nature has 
so conditioned them near to the starting point of these wings by the 
fourth proposition concerning weight where it is stated that that part 
of the support is most powerful which is nearest to its starting point. 

c.a. 395 r. b 

No element when united will have weight within its element; there- 
fore the parts of the air do not weigh upon the lower parts. 

No body of dissimilar quality will come to rest within this if it is at 
liberty, because as this body has not the same quality as the air it must 
needs be either heavier or lighter, and if it is heavier it will drop down, 
and if lighter it will penetrate upward. 



That thing which has most conformity with the clement that sur- 
rounds it will issue forth lrom it with the slowest movement. 

And the thing which is most unlike will separate itself from it with 
more impetuous movement. 

When the force generates swifter movement than the flight of the 
unresisting air this air becomes compressed after the manner of feathers 
pressed and crushed by the weight of the sleeper. And that thing which 
drove the air finding resistance in it rebounds after the manner of a 
ball struck against a wall. TV. 10 a 

The line that is straightest offers most resistance. Tr. 24 a 

That thing which within the line of equality shall find itself at a 
greater distance from its support will be less sustained by it, as is shown 
below in m n [diagram]. 

That thing which is at a greater distance from its support will be less 
sustained by it, and consequently will fulfil its natural desire with 
greater liberty. 

Violent movement the more it is exerted the more it grows weaker : 
natural movement does the opposite. 

That thing which is at a greater distance from its support will be less 
sustained by it: being less sustained it will partake more of its liberty, 
and since the weight that is free always descends the thing therefore 
being weighty will descend more swiftly. 

That part of the pole which is farther from its support will be less 
sustained by this support. Being less sustained it continues to follow its 
nature more freely, and this being heavy and the nature of heavy things 
being to desire to descend it will therefore descend more swiftly than 
any other part. Tr. 30 a 

The air is capable of compression and water is not; and when the 
movements which drive it are swifter than the flight of this air, as the 
part which is more caught by its mover becomes denser and conse- 
quently offers more resistance; and when the movement made in it is 
more rapid than the escaping power of this air its mover comes to take 
a contrary movement. As is shown in the case of the birds which are 
not able to drive the points of their wings downwards with the speed 
with which they are moving, because their motive power moves them 



so much the less as the bird raises itself up as the extremity of the wing 
fails to go down. After the fashion of the man who keeps hands and 
breast close to a wall and presses with his hands upon this wall so that 
if the wall does not give way the man must needs turn back. 

Tr. 42 a 

That part of the cloth that is farthest away from its support will sur- 
render itself most to the movement of the wind. 

That earth which is most mixed with water will offer least resistance 
to weights placed upon it. 

That water which is most intermingled with earth will offer resist- 
ance to greatest weight. Tr. 60 a 


Everything hit against a resisting object leaps back from this object 
at an angle equal to that of the percussion. 

Note concerning water. [Diagram] 

The same is proved in the tenth proposition of the book concerning 
the nature of the blow where it treats of the ball struck against a wall. 
And if you wish to know the depth of a fall of water observe the line 
of the fall in c b, of what degree of slant it shows itself; then observe 
the part that lies between the point of impact b and the point a to 
which it rises; and make the angle a b d and measure how it is shown 
in the tenth of Percussion. And if you should be of opinion that the 
water in this case would not be able to deflect owing to some resisting 
object intervening in the line of its recoil know that if the fall is of long 
continuance it will have worn away every obstacle which was set in the 
path of its springing force. Tr. 66 a 


No inanimate object will move of its own accord; consequently 
when in motion it will be moved by unequal power, unequal that is in 
time and velocity, or unequal in weight; and when the impulse of the 
first motive power ceases the second will cease abruptly. a 22 v. 



[Of force and spherical body \ 

Every spherical body of thick and resisting surface when moved by 
a like force, will make as much movement in the rebounds caused by 
its impact upon a concrete ground as if it were thrown freely through 
the air. 

How admirable Thy justice, O Thou First Mover! Thou hast not 
willed that any power should lack the processes or qualities necessary 
for its results; for if a force have the capacity of driving an object con- 
quered by it a hundred braccia, and this object while obeying it meets 
with some obstacle, Thou hast ordained that the force of the impact 
will cause a new movement which by divers rebounds will recover the 
entire amount of the distance it should have traversed. 

And if you were to measure the track made by these bounds you 
will discover it to be of the same length as it would be if a similar ob- 
ject were impelled freely through the air by the same force. 

You may make an experiment of this with a small glass ball as it 
strikes upon a surface of smooth polished stone. Take a long staff and 
mark it with different colours from end to end, and then give it some- 
one to hold, and set yourself at some distance away [to watch] the re- 
bounds [and see] against the height of the staff to what colours the ball 
rises successively with each rebound, and make a note of them. If 
there are as many observers as the number of times the ball rebounds 
each will keep it more easily in memory. But either have the staff fixed 
at the top or with the end in a hole, for if anyone held it with his hand 
he would interrupt the line of sight of the judge. Arrange that the first 
bound be made between two right angles so that the ball may always 
fall in the same spot, because then the height of the rebounds against 
the staff may be more accurately discerned. 

Then have this ball discharged by the same power in free course and 
make a note of the spot where it strikes; and measure it and you will 
find that the length of the second course is identical with the first. 

a 24 r. 

If you should be in a boat, and you there exert your utmost force, 
the boat will never stir from its position unless the said force has a 
greater obstacle outside this boat than that made within it. 

Again if you are all huddled up in a sack and within it make efforts 



to move yourself you will find it impossible to change your position, 
but if you draw a foot out of the sack and use it as a lever on the 
ground putting your head to the bottom of the sack you will be able 
to draw it ofT backwards. 

The flame also does the same with its desire to multiply and extend 
itself in the bombard, for while it is entirely inside it the bombard does 
not recoil. But when this flame strikes and pushes the resisting air while 
remaining united to that which pushes on the bottom, it is the cause of 
the bombard recoiling; for that portion of the flame that strikes, not 
being able to find in the air that instant passage that it requires, throws 
its force upon the opposite side. a 28 r. 


Force I define as an incorporeal agency, an invisible power, which by 
means of unforeseen external pressure is caused by the movement 
stored up and diffused within bodies which are withheld and turned 
aside from their natural uses; imparting to these an active life of 
marvellous power it constrains all created things to change of form and 
position, and hastens furiously to its desired death, changing as it goes 
according to circumstances. When it is slow its strength is increased, 
and speed enfeebles it. It is born in violence and dies in liberty; and 
the greater it is the more quickly it is consumed. It drives away in fury 
whatever opposes its destruction. It desires to conquer and slay the 
cause of opposition, and in conquering destroys itself. It waxes more 
powerful where it finds the greater obstacle. Everything instinctively 
flees from death. Everything when under constraint itself constrains 
other things. Without force nothing moves. 

The body in which it is born neither grows in weight nor in form. 
None of the movements that it makes are lasting. 

It increases by effort and disappears when at rest. The body within 
which it is confined is deprived of liberty. Often also by its movement 
it generates new force. a 34 v. 

Every weight desires to descend to the centre by the shortest way; 
and where there is the greater weight there there is the greater de- 
sire, and that thing which weighs the most if it is left free falls mosr 




rapidly. The less the slant ot the opposing substance the greater its re- 
sistance. But the weight passes by nature into all that supports it, and 
thus penetrating from support to support it grows heavier as it passes 
from body to body until it realises its desire. Necessity draws it and 
abundance drives it away. It is all in all its vertical opposition and all 
in each of its degrees. And that opposition which slants the most will 
not ofler resistance to its descent, but, being free, will fall together 
with it. In its function of pressing and making heavy it is like force. 
Weight is subdued by force, as force is by weight. One can see weight 
without force, but one cannot see force without weight. If weight has 
no neighbour it seeks one with fury and force drives it away with 
fury. If weight desires an unchangeable position force readily flies from 
it. If weight desires stability and force is always desirous of flight, 
weight of itself is without fatigue, while force is never exempt from it. 
The more weight falls the more it increases and the more force falls 
the more it diminishes. If one is eternal, the other is mortal. Weight is 
natural and force is accidental. Weight desires stability and perma- 
nence, and force desires flight and death of itself. Weight, force and a 
blow resemble each other in respect of pressure. a 35 r. 

In the centre of the direct path taken by heavy bodies which traverse 
the air with violent movement, there is greater power and greater strik- 
ing force when an obstacle is met with than in any other part of its 

The reason of this is that when the weight parts from the force of its 
mover, although this separation is in the initial stage of its power, it 
finds nevertheless the air without movement, and finds it in the initial 
stage of its resistance, and although the sum total of the resistance of 
the air is greater than the power of the weight which is pushed upon it, 
nevertheless as it strikes only a small part it succeeds in remaining the 
conqueror. Consequently it drives it from its place and in so driving it 
it somewhat impedes its own velocity. This air therefore after having 
been pushed pushes and drives the other, and generates revolving 
movements in its wake, of which the weight that is moved within it is 
always the centre, after the fashion of circles formed in the water, 
which have their centre at the spot struck by the stone. And so as the 
one circle drives the other, the whole air that is along the line in front 



of its mover becomes prepared for movement, and this increases in pro- 
portion as the weight that drives it presses the more. In consequence 
this weight finding less resistance in the air redoubles the speed of its 
course, the same as a barge drawn through the water, which moves 
with difficulty at the beginning of the movement although the force of 
its mover may be at its maximum, but as with arched waves this water 
commences to take its movement the barge in following this movement 
meets only with slight resistance and therefore moves with greater ease. 
The bullet likewise finding but slight resistance follows the course it 
has begun until the point at which abandoned in part by its first force 
it commences to grow weak and to drop, and as its course changes the 
way already prepared for its flight by the fleeting air contains it no 
longer; the more it drops however the more it finds fresh resistance in 
the air and the more it delays, up to the point at which resuming its 
natural movement it acquires fresh speed, and even so the barge as it 
turns delays its course. Now therefore I conclude according to what is 
demonstrated in the eighth proposition that that part of the movement 
which occurs between the first resistance of the air and the beginning 
of its drop is of the greater power, and it is the centre of the course 
made in the air in a straight direct line. a 43 v. 


I maintain that the displacement caused by the weight which falls 
is equal to the displacement caused by the force. 

The body that receives the blow is not injured in the part opposite 
as it is in the part which is struck. The proof of this is shown when a 
stone is struck while lying in a man's hands, for the hand is not in- 
jured when it is holding the stone that is struck as much as it would be 
injured if it actually received the blow. a 53 v. 


I affirm that the said movement is based upon several points of 
Force is produced by the lessening and contraction of the muscles 



which draw hack, and of the nerves which stretch as far as the sensa- 
tion communicated by the empty cords dictates. b 3 v. 


If a wheel of which the movement has become very rapid continues 
to make many revolutions after its motive power has abandoned it, 
then if this motive power continues to cause it to turn with the same 
quickness of movement, it would seem that this continuance necessi- 
tates but little force. 

And I conclude that in order to maintain this movement only a 
slight effort by the motive power would be needed, and so much more 
as by nature it tends to become fixed. b 26 v. 


I ask if a weight of a pound falling two braccia bury itself in the 
earth the depth of a hand how deeply will it bury itself if it falls forty 
braccia, and how far a weight of two pounds will bury itself if it falls 
two braccia? 

One may ask also if the size of this weight be represented by a 
quantity a and then this quantity be doubled, its weight remaining the 
same and falling from the same height, how much greater impress the 
lesser bulk will make than the greater if the ground offer equal re- 
sistance? b 61 r. 



The blow since it is of very short and even of indivisible life pro- 
duces suddenly its full and quick effect upon what is opposed to it, and 
this effect ends before it reaches the base of the thing struck. For this 
reason therefore you will find more enlargement at the summit of the 
thing struck than at its base. And if you wish to ascertain how much 
greater the power of the blow is upon the thing struck at its summit 
than at its base* calculate how many times the circumference of the 
base m n will go into that of the summit a c; as many times as m n 



goes into a c so many times will a c receive into itself greater force than 
m n. But if this support a m is pressed down by weight or by force, 
m n will be as much enlarged as a c, because their powers move more 
slowly than that of the blow. c 6 v. 


The Blow 

Since the blow is more swift than the movement, the thing touched 
by the blow although it may be in movement will rather obey the effect 
of this blow than that which accelerates the movement. c 7 r. 

If two make the same journey in the same time he who runs often, 
with frequent intervals for rest, will undergo as much fatigue as he 
who goes gently and continuously. c 7 v. 



If two balls strike together at a right angle one will deviate more 
from its first course than the other in proportion as it is less than the 
other. c 15 r. 

The part of a log first severed from the end of it by the stroke of the 
axe flies off to a greater distance than any other part carried away by 
the same blow. 

This is because the part of the log that first receives the blow receives 
it in the first stage of its power and consequently goes farther. The 
second part flies a less distance because the fury of the blow has already 
subsided, the third still less and so also the fourth. 

The wood which is divided from the rest by the stroke of the axe 
will fly oflf with greater violence at one time than at another as the 
stroke is more powerful at one time than another, and the piece will 
fly of? to a greater distance. This is because as the blow is the most 
powerful and instant thing that a man can do, as is shown in the fourth 
proposition which treats of the nature of the blow, when the axe, 
driven by man's strength and by the movement of the hands in falling, 
from the weight and blow of the hatchet has entered within the surface 
of the close-grained wood, so soon as this fine edge has entered imme- 
diately the thick part of the hatchet follows and proceeds with such 



vigour and swiftness to widen and enlarge the edges of the cut that 
it pulls it asunder with great force, and the quicker it is the more the 
cut will be enlarged and deepened, and if (part) is entirely severed it 
flies from the blow with great swiftness, as may be shown by experi- 

Water air and fire produce the same effect in their rebound from 
objects that oppose their course: 

A piece of wood separated from the rest by the blow of an axe will 
fly of? from it at an equal angle to that of the blow. 

Everything of a compact surface that falls upon a resisting object 
will have the line of its rebound at the same angle as the line of its 
incidence. c 22 v. 

Movement and percussion 

Among bodies of equal movement and size that which is of greater 
weight will give a greater blow to the thing that opposes its course, 
and since turbid water is heavier than clear, the blow which it gives 
upon the thing that opposes its course will be greater. 


Movement of water 

A body with a thicker harder surface will cause the objects that 
strike against it to separate from it with a more powerful and rapid 


Water that falls upon gravel mixed with sand and earth will hollow 
it out more deeply and more rapidly for the aforesaid cause than if it 
fell upon plain soft mud, for as it falls upon gravel it takes a swift 
powerful leap and gnaws away more of what first opposes its bound 
and rises more. 

The angle caused by the percussion of equal spherical bodies is al- 
ways equal to that of the rebound. c 28 r. 

Every weight tends to fall towards the centre by the shortest way. 

c 28 v. 




When you wish to represent a man in the act of moving some weight 
reflect that these movements would be made in different directions, 
that is in the case of simple movement from below upwards, as that 
which a man makes when he stoops to lift a weight with the intention 
of raising it as he straightens himself; or when he wishes to pull some- 
thing backward or push it forward or draw it down with a cord that 
passes over a pulley. Here one should remember that a man's weight 
drags in proportion as the centre of his gravity is distant from that of 
his support, and to this must be added the force exerted by his legs and 
bent spine as he straightens himself. e 15 r. 

The mover is always more powerful than the thing moved. 

e 20 v. 

Of the knowledge of the weights proportioned to the forces of their 
movers : 

The force of the mover ought always to be in proportion to the 
weight of its movable thing and to the resistance of the medium in 
which the weight moves. But one cannot deduce the law of this action 
unless one first gives the quantity of the condensation of the air when 
struck by any movable thing whatever; and this condensation will be 
of greater or less density according to the greater or less speed of the 
mobile thing pressing on it, as is shown in the flight of birds, for the 
sound that they make with their wings in beating the air is deeper or 
more shrill according to whether the movement of the wings is slower 
or swifter. e 28 v. 

The weight of every heavy thing suspended is all in the whole 
length of the cord that supports it and all in each part of it. e 32 v. 


Compound impetus is the name given to that which participates in 
the impetus of the mover and the impetus of the thing moved, as is the 
movement / h c which is between two simple movements one of which 
is near the beginning of the movement and the other near the end: a g 



is the first and d e c is at the end. But the first only obeys the mover 
and the last is only of the semblance of the thing moved. 


The irregular fdechonpossto] impetus accompanies the thing moved 
with three kinds of impetus, of which two spring from the mover and 
the third from the thing moved. The two that originate in the mover 
are the straight movement of the mover mingled with the curved move- 
ment of the thing moved, and the third is the simple movement of the 
thing moved which tends merely to turn in the middle of its con- 
vexity at contact with the plane where it turns and lays itself down. 


Friction is divided into three parts: these are simple, compound and 

Simple friction is that made by the thing moved upon the place 
where it is dragged. Compound is that which the thing moved makes 
between two immovable things. Irregular is that made by corners of 
different sides. e 35 r. 


The weight distributed over the whole length of the cord which sup- 
ports it will give less strain to this cord than if it was suspended to its 
lowest part, and this is proved by one of the Elements which says 
'Among cords of equal thickness the longest is the least strong'. 

Consequently the cord a b which supports the weight distributed 
over all the remainder of the cord b t is so much stronger than the 
opposite part of the cord a c as it is shorter. 

One cord supports as many times the weight of another as the num- 
ber of the curves is greater in the one than the other. 

The division made by the cord with its pulley is never rectangular; 
this is proved from the two simple cords that hang from the same pul- 
ley which would meet at the centre of the earth. 




There are three conditions of gravity, of which the one is its simple 
natural gravity, the second is its accidental gravity, the third the fric- 
tion produced by it. But the natural weight is in itself unchangeable, 
the accidental which is joined to it is of infinite force, and the friction 
varies according to the places wherein it occurs, namely rough or 
smooth places. e 54 v. 


We may define the nature of compound balances both as regards 
circular balances that is to say pulleys and wheels and also rectilinear 
balances. But first I will make some experiment before proceeding 
farther because it is my intention first to cite experience then to show 
by reasoning why this experience is constrained to act in this manner. 
And this is the rule according to which speculators as to natural effects 
have to proceed. And although nature commences with reason and 
ends in experience it is necessary for us to do the opposite, that is to 
commence as I said before with experience and from this to proceed to 
investigate the reason. 

I see that it is necessary in the compound rectilinear balance in the 
second demonstration that as much as the one of the extremities de- 
scends so much the opposite extremity rises, and the cause of this is the 
equality of their arms. e 55 r. 


Gravity suspended or supported is all in all its support and all in 
each of its parts. 

The cord bent over its pulley supports more weight in its pendent 
extremities than when it is stretched out in a continuous straight line. 
This may be proved thus: suppose the cord bent over the said pulley to 
be d c e f and the ultimate strength of its resistance to be represented 
by 10; I affirm that if the same cord be stretched straight as is shown at 
a b it will not support more than five. 

And this proceeds from the seventh of this where it is stated: 'Each 



cord gains as much in strength as it loses in length; consequently the 
cords c d and e f in order each to have their length double that of the 
cord a b must necessarily have double the strength of the cord a b? 


The maximum strength of the curved cord is in the middle of its 
bend: this is proved by the eighth of this which says 'That cord is 
strongest which is thickest'. It follows that as the cord is compressed in 
its fold over the pulley where it is bent it becomes widened and lowers 
itself a little, and for this cause necessity constrains it to become thus 
compressed. e 55 v. 


Every heavy substance moves on that side on which it weighs most. 

And the movement of the heavy substance is made on that side 
where it encounters least resistance. 

The heaviest part of bodies that move in the air becomes the guide 
of their movements. 

That heavy substance is of more slow descent in the air which falls 
in greater width. 

It follows that that heavy substance will have the swiftest descent 
which confines itself within the least width. 

The free descent of every heavy substance is made along the line of 
its greatest diameter. 

That heavy substance will be swiftest in movement which reduces 
itself to the smallest bulk. 

The descent of a heavy substance is as much slower as it extends in 
greater breadth. e 57 r. 


Every liquid heavy substance settles down with its opposite extremi- 
ties in a state of equilibrium when it is of natural uniform weight. 
And it is lowered as much on one side as it is raised on the other and 



acts round its centre, as one sees with the extremities of the balance 
round its axis with their oscillations upwards and downwards until the 
impetus is consumed; and this is brought about solely through the in- 
equality of the opposite sides round the centre of the water or of the 
balance. e 57 v. 

[Gravity and movement. Balances] 

By what is said below the balance does not have all its natural weight 
upon the centre of its revolution, but it has as much less as the weight 
that moves the upper arm has the more slanting movement, as is 
proved in this discourse. 

The heavy substance in suspension is all in all and all in every part 
of the centre line of its support. 

The staff placed slantwise has two kinds of gravity of which one 
weighs slantwise between the centre of the earth and the horizon. The 
other weighs vertically upon the centre of the earth. And of these one 
is accidental and the other natural. And this occurs where the mathe- 
matical centre is not the centre of the revolution of the balance. This is 
proved thus, let a b c d be the balance and s its mathematical centre; 
the centre of the revolution will be /. I affirm that when the balance 
is in such a position the mathematical centre s is the same in the line 
that points towards the centre of the earth, that is g h, as the centre of 
the revolution /, and this line g h divides the staff of the balance into 
two equal and similar parts namely the part a b e f and the part c d e f. 
Whether one wishes to rest the balance upon the point s or the point / 
is immaterial for both the one point and the other are in the central line 
g h which divides the weight equally. 

There still remains the above-mentioned slanting weight which is 
above the centre of the revolution /, that is the weight which is above 
the line n o that is a b r /, to which the counter weight c d r f offers no 
resistance in the above-mentioned slanting movement. e 58 r. 



The power of the mover is always greater than the resistance of the 
thing moved. 




There is added as much accidental weight to the mover placed at the 
extremity of the lever as the movable thing placed at the extremity of 
the counterlever exceeds it in natural weight. 

And the movement of the mover is as much greater than that of the 
thing moved as the accidental weight of this mover exceeds its natural 

This may be proved; for let us say that the movement of the mover 
is from b to d and of the thing moved from a to c\ I maintain that the 
movement b d will be as much greater than the movement a c as the 
accidental weight of b exceeds the [natural] weight b; and as this ex- 
ceeds it by one, the natural weight therefore also ... e 58 v. 


A balance of equal arms and weights when removed from a position 
of equality will have its arms and bows unequal because it changes 
the mathematical centre, and consequently necessity constrains it to re- 
gain the lost equality of arms and weight. This is proved by the second 

Transcript of the above 

A balance with equal arms and weights removed from a position 
of equality will make arms and weights unequal, and consequently 
necessity constrains it to regain the lost equality of arms and weights. 
This is proved by the second of this, and it is proved because the higher 
weight is more removed from the centre of the revolution than the 
lower weight, and consequently having a more feeble support it de- 
scends more easily and lifts up the opposite side of the weight joined 
to the extremity of the lesser arm. 


The accidental weight set in the balance against the natural weight 
is worth as much as this natural weight, and this is proved by means 
of the weight that the pole of the balance receives from it, for it loads 



itself so much more with the accidental than the natural weight in 
proportion as the greater arm of this balance exceeds the smaller in 
length. e 59 r. 




First. If the weights, arms and movements slant equally these 
weights will not move each other. 

Second. If the weights equal in slant, and equal, move each other 
the arms of the balance will be unequal; for 'Equal weights maintain 
equal gravity in an equal slant'. 

Fourth. If the weights and the arms of the balance with the slant of 
the movements of these weights are equal, then these weights will show 
themselves unequal if their appendices have their slants unequal. 

Third. But if the equal weights in the arm and the balance move 
one another then the movements of the weights will be of unequal 

First. Why it is a definition. The cord that hangs from the opposite 
sides of the pulley or beam where it rests, is always divided and joined 
in rectangular division and union by the opposite ends of the half 
diameters of this pulley or beam or other round instrument, no matter 
what the slant of the cords may be. e 59 v. 


The potential lever will never be consumed by any power. 

This is proved by the first which says: — 'Every continuous quantity 
is divisible to infinity'. But that which is divisible in act is also divisible 
in power : but it is not the case that that which is divisible in power is 
divisible in act. And if the divisions made potentially towards the in- 
finite change the substance of the matter divided, these divisions will 
return to the composition of their whole, the parts reuniting in the same 
stages in which they were divided. For example let us take ice and 
divide it towards infinity; it will become changed into water, and from 
water into air, and from air into fire 1 ', and if the air should come to 

1 Words crossed out in MS. 



thicken again it will change itself into water, and from water into hail, 
etc. E 60 r. 

A cord of any thickness or strength whatever placed in a level posi- 
tion as regards its opposite extremities will never be able to straighten 
itself if it has any weight placed in the centre of its length. 


Given the straight cord suspended by one of its ends which breaks 
itself exactly where it is fastened by its own weight, one asks what 
weight it will support in any arch that may be made of this cord, this 
arch having its extremities in the position of equality. 

Where the potential lever is in existence the force will also be in 

The force will be of so much the greater excellence as the potential 
lever is less in quantity. 

The force is always created at the same time as the potential lever 
and so it dies when this lever fails. e 60 v. 


[Levers real and potential] 

It is necessary first to describe the real powers in whatever aspect, 
second the semi-real powers, third the potential virtue. Next define how 
the centre of the circumvolution is that which divides the power of the 
lever from the power of its counterlever. 

And the movements of the lever and of its counterlever are always 
contrary in their movement of circumvolution round the above-men- 
tioned centre. And all the powers joined to the lever and counterlever 
are always in rectangular conjunction with this lever both real and po- 
tential. And this angle has always one of its sides which proceeds from 
the centre of the circumvolution, and the real arms of the balance will 
never contain within themselves the potential arms unless they are in a 
position of equality. And always the junction of the real or potential ap- 
pendix with the arm of the balance is the nearest part of this appendix. 

The first direction taken by the appendix after its junction with the 



arm of the balance shows the direction of the potential appendix which 
in rectangular conjunction meets the extremity of the potential lever. 

e 65 v. 

Always the cords folded in an angle in which the heavy substance is 
supported joined to the ring will bear equally the burden of this weight 
at their extremities, and this comes from the fact that the cord is of 
uniform slant. 

The cords which with equal slant meet at the point of suspension of 
a heavy substance always support equally the weight of this substance. 

If two cords converging or diverging descend to a suspended beam 
situated at any angle and are joined to it in any part of its length, so 
that the centre of the beam is placed between them, when these parts 
find themselves in these conditions the centre of gravity of the beam 
will be in the intercentric line which passes through this beam. 


It is impossible that the power of any motive force should be able at 
the same time and with the same movement to create a power greater 
than itself. This is proved by the third of this which says: — Towers 
which are equal to each other do not overcome each other', e 66 r. 

If two cords descend with different lengths and with their slants con- 
verging or diverging to the point of suspension of the extremities of the 
beam, then if this beam be equijacent the slants of the two aforesaid 
cords will be equal one to the other. e 66 v. 

What is gravity and whether it is natural or accidental, and one may 
ask the same concerning levity: 

The answer is that both are accidental powers because each always 
waits for its destruction and one is never born without the other or dies 
without the other. This is proved by the air which forms in the shape 
of a bubble or bladder at the bottom of the water, where the fact of its 
formation immediately creates its levity and creates the weight of the 
water that is above it. And as soon as the bubble arrives at the surface 
its levity dies together with the gravity of the water that was above it. 

The stone that descends through the water first makes the water 



heavy that closes up the beginning of the entrance made by the stone, 
and makes light the water that rises to fill up the space that the stone 
leaves as it descends, because that which moves upwards is light. 

Whether the space of the water penetrated by the stone is filled by 
water descending or by water from the side or by water that is below. 

e 67 r. 

[ Of a heavy body] 

If the angle that is formed by the meeting of the two cords that sup- 
port a weight is cut by the intercentric line of this weight, then this 
angle is divided into two other angles, and as these are divided anew by 
the line of the equality two triangles are then produced, and these will 
have the same proportion between base and base as there is between 
angle and angle, and the same proportion between angle and angle as 
there is between triangle and triangle, and the proportion of triangle to 
triangle is the same as that of gravity to gravity [the same as those] in 
which the heavy substance is divided in relation to the two cords by 
which it is suspended; but the proportion is in inverse ratio because the 
greatest weight falls on the cord which makes itself the outer side of 
the lesser triangle. 

How many are tiie centres of a heavy body which is not uniform? 

There are three centres of a gravity that is uniformly irregular. 

Of which the first is the centre of the natural gravity, the second is 
the centre of the accidental gravity and the third is the centre of the 
magnitude of this heavy body. 

But the centre of the natural gravity does not lie within the position 
of equilibrium if the heavy body is not uniform in weight and of suit- 
able shape, such as the spherical or parallel body or others like these. 

e 68 v. 


The air becomes condensed before bodies that penetrate it swiftly, 
acquiring so much more or less density as the speed is more violent or 

A plank that is uniform in breadth, length, depth and weight will 
not preserve the slanting movement with which it started through the 
air that it penetrates for a long space, but will turn back and then again 
forward and so end its descent with a fluctuating movement. This 



springs from the fact that the uniform natural thickness of the air is 
destroyed because it is condensed under the right angle of the surface 
which strikes the air and cleaves it open. But on the opposite face of 
this plank it does the contrary in that it becomes rarefied, and as a con- 
sequence the rarefied air is of less resistance, and for this reason this 
surface shows itself heavier. The rarefaction acquired by the air that is 
behind this plank is much greater than the density that is produced in 
front of it. It may be proved why the air is condensed : — the air is con- 
densed before the bodies that penetrate it for when one pushes a part 
one does not push the whole of that which is in front. This is demon- 
strated by the flooding that is produced before the prow of a ship. 

e 70 v. 

[Of the descent of heavy bodies in the air] 


The air becomes as much more rarefied behind the movement of the 
movable thing as it becomes denser in front of the same movable thing. 

Why the slanting descent does not keep its straightness. 

The straight line of the oblique descent made by bodies of uniform 
thickness and weight in air of equal resistance will not be continued by 
a heavy substance that descends. And this is due to the fact of the air 
being pressed by the surface of the heavy substance that is penetrating 
it, and becoming condensed resisting and stopping this surface; whence 
of necessity the opposite surface of this heavy substance finding itself in 
rarefied air immediately acquires gravity and falls with more speed 
than that which is retarded by the thickness of the air condensed by it. 
And for this reason the impetus to the right made by the movable thing 
is turned to the left, preserving its slant, up to the point at which the 
other air is condensed anew beneath it; this air again resists and again 
turns the left slanting descent to a right descent, then from right to left 
and from left to right until the point at which the movement ends. 

The descent of the beam placed in any slanting position will always 
be made by a straight line. This is proved by the seventh of this which 
says: — 'Heavy substances of uniform shape and weight which descend 
through an equal medium will have the same rate of speed'. If therefore 
a beam of uniform shape and weight be divided into equal and similar 



parts their descent will be of equal and similar speed, and what the part 
docs the whole will do. 

The adversary says that the whole beam united will not have a de- 
scent similar to the descent of its divided parts because the whole gives 
the whole of its weight slanting to the lower surface, and the part gives 
the whole of its weight to the surface of the part and there is such 
speed from surface to surface as there is from the whole to the part. 

E73 r. 


Of heavy substances not uniform in shape the heavier part will al- 
ways become the guide of their descent through the air. 

With beams of uniform shape at the end of the movement the mov- 
able thing will have always preserved the same slanting position that it 
had at the beginning of the movement. 

This is proved by means of the beam suspended on the balance n m. 

The heavy substance weighs so much less in the air as its movement 
is more slanting. 

And the straight descent of the beam weighs as much less in the air 
as this beam is less slanting. e 73 v. 


A heavy substance of uniform thickness and weight, placed in a posi- 
tion of equilibrium will have a straight descent with equal height in 
each of its parts without ever deviating from the position of its first 
equilibrium, if the air be motionless and of uniform resistance, and this 
movement will be very slow as will be proved. 

But if the heavy substance of uniform thickness be situated slantwise 
in air of uniform resistance then its descent will be made slantwise and 
it will be more rapid than the first aforesaid. e 74 r. 



The balance a e g is formed of two tubes joined at an angle in the 
lower part, and the water that is enclosed within them is joined hav- 
ing in the one arm a quantity of oil and in the other plain water. 



I say that the level of the water in one tube and the other will not 
remain in a position of equality, nor will the surface of the oil find 
itself in a position of equality with the surface of the water placed in 
the opposite tube. This is proved because the oil is less heavy than the 
water and for this reason it remains above the water, and its heaviness 
united in the same tube with the heaviness of the water that lies be- 
neath it makes itself equal to the weight of the water that is united 
to it as a counterweight in the opposite tube. But since it is said that 
oil is less heavy than water it is necessary, if one should desire to create 
an equivalent to the weight of the water that is lacking beneath it, that 
there should be a greater quantity than of this water that is lacking, 
and that as a consequence it occupies more space in this tube than an 
equivalent weight of water would have occupied; and therefore the 
surface of the oil in its tube is higher than the surface of the water in 
the opposite tube, and the surface of the water that is beneath the oil 
is lower than the surface of the water opposite. e 74 v. 


Of heavy bodies which are not flexible and are of equal weight one 
with another, there will be the same proportion between the speed of 
their descent as is that of their uniform bulk. 

Whether the air which clothes bodies with itself moves together 
with these bodies. 

The air that clothes bodies with itself moves together with these 
bodies: this experience shows us when the horse runs along dusty roads. 

Whether the movement of the air is as swift as its mover. 

The air will never have swiftness equal to that of its mover; and this 
is shown us by the movements of the dust that I have already men- 
tioned which follows the course of the horse, for after having moved 
a very short distance it turns back with an eddying movement and 
thereby consumes its impetus. e 80 r. 

[Of movement] 

First: If a power move a body through a certain space in a certain 
time the same power will move the half the body in the same time 
twice the space. Second: Or the same virtue will move the half of this 



body through this whole space in half this time. Third (as Second). 
Fourth: And the half of this virtue will move the half of this body 
through all this space in the same time. Fifth: And this virtue will 
move twice this movable body through the whole of this space in 
twice this time, and a thousand times this movable body through the 
whole of this space in a thousand such periods of time. Sixth: And the 
half of this virtue will move this whole body through half of this space 
in this whole time, and a hundred times this body through the hun- 
dredth part of this space in the same time. Seventh: And if separate 
virtues move two separate movable things though a given space in a 
given time, the same virtues united will move the same bodies united 
through this same space in this same time, because in this case the first 
proportions remain always the same. f 26 r. 



Whether the air escapes from beneath the water by its nature or 
through its being pressed and driven by the water. 

The reply is that since a heavy substance cannot be supported by a 
light one this heavy substance will proceed to fall and seek what may 
support it, because every natural action seeks to be at rest; consequently 
that water which surrounds this air above, on the sides and below finds 
itself all spread against the air enclosed by it, and all that which is 
above d e n m, pushes this air downwards, and would keep it below 
itself if it were not that the laterals a b e f and abed which surround 
this air and rest upon its sides came to be a more preponderant weight 
than the water which is above it; consequently this air escapes by the 
angles n m either on one side or on the other, and goes winding as 
it rises. 

As much force is exerted when an object is moved against the mo- 
tionless air as when the air is moved against a motionless object. 

I have seen movements of the air so violent as to carry away and 
strew in their course immense forest trees and whole roofs of great 
palaces; and I have seen this same fury with its whirling movement 
bore a hole in and hollow out a bank of shingle and carry away in the 
air gravel, sand and water for more than half a mile. 



The same weight will be sustained in the air without movement, if 
falling there with slanting movement, it is able afterwards to raise 
itself up very high with a reflex movement. f 37 v. 

How much air is required to raise various heavy objects of different 
material ? 

How much water entering into the boat will cause it to sink? 

Which air supports more [?less]? That enclosed or rarefied as is 
the case in cupping glasses? Or in its natural state? Or when com- 
pressed, as it is in balls which are inflated by the force of a screw? 
There can be no doubt that it is the rarefied, then that in its natural 
state, and the compressed air resists least [?most]. 

Each part of the volume of the water which falls from the river 
through the air follows the line in which the impetus was commenced 
which led it to this fall. f 47 v. 

[Movement of liquids] 

The natural movements of liquids in the air are swifter and more 
diffused at the end than at the beginning. 

The seminatural movements made by the water between the bed 
of the river and the air will be of equal speed if the bed of this river 
is straight and equal in slant and breadth. 

The accidental movements made within the air become slower at 
every stage of height. 

The semiaccidental movements made between the bed of the canal 
and the air upon a bed of uniform slant and width always tend to 
become slower, but are longer than the simple-accidental because they 
proceed to support themselves and always lighten themselves of part 
of their weight. f 50 v. 

First. If a power move a body a certain space in a certain time the 
same power will move the half of this body in the same time twice 
this space. 

Second. If any force move any movable thing through a certain 
space in a certain time the same force will move the half of this 
movable thing through the whole of this space in half this time. 

Third. If a force move a body in a certain time a certain space the 



same force will move the half of this body in the same time the half 
of this space. 

[Fourth: | If a force move a body in a certain time a certain space 
it is not necessary that this power move twice this weight in twice the 
time twice this space, because it might be that this force would not be 
sufficient to move this movable thing. 

[Fifth?] If a force move a body in a particular time a particular 
space it is not necessary that the half of this force move this same 
movable body in the same time the half of this space for perhaps it 
would not be able to move it. 

Sixth. If two separate forces move two separate movable things the 
same forces united will move in the same time the two movable things 
joined together for the same space because there remains still the same 
proportion. f 51 v. 

Fourth. If a power move a body in a particular time a particular 
space half the force will move in the same time half the movable thing 
half this space. 

If every movable thing pursues its movement along the line of its 
commencement what is it that causes the movement of the arrow or 
thunderbolt to swerve and bend in so many directions whilst still in 
the air? 

What has been said may spring from two causes one of which is 
that the air which is compressed before the fury of its onset offers re- 
sistance to it, and consequently this movement becomes bent and as- 
sumes the nature of a reflex movement though it does not proceed in 
straight lines. Its action is as in the third of the fifth concerning water, 
where it is shown how sometimes the air issuing out of the beds of the 
swamps in the form of bubbles comes to the surface of the water with 
sinuous curving movement. The second manner of sinuous movement 
of the flash of lightning may arise from the fact that the substance of 
the thunderbolt discharges itself now to the right and now to the left, 
now upwards and now downwards, acting in the same way as the 
spark that leaps from the lighted coal; for if the coal exhales gas from 
one of its sides it becomes disintegrated by the damp spreading within 
it and bursting into flame separates these pieces of coal and produces 
another spark, which at its birth strikes against the rest and drives it 



back; and this then does the same again in different directions throw- 
ing out a succession of sparks into the air until it is itself consumed. 
But to me the first explanation pleases most because if the second were 
true you would see that a single thunderbolt would produce many 
just as this spark does. f 52 r. 



I say that when a table is struck in different places the dust that is 
upon it is reduced to various shapes of mounds and tiny hillocks — and 
this arises from . . . 

The dust which when the table is struck is divided into various 
hillocks descends from the hypotenuse of these hillocks, enters beneath 
their base and raises itself again round the axis of the point of the 
hillock, and so moves as to seem a right-angled triangle; and this arises 
from . . . 

When the dusty table is struck at one side observe the manner in 
which the movement of the dust commences to create the aforesaid 
hillocks, and how this dust rises to the top of the hillock. f 61 r. 


The movement of the air is less in front of the movable thing that 
penetrates through it than it is behind this movable thing. 

The air that fills the void which the movable thing leaves of itself 
as it penetrates through this air has its whole mass of equal speed to 
that possessed by this movable thing; but the parts of this air because 
it is of the nature of a vortex, that is with circling movement in the 
form of eddies, is much swifter in itself than the movement of the 
aforesaid movable thing. 

Here it seems that because the movable thing has more swiftness of 
air behind it than in front this air is the cause of the movement of 
this movable thing, and by the seventh this cannot be. 

No movable thing is ever swifter than the swiftness of the power 
which moves it. 

The wave that the air makes before the movable thing which pene- 



(rates it docs not pass almost in front of this movable thing, because 
thft would be contrary to the seventh, the last but one. 

The air behind the movable thing turns back revolving in those 
parts which border on that which flows behind the movable thing. 

The air that flows behind the moveable thing which wanders through 
it is moved by the impetus afforded it by this movable thing; and 
striking with its great expanding wave upon the other air it turns 
back, and with a great revolving movement which grows less at its 
extremities it finally comes to stop and does not follow this movable 
thing. f 74 r. 


No impulse can end immediately but proceeds to consume itself 
through stages of movement. 

The air which was at first behind the hole made by the movable 
thing in the air accompanies this movable thing only a little way, 
according to the eighth. 

Eighth. The air which successively surrounds the movable thing 
that is moving through it makes divers movements in itself. This is 
seen in the atoms that are found in the sphere of the sun when they 
penetrate through some window into a dark place. If among these 
atoms one throws a stone in the length of the solar ray one sees the 
atoms range themselves about the position where the course taken in 
this air by the movable thing was filled by the air, as is proved in the 

Fifth. Nothing that is not provided with power of sensation moves 
of itself, but its movement is made by others; and the movement pro- 
duced acts very briefly in the time and in the space that necessity gives, 
as is shown in the fourth} 

Fourth. The air which moves to fill up the vacuum made in it by 
the movable thing has in itself varying degrees of speed density and 
movement. f 74 v. 

Every movable thing that creates a reflex action ends it course in 
the line of its incidence. 

1 Words crossed out in MS. 



This happens because the movement of its incidence is of greater 
power than the reflex movement and that which is more powerful has 
more duration than the less powerful. 

The movement of incidence of the movable thing will be more 
powerful than its reflex movement, because the percussion of the 
incidence made upon the dense object diminishes in part the impetus 
united to this movable object, and this diminution does not leave this 
reflex movement as powerful as it has been said the movement of inci- 
dence is. In every stage of movement however the impetus of the 
movable thing is diminished of itself apart from its percussion with 
a dense object, and it does not follow that this percussion will not 
lessen it much more, seeing that if you measure the movement which 
this movable thing would have made without incidence and the move- 
ments produced by many bounds up and down, you will find that 
the continuous movement in the same spot will be longer than that 
which is frequently broken by the incidences, even though the begin- 
nings of the impetus in each of them were of equal power one with 
another. f 75 v. 

The more deeply an object is sunk in water the less is it moved 
by the wind which strikes the part of it that is above the water. This 
is contrary to Battista Alberti who gives a general rule of how much 
the wind drives a ship in an hour. f 82 r. 

The very rapid friction of two thick bodies produce fire, f 85 v. 

An object that has its sides set slantwise in the middle of the course 
of the water, although the water strikes upon its smooth side, will go 
in greater bulk towards the side of the slant that is lower. 



And because as is proved in the seventh, it is not alien to the nature 
of the air to become compressed and rarefied almost in a moment, — 
and this is not found possible with water, which keeps its first form, — 
it is therefore easier for the air at the side of and above the movable 
thing to descend there in order to fill the vacuum of itself left by this 
movable thing above itself, than for the air beneath to bend and move 



in a long curving line in order to fill up this vacuum; and this is also 

impossible by the eighth, which proves that every impetus moves this 
air with it along the line in which this impetus is created, as the wind 
which moves as much air as its impetus moves, as is seen with the 
dust stirred by these winds or with the atoms floating in the sun's rays 
when they are blown about by it. So therefore the air, being driven by 
the impetus of the heavy substance which descends there, flies by the 
line of the movement made by its mover, that at the side becomes 
changed into lateral eddies, and the upper air descends there from 
above, always filling up the vacuum that the movable thing leaves 
above of itself at each stage of its movement. 

The air below the movable thing which descends through it becomes 
dense and above it becomes rarefied. f 87 r. 

The movement that the air makes in the air compresses itself and 
the air that it strikes. 

Air moved in a body that is thicker than itself compresses itself more 
than when it moves in other air. 

Air moved within a body lighter than itself becomes rarefied. 

Water moved within a body lighter than itself comes to be rarefied 
not in its quality but in the quantity of its dispersion and extension. 

f 88 r. 

Of the proportion that the movement of water has which is poured 
out of the bottom of a very long trench, the exit of which is a hundred 
times narrower than the breadth and depth of the trench: 

It may be asked how much slower will be the movement of the 
water in the upper part of the trench than the movement of the 'rozza' 
which is formed of the same breadth as the mouth by which the water 
issues from the trench. f 95 r. 

I have learnt from percussion that the falling movement exceeds the 
reflex movement. c 1 r. 



The heavy object which descends freely does not give its weight to 
any support. This may be proved: a is one and b is two; it follows that 



m supports only two because the excess that b which is two has over 
one is one. And this one finding no resistance in a descends freely, for 
it has no support and not having any support does not have its move- 
ment impeded. Therefore m the extremity of the balance is not sensible 
of this excess because that which falls is not supported. g 13 v. 



The cords of the tackles will be broken in the contact of the cord of 
the motive power with the first pulley. This is proved by the ninth of 
this which says: 'The cords in the tackles which descend always 
undergo greater strain than those that rise'. And 'Of the cords that 
descend the last feels less of the force of the motive power than the 
first'. And 'The cords of the tackles feel more weight in proportion 
as they are swifter: of the cords that move within the tackles the last 
is swifter than any of the others.' 


The question is whether the weights descending in the pulleys give 
more or less of their weight to the pivots of the tackles as they descend 
than when they are stationary. g 17 v. 


Of the spring and counterpoise of equal powers it is always the 
spring which is worth more, seeing that its power is pyramidal; and 
its greatest power is at the commencement of its movement. But the 
counterpoise has a compound power, one part of which is cylindrical 
and the other pyramidal. The cylindrical is such that the weight is 
always equal in itself and draws with an equal power both at the 
beginning of the movement and at the end. But the pyramidal com- 
mences in an instant and at a point, and with each degree of move- 
ment and of time it acquires volume and speed, its movement being 
free and swift. But in the slow movement made by the heavy substance 
the pyramidal power ceases and there only remains the cylindrical 



power, which as has been said is worth at the beginning as much as 
I may be worth at the middle or at the end or in any other part of its 
movement. c 30 r. 


If the angle formed by the meeting of two slanting cords which 
descend to the point of suspension of a heavy body is divided by the 
centre line of the heavy body, this angle is divided into two parts 
which will have the same proportion between them as that in which 
the said heavy body is divided within the two cords. 

If the angle formed by the meeting of the two cords that descend to 
the point of suspension of a heavy body is divided by the intercentric 
line which passes through this heavy body, this angle is then divided 
into two other angles, between which there is the same proportion as 
that from base to base and from angle to angle, and equally from 
whole triangle to whole triangle; and these proportions resemble those 
of the weights which the heavy body gives of itself to its supports. 

G39 v. 

The stafT most uniform in thickness bends with the most perfect 
curve. g 45 r. 

OF THE MOVEMENT OF SHIPS [Drawings] b a, d c, f e 

These three ships of uniform breadth, length and depth when pro- 
pelled by equal powers will have different speed of movement; for the 
ship that presents its widest part in front is swifter, and it resembles 
the shape of birds and fishes such as the mullet. And this ship opens 
with its side and in front of it a great quantity of water, which after- 
wards with its revolutions presses against the last two thirds of the 
ship. The ship d c does the opposite, and / e has a movement midway 
between the two aforesaid. • g 50 v. 


In movable things uniform in substance but not of uniform weight 
the heaviest part always serves as a guide. 



The pyramidal weight uniformly irregular in size which is pushed 
by the bow with the point forward will turn its base immediately 
towards the place where the whole is moved. g 51 r. 



An arrow shot from the prow of a ship in the direction in which 
the ship is moving will not appear to stir from the place at which it 
was shot if the ship's movement be equal to that of the arrow. 

But if the arrow from such a ship be shot in the direction from 
whence it is going away with the above-mentioned rate of speed this 
arrow will be separated from the ship with twice its movement. 

g 54 r. 

Of the movement of the movable thing which glides with continu- 
ous movement over a movable spot or which being movable flows 

The movement of the liquid which flows through the bottom of the 
movable vessel will be in a straight line situated slantwise, the slant 
being at a greater or less angle according as the movement of the vessel 
is swifter or slower. 

Of the movement made by the place that receives the thing poured 
out from the vessel. 

There is as much force necessary to receive upon the moving place 
the thing which is poured from the immovable vessel as there is to 
move the vessel which causes the thing to pour upon an immovable 

But if the movement of the vessel that pours equals the movement 
of the place which receives upon it the thing that is poured the move- 
ment of the thing that descends is a slanting straight line, as is shown 


The arrow shot from the centre of the earth to the highest part of 
the elements will ascend and descend by the same straight line al- 
though the elements may be in a movement of circumvolution round 
their centre. 



The gravity which descends through the elements when they are in 
circumvolution always has its movement to correspond to the direction 
of the line that extends from the commencing point of the movement 
towards the centre of the world. g 54 v. 



Of the heavy substance descending through the air, the elements 
that revolve making their entire revolution in twenty-four hours: 

The moving substance that descends from the uppermost part of the 
sphere of fire will make a straight movement as far as the earth al- 
though the elements are in perpetual revolving movement round the 
centre of the world. This is proved: — let b be the heavy substance 
which descends through the elements which moves from a to descend 
to the centre of the world m ; I say that such heavy substance, although 
it may make a curved descent in the form of a spiral line, will never 
deviate from its rectilinear descent by which it advances continually 
between the place from whence it is separated and the centre of the 
world; for if it were parted from the point a and descended to b y dur- 
ing the time in which it has descended to b it has been carried to d, 
the position of the a has become changed to c, and so the movable 
thing finds itself in the [line of] direction that extends between c and 
the centre of the world m. If the movable thing descends from d to /, 
the beginning of the movement moves in the same time from c to /, 
and if / descends to h it turns at g, and so in twenty-four hours the 
movable thing drops to the earth below the place from which it was 
first separated, and such a movement is composite. 

If the movable thing descends from the highest to the lowest part 
of the elements in twenty-four hours its movement is formed of 
straight and curve. I say straight because it will never deviate from the 
very short line which extends from the place from which it is sep- 
arated to the centre of the elements, and it will stop at the lowest 
extremity of such line of direction, which is always found according 
to the zenith beneath the place from whence this movable thing is 
separated. And this movement is curved in itself in all the parts of the 
line and as a consequence at the end it is curved in all the line. And 



from this it comes about that the stone thrown from a tower does not 
strike the side of the tower before reaching the ground. g 55 r. 


Simple impetus is that which moves the arrow or the dart through 
the air. 

Compound impetus is that which moves the stone when it issues 
from the sling, and this impetus is not of long duration because the 
noise produced by the revolving movement of the movable thing re- 
veals to us that this movable thing meets with resistance in the air that 
it penetrates. g 72 v. 


Impetus is the impression of movement transmitted by the mover to 
the movable thing. 

Impetus is a power impressed by the mover on the movable thing. 

Every impression tends to permanence or desires permanence. 

This is proved in the impression made by the sun in the eye of the 
spectator and in the impression of the sound made by the clapper as 
it strikes the bell. 

Every impression desires permanence as is shown by the image of 
the movement impressed upon the movable thing. g 73 r. 


The air that is compressed beneath the movable thing as it descends 
through it in a slanting position flees more from the upper than from 
the lower part of this movable thing. 

Continuous tracts of air are as much rarefied on the one side as they 
are compressed on the other. 

The rarefied part of the air offers so much less resistance as the com- 
pressed part offers more resistance. Therefore the back part of the 
movable thing, b, will descend with greater impetus than its front part, 
and this is the reason why the front a which at the outset was below, 
at the end of the reflex movement is raised up. g 73 v. 




Every natural action is made in the shortest way: this is why the free 
descent of the heavy body is made towards the centre of the world 
because it is the shortest space between the movable thing and the 
lowest depth of the universe. g 75 r. 



Every heavy substance when it moves horizontally has only weight 
in the line of its movement. This is shown by the first part of the 
movement made by the ball from a mortar, this movement being in 
a horizontal direction. 

But the heavy substance floating in the wind in any direction will 
have so much more or less gravity round the front than in the beam 
of the balance, according as the junction of the pendulum of the 
weight with the arm of the balance is nearer to a right angle. 

The revolving movement made rapidly by the weight round the 
fixed point of its axis will have so much more heaviness in this weight 
as this revolving movement is more rapid. g 77 r. 


The weight that the beam of the balance has is divided in two parts, 
of which one tends towards the centre of the world and the other is 
accidental, because it moves by transverse movement. But the first, 
which casts its weight towards the centre of the world, has equal 
lateral weights on either side, and these fix in accordance with their 
centres of gravity and their distances the mathematical centre of this 
balance. The second mathematical centre may rather be termed the 
point of mathematical contact of the pole of the balance with its sup- 
port, and this is away from the centre of the natural gravity of the 
balance by as much space as the upper part of this balance exceeds the 
lower part in weight; for which reason the transverse weight of this 
balance does not of itself give weight to either of the two above- 
mentioned centres; and this is proved in the sixth of this which says: 
— 'Every parallel body of uniform thickness and weight placed slant- 



wise has in it two divided gravities of which the one tends towards the 
centre of the world and the other is transverse.' But the one is natural 
and simple and the other accidental and compound. 

But if such body situated in such a way has a free descent in the air 
the centres of the two gravities will become transformed one into the 
other during some period of movement; and at the end there will 
remain one single centre common to all the heavy substance that de- 
scends; and thus with straight movement it will penetrate all the air 
that is below it. g 79 v. 


Impetus is that which under another name is termed derived move- 
ment, which arises out of primary movement, that is to say when the 
movable thing is joined to its mover. 

In no part of the derived movement will one ever find a velocity 
equal to that of the primary movement. This is proved, because at 
every stage of movement as with the cord of the bow there is a loss of 
the acquired power which has been communicated to it by its mover. 
And because every effect partakes of its cause the derived movement 
of the arrow goes lessening its power by degrees, and thus participates 
in the power of the bow which as it was produced by degrees is so 

The impetus impressed by the mover on the movable thing is in- 
fused in all the united parts of this movable thing. 

And this is shown because all the parts both those internal and those 
of the surface are of equal movement except as regards the movement 
of circumvolution, for in this the more impetuous part always revolves 
round the less impetuous, that is those which are nearer the centre of 
the movable thing. And the part that was first moved always remains 
more distant from the beginning of its movement if it is not checked, 
and this is admitted because it is more potent in its capacity to revolve. 

And if one were to say with the adversary that the impetus which 
moves the movable thing is in the air that surrounds it from the mid- 
dle backwards, one would deny this, because the air that follows the 
movable thing is drawn by the movable thing to fill the void left by 
it, and because also the air that is compressed before the movable thing 
escapes backwards in the opposite direction. 



And it the air turns back it is a manifest proof that it strikes against 
that which the movable thing draws behind it; and when two things 
collide the reflex movement of each starts, and these reflex movements 
are converted into whirling movements which are carried by the air 
that fills up the vacuum left by the movable body, and it is impossible 
for the movement of the mover to be increased by the movement of 
the movable body in the same time, because the mover is always more 
powerful than the movable thing. g 85 v. 



Which will remove the same movable thing a farther distance, a 
great power with a small movement or a lesser power with a greater 
movement ? 


The derived movement made by the same movable thing will be of 
greater length which has a greater primary movement from the same 

This is proved because experience shows us that the same power 
always has such proportion between its primary movement and the 
derived movement of its movable thing which . . . 

It is proved by the fifth of this which says: — between the various 
lengths of the primary movement one will find that the various lengths 
of the derived movement of the same movable thing are in the same 
proportions as their primary movements, because if the power of the 
same mover separates the movable thing from itself a space of a finger 
in one interval of harmonic time, the same power in two intervals of 
harmonic time will separate the same movable thing twice the same 
finger's space from itself. And this arises from the fact that the derived 
movement always has the same proportion as the primary movement. 

The impetus is not always produced in the movable thing because 
not even the mover has always an impetuous movement. 

As is shown by the light chariot drawn by oxen over a level tract, 
for so soon as the oxen end their movement the movement of the 
chariot is ended. g 86 r. 




There are five varieties of local movements of which the first is up- 
wards, the second downwards, the third in a horizontal direction, the 
fourth slanting upwards and the fifth and last slanting downwards. 

How the impetus of movable things joined by a cord passes from 
one movable thing to the other. 

The impetus produced by the movable things joined with a cord 
each of which is reciprocally the mover, of the other will remove the 
two movable things to a short distance from their first mover. 

When of two movable things joined by their two opposite extremi- 
ties to the same cord the one is less than the other, the sum of their 
movement will be less than if these movements were equal to each 

When the larger of two weights joined to a cord is first in move- 
ment, the movement of the two joined together will be greater than 
if the beginning of the movement had originated with the lesser mov- 
able thing. g 86 v. 


Primary movement is that which is made by the movable thing 
during the time when it is joined to its mover. 



Derived movement is that which the movable thing makes in the 
air after it is separated from its mover. 

Derived movement has its origin in primary movement and it never 
has swiftness or power equal to the swiftness or power of the said 
primary movement. 

The course of this movable thing will be in conformity with the 
direction of the course of its mover when all the parts of this movable 
thing have movement equal to the primary movement of their mover. 

If all the parts of the movement made by the part of one whole are 
of equal movement then this movable thing will not revolve; and this 
movement will receive the whole power of its mover, and it will ob- 



serve the proper length that its movement requires, the weight of the 
movable thing being proportioned to the power of its mover, c 87 r. 



The ropes of the tackle share in equal parts the weight that they 

The power that moves the tackle is pyramidal since it proceeds to 
delay with uniform lack of uniformity down to the last rope. 

And the movement of the ropes of this tackle is pyramidal because 
it proceeds to delay with uniform lack of uniformity from the first 
cord to the last. 

Therefore the rope feels the power of the mover so much more 
when it is swifter and so much less when it is slower. 

The ropes feel the power of their mover so much the more when 
they are nearer and so much less when they are farther away. 

G 87 V. 


The same virtue is so much more powerful as it is more concen- 

This is the case with heat, percussion, weight, force and many other 
things. And let us speak first of the heat of the sun which imprints 
itself in a concave mirror and is reflected by it in pyramidal figure, 
which pyramid acquires proportionately so much more power as it 
is more constricted. That is that if the pyramid strikes the object with 
half its length it contracts half its thickness at its foot; and if it strikes 
it at ninety-nine hundredths of its length it contracts its base by ninety- 
nine hundredths and increases by ninety-nine hundredths the heat 
which this base receives from the above-mentioned heat of the sun or 
of the fire. 

Furthermore the percussion of the pyramidal iron will penetrate 
the penetrable thing struck by its point to a greater extent according 
as this point is narrower. 

The heavy substance also when constricted in less space is of greater 
weight because a less quantity of air offers resistance to it. Of move- 
ment and force we shall speak elsewhere. 



So also such other qualities as sweetness, bitterness, sharpness, rough- 
ness do the same as has been stated above; and an example of this is 
shown when any of these increases in quantity mixing itself with snow 
or water which neither gives it flavour nor takes it away from it but 
completely deprives it of power. g 89 v. 

The greatest strength of the tackle is in the rope that is joined to its 
moving power. And the least strength will be in the rope that is joined 
to one of the tackles. 

The weight drawn by the ropes that pass through the tackles is 
divided in equal parts between the ropes joined to these tackles. 

That rope of the tackles will be swifter which is nearer to its mover; 
it follows that the slowest will be that which is farther away from this 
mover. g 95 v. 

If a man be at the bottom of a well which contains twenty thousand 
braccia of water he will not be conscious of any weight, h 49 [1] v. 

Why the movement made by the sieve collects together at the top 
all the lightest parts? And it is the same with the dredger when one 
searches for gold in the Ticino by means of a blow, and also with the 
sweepings of the goldsmith's workshop when they are washed. 

H 52 [4] V. 

If the part of any substance in the air is greater than that in the 
water its movement will follow the course of the air. h 59 [n] v. 

A straight movement transformed into another without rebound 
loses its power. 

The natural movement will impart the greatest blow which main- 
tains in a straight line the course it has begun. h 61 [13] r. 

Such part of water as is in contact with the air will move according 
to the course of this air although the water upon the bottom move in 
a contrary direction. h 61 [13] v. 

Water which descends from a wide expanse by a straight channel 
will intersect at its entry from right to left. 

And after its entry the part in the centre of the channel will be 
higher than all the rest of the expanse. 



In such another course made by the water in its channel the part 
of the centre will be lower than the rest of the expanse. 

As it follows the same space the depression in the centre changes to 
a greater height. h 62 [14] r. 

The blow is more powerful than the movement and where it is the 
blow of the water every obstacle is removed: then when the movement 
is afterwards created it carries with it all the gravity occasioned by the 
blow and discharges it in the blow made by the rebound, and so from 
rebound to rebound the force of the violent movement diminishes. 
Consequently the heaviest of the stones are deposited and not being 
able to be borne along by the feeble movements which follow they 
remain there; and the last things which are at the furthest distance 
from the point at which they started are the lightest things. 

h 62 [14] v. 

Water that falls nearest the perpendicular has the least power to 
drive big gravel before it. h 66 [18] r. 

All violent movements grow feebler the more they are separated 
from their cause. h 77 [29] v. 

In proportion as the natural movement is separated from its cause 
it becomes more rapid. h 78 [30] r. 

The weight which is at the greatest distance from the perpendicular 
of its support weighs less. h 80 [32] v. 

That part of the blow produced by a continuing cause will be so 
much the more powerful as it is more distant from the cause of its 
movement. h 81 [33] v. 

[Weight and Water] 

As much weight of water will escape from its position as the sum 
of the weight that drives this water. 

The weight supported on the water will be as great as the sum of 
the weight of the water which gives place to these weights. 

H 92 [44] r. 

The centre of every weight rests under the centre of its support. 

h 105 [38 v.] r. 




If someone descends from one step to another by jumping from one 
to the other and then you add together all the forces of the percussions 
and the weights of these jumps, you will find that they are equal to 
the entire percussion and weight that such a man would produce if 
he fell by a perpendicular line from the top to the bottom of the height 
of this staircase. 

Furthermore if this man were to fall from a height, striking stage 
by stage upon objects which would bend in the manner of a spring, 
in such a way that the percussion from the one to the other was slight, 
you will find that at the last part of his descent this man will have his 
percussion as much diminished by comparison with what it would 
have been in a free and perpendicular line, as it would be if there were 
taken from it all the percussions joined together which were given at 
each stage of the said descent upon the aforesaid springs, i 14 v. 

Pagolo says that no instrument that moves another instrument in 
contact with it can avoid being moved by it. So if the wheel moves its 
pinion the pinion will also move this wheel. But such a thing is not 
general, for though the cog n moves the wheel the wheel will not move 
the cog n turned against the ground. 

The line of the movement made by the course of two objects that 
strike which does not bend in meeting with the first object is that 
which strikes the second object more with its reflex movement; and 
so that will act conversely which bends more in its impact with the 
first object. 1 28 r. 

That body which when moved strikes the first object with a greater 
blow consumes more of its impetus in the percussion, with result that 
the blow made by the reflex movement will be weaker. And the line 
of the movement which does not bend at all at its first percussion 
strikes the second object most. 1 28 v. 

The leap is always smaller than the descent made by the thing which 
leaps; and this leap is termed reflex movement, which is always weaker 
than the straight movement. 1 43 v. 



I The kjiije- grinder's wheel] 

The pole is worn most on the course in which its mover exerts the 
greatest force; as is shown by the knife-grinders who go through the 

Every pole becomes worn on the side in which its mover applies 
most force. See a man sharpening knives by turning a wheel with his 
feet: as he drives his foot down he starts the force which then pro- 
duces the movement of an entire revolution. One understands clearly 
that in proportion as this movement is farther removed from its cause 
it becomes slower and wears away its support less; and each time that 
the movement is repeated the pressure of the feet renews it with fresh 
impetus, and again under this impetus the support is consumed, and 
so it comes about that the pole is worn unequally. 1 45 r. 

{Movement — jailing and reflex] 

I ask whether the movement made by the stone in a continuous line 
is equal to that movement which is in a reflex line, that is before the 
rebound and after the rebound. 1 61 [13] r. 



For what reason the mortar does not follow the rule of the carbine: 
If for a ball that weighs an ounce there be allowed an ounce of powder 
with which this ball is shot a mile, then the ball constantly increasing 
in the same substance with each degree of weight, one gives it a pro- 
portionate quantity of powder up to a thousand pounds, and increases 
the size of the machine so that it always takes forty balls, that is to 
forty times the thickness of this ball, the metal being always half the 
thickness of this ball; then the carbine increasing in volume in all its 
particulars from a ball of an ounce to one of a thousand pounds, you 
will find that the more the ball weighs the farther it travels. 

1 84 [36] v. 

The movement made by the arm in throwing the stone is twofold; 
for when the elbow goes forward and the fist with the stone turns 
back with a circling movement and then goes forward and drops its 
stone with a sudden stoppage of the arm, this stoppage by its sudden- 
ness is followed by a recoil, and produces the effect of percussion in the 




air; consequently the movement is much greater than if one were to 
move the arm in one action and it followed a circular movement in 
leaving the stone. 


Why the short mortar makes a louder explosion when fired than the 
long one, as one hears it in drawing the breeches of the small cannon. 

. 85 [37] r. 

The movement made by the cord is much more rapid near the be- 
ginning than near the end. Consequently we may say that as the arrow 
follows the nature of the greatest movement of the power that drives 
it, when such power diminishes the arrow is already separated from 
the cord, and the percussion made by the arms of the cross-bow upon 
the cord that holds it is made after the departure of the arrow, and 
therefore this percussion does not increase the movement of this arrow. 

1 85 [37] v. 



This movement is at first increased by impetus, because the more 
the stone which produces it falls the more speed and gravity it ac- 
quires; secondly the more the cord that is unrolled from the part of 
the circle is unrolled the more distant it becomes from the centre of 
the movement; consequently the more distant it becomes the more it 
acquires weight and impetus. 1 91 [43] v. 

There are some movements of moving things which continue their 
direction while receiving percussion in the middle of this movement, 
and these are of great force. 

I ask why field-lances or hunting-whips have a greater movement 
that the arm. I say that this happens because the hand describes a much 
wider circle as the arm moves than does the elbow; and in conse- 
quence moving at the same time the hand covers twice as much space 
as does the elbow, and therefore it may be said to be of a speed double 
that of trie movement of the elbow and so it sends things when thrown 
a greater distance from itself. 



Thus you sec clearly that the circuit described by the elbow is less 
by half and its speed is slower by half. 

It is true that if one takes from the movement made by the hand 
an amount equal to that made by the elbow they become of equal 
slowness. 1 99 [5 1 ! v - 

If the stone moved by the power of its engine follows in the begin- 
ning of its movement the greatest power and speed of its mover, why 
does it not follow this same equality of the first movement without 
waiting for it to separate as it reduces its speed? If however it should 
separate immediately what does it profit it to be closely attended by 
such power? 1 100 [52] r. 

The power that moves which accompanies the movable thing 
farthest will cause it to move farthest from the boundary where it is 
separated from it. 

It is almost universally the case that everything which is the cause 
of movements decreases its power before it separates itself from the 
thing moved by it. 

The mortar only increases its force because in the longer movement 
which the ball makes in its body more powder is ignited, for it must 
needs be admitted that this setting on fire occupies a divisible period 
of time; and the more periods of time it lasts the more powder is 
ignited, the more the fire is driven through this machine, and the 
greater the impetus and fury with which it expels the ball. 

One asks with regard to the same course of the mover whether it 
removes the thing moved farthest if it commences with slow move- 
ment and continually increases the impetus, or if it commences quickly 
and then proceeds to slacken, or if it goes at an even pace. 

1 100 [52] v. 

I ask whether if two movable things equal in shape weight and sub- 
stance are of double speed their course will be double the one of the 
other or no? 

Because one sees bows and cross-bows with long arms that have a 
long and slow range and one sees cross-bows with short thick arms 
that have a rapid and short range; the percussions of the short make 
a quick passage in the third part of their movement and the long make 
a slower one. 



In the consideration of this it is necessary to take into account all 
the mathematical forces; seeing that in the case of these cross-bows 
there are at work different causes which produce many different 
effects, for there are found among them some with a great ascent, 
some with a short one, some long and thin and some fat and short, 
some wide and some narrow: so they proceed to vary in shape and in 
power in many different ways. 1 101 [53] r. 


To test the residuum of power of the things that move and draw 
the weights by giving them a greater or a less weight see what is the 
weight which is most distant from its mover. 

And let it always be round in shape and of uniform substance, and 
the balance m n should have its arms of equal length and weight, and 
the centre of the weight which strikes the centre of the movable weight 
should be always raised to the . . . . , and moreover when you have 
found a weight which recedes more from the beginning of the move- 
ment going as rapidly as possible by way of percussion and you re- 
produce it with a simple weight without percussion, you will be able 
to discern what difference there is between the causes of the move- 
ments when they are due to weight and when merely to percussion. 
And so with regard to this weight you will proceed to change the 
things that it moves until you find a weight which is proportionate to 
the power of this mover, that is the ball which is propelled as far as 
possible from its motive power. Then weigh one movable thing and 
the other, and consider the distances where they have been moved, and 
you will be able to deduce with accuracy the general rule between these 
two powers. 

Then diminish the movable thing by half as those agree to do who 
have written upon proportions, and you will see that it cannot be that 
the half less weight will be moved twice as quickly by the same power, 
that is to say that if it was twice as swift it would go twice as far be- 
cause the proportion of the movements is as that of the speeds. And 
if some have said that the more the movable thing is diminished the 
more rapidly its mover drives it in proportion to its diminution on to 
infinity, constantly acquiring speed of movement; it would follow that 



an atom would be almost as rapid as thought itself, or as the eye which 
roves in an instant to die height of the stars; and as a consequence its 
journey would be infinite, because the thing which can diminish in- 
finitely would increase infinitely in swiftness and traverse an infinite 
distance, because every continuous quantity is divisible to infinity. This 
opinion however is condemned by reason and also by experience. 

It would follow also that if the mortar throws its ball three miles 
from itself in twenty divisions of musical or harmonic time with a 
hundred pounds of powder and a thousand balls, that taking a pound 
of balls it would make with the said powder ... in the same time. 
Work by the rule of three, saying if a thousand pounds of stone are 
thrown to me in two divisions of time, you keeping in imagination the 
three thousand, saying also if a thousand pounds of balls are thrown 
to me, at the said distance in twenty divisions of time, in how many 
divisions of time will the same power throw me a pound of balls. 
And reckon if a thousand give me twenty what will give me one : and 
you will find that it will give you 2 %ooo divisions of time which make 
about /'50 of time. Now if one shoots with powder the weight of a 
small grain, the experiment will not send the ball farther than the 
mortar sends its smoke when one begins to fire, and by this reasoning 
it would be sent a million miles in the time when the thousand pounds 
of balls go three miles. You investigators therefore should not trust 
yourselves to the authors who by employing only their imagination 
have wished to make themselves interpreters between nature and man, 
but only [to the guidance] of those who have exercised their intellects 
not with the signs of nature but with the results of their experiments. 

1 102 [54] r. and v., 103 [55] r. and v. 



The heavier the thing the more power attends its movement. 

This is seen with jumpers who have their feet joined, who in order 
to make a greater jump throw back their clenched hands and then 
move them forward violently as they take of! for the jump, finding 
that by this movement the jump becomes greater. 

And there are many who to increase this jump take two heavy stones 



in their two hands and use them for the same purpose as they used to 
use their fists; their leap becomes much greater. i 104 [56J v. 

If a weight falling a distance of ten braccia buries itself a span in the 
earth how far will it bury itself when it falls two braccia? 

1 no [62] v. 


Albert of Saxony x in his 'De Proportione' says that if a power moves 
a movable thing with a certain speed it will move the half of this 
movable thing twice as swiftly. This does not appear so to me, for the 
reason that he does not take into account that this power exerts its ulti- 
mate force, and unless it did this, the thing which weighed less would 
not be in proportion to the force of the mover or of the medium 
through which it has passed. Consequently it would be a thing floating 
in the wind and not in straight movement, and it would go less far. 

1 120 [72] r. 

I ask in what part of its curving movement will the cause that moves 
leave the thing moved or movable. 

Speak with Pietro Monti of these ways of throwing spears. 

1 120 [72] v. 


What is the cause of movement. What movement is in itself. W r hat 
it is which is most adapted for movement. What is impetus; what is the 
cause of impetus, and of the medium in which it is created. What is 
percussion; what is its cause. What is rebound. What is the curve of 
straight movement and its cause. 

Aristotle, Third of the Physics, and Albertus [Magnus] and Thomas 
[Aquinas] and the others upon the rebound, in the Seventh of the 
Physics, 'De Ccelo et Mundo'. 1 130 [82] v. 

[Weight and movement — experiment] 

There are two balls of the same substance and shape but the one 
double the other in weight, and I wish to let them fall through two 

1 Albert of Saxony, bishop of Halberstadt (14th century), was a commentator of 



tubes so situated that the balls clash together at the end of these tubes. 
I ask how Ear each at the beginning of their movement ought to be dis- 
tant from the meeting of these tubes. 1 131 [83 J r. 

If the stone or the water struck by the movable thing falling upon 
it follows the reflex movement in the manner that this movable thing 
would follow by itself after its percussion or no. k i v. 

A drop that falls upon a place of uniform density and smoothness 
will in rebounding scatter its extreme particles in an exact circle; and 
so conversely ... k 56 [7] r. 

Of movements there are two kinds namely simple and composite. 

Of the simple movements none is slower or swifter than the slow- 
ness or swiftness of its mover. Composite movements may be either 
slower or swifter infinitely more so than their mover; and also they 
may be equal to this mover. k 107 [27] r. 

In the case of a stick used to beat with, the slowest movement is in 
the centre of its length. 

When one weight falls to the ground at the same time as another 
weight, in the percussion which they make the lesser weight leaps into 
the air. k 107 [27] v. 

Nothing movable will ever be swifter than the part of its mover that 
touches it. 


That thing moves more after it is separated from its mover which is 
moved by a greater power. k no [30] r. 

Every small movement made by the movable thing surrounded by 
air maintains itself as it goes by impetus. 

A movable thing moved by a slow mover, if it has to move a thing 
by rubbing it, only moves when joined to its mover. k hi [31] r. 

[With diagrams] 

Why every substance that possesses gravity either free altogether or 
in part shows in its whole or in part the natural desire to descend. 

The wheel a b being fixed in the position that you see, the heavy 
substance a will descend at b\ and below for this reason the heavy sub- 



stance c placed above the centre of its axis, will go as near as it can to 
the centre of the earth; and m n does exactly the same below. 

l 40 r. 
[Reflex movements} 

The reflex movement will be of greater strength when it is longer. 

And that reflex movement will be longer which is produced between 
more diverse angles. 

The reflex movement which is longer is less impeded because it 
differs little from the movement of percussion, and this percussion has 
little strength and consequently it loses little of the power of its first 

But if the reflex movement is shorter it is a sign that it is more im- 
peded at the place of the percussion, and it differs much from the 
movement of incidence and consequently the power of the first move- 
ment is greatly diminished. l 42 r. 


The reflex movement will be as much weaker as it is shorter. 

That reflex movement will be shorter which is cause by a greater 

That percussion will be of greater power which is made between 
more equal angles. 

Of the percussions made between equal angles that will be of the 
greater strength which is caused against a more compact object. 

And in the percussions made on objects of equal compactness that 
will be more powerful which has its object of greater resistance. 

The spherical body turns so much more in the reflex movement as 
the percussion is made between the more unequal angles. l 42 v. 

[Movement — cannonball} 

The smoother the surface of the cannonball the greater the ease with 
which it will turn in the air as it moves. 

In a proposition of this kind one imagines that the cannonball shot 
from the mortar has to turn itself in the air which it compresses, and 
if this cannonball is not altogether smooth its curve may occasion diffi- 
culty in friction with the air which surrounds it, as I have proved in 
the fourth concerning frictions. 



So when the rebound of this cannonball is made at a more obtuse 
angle this cannonball will turn more upon itself; and if the ball be 
lacking in smoothness it will come to fail in speed much more than if 
it was smooth. l 43 r. 

The percussion of each heavy spherical substance will not occasion 
scars which have between them a proportion resembling that of the 
slant of the places where they strike. 

This proposition would be entirely confirmed by experience if there 
were not the firm compression of the air driven by the fury of the ball, 
for this not being of itself swift as is the movement by the mover which 
drives it becomes compressed, and is the more compressed as it is the 
more driven. And this is how it comes about that this ball afterwards 
striking by a line that is not central within the range of the perpendic- 
ular [line] a c commences the first stage of its slant, and the last ends 
in the range of the horizontal [line] a b. l 43 v. and 44 r. 

[Drawing — ship with sail] 

If the water here were to move as quickly as the air the ship would 
move like the wind, without a sail; but because the wind is swifter high 
up than low down therefore the wind has more power on the sail than 
on the water. l 47 v. 

Corn tossed up with a sieve leaps up in the form of a pyramind. 

[Movement, percussion] 

That thing moves more in derived movement which is more accom- 
panied by its mover. 

What difference there is between the percussion of the united thing 
and that which is disunited. l 64 v. 

[Weight and movement] 

If the heavier part of bodies makes itself the guide of their move- 
ment, and an arrow be pierced and a portion of quicksilver be placed 
within it, how will the arrow act and what course will it take on being 
drawn to a height? 

If a lance be made up of pieces arranged together after the manner 
of an indented box why does it deal a harder blow than a lance formed 
of a single piece? l 65 v. 




If a power move a weight a certain space in a certain time, the half 
of that power will move that whole body the half of that space in the 
said time, or all the space in double the time. 

Or the whole of that power will move a weight double that first 
weight half that distance in the same time. 

Or it will move the said weight in half the said time half the space. 

L 78 V. 

Man and every animal undergoes more fatigue in going upwards 
than downwards, for as he ascends he bears his weight with him and 
as he descends he simply lets it go. l 84 v. 

That body will show itself heaviest which meets with the most feeble 
resistance, and that heavy body will meet with the most feeble resist- 
ance in which the centre is farthest removed from the centre-line of 
its support. l 85 v. 

The centre of every gravity that is suspended stands below the 
central line of the cord that sustains it. 

If from the two equal arms of the balance its two cords proceed, the 
one being double the other in length, as regards the weight and be- 
cause these cords meet in the same spot in order to support a weight 
one cord will feel as much more weight than the other as it is longer 
than the other. m 37 r. 

Gravity suspended to a cord is all in all the length of the cord and 
all in each of its parts. m 40 v. 

Why the flat sand being made up of grains dissimilar in shape and 
size the water that flows above it drives these grains with different 
strengths of movement: 

Just as the bodies differing in weight and shape make different move- 
ments in the still air so do the air and water which move between 
bodies at rest. And this is the reason why the sand loses its flatness 
through the movement of the water that passes over it; for the water 
that is moved over the sand performs the same function as the air that 
is moved over the water. And if one should prove that the bottom of 
the flat sand makes its waves and becomes uneven through the uneven- 



ness of its granules, and that this uncvenness cannot occur on the sur- 
Eace of the water which is smitten and is of uniform body, I maintain 
that the air is full of parts which have dissimilar movement and there- 
fore there is no uniformity in the movement of the parts moved by the 
contact of the air. m 41 r. 

[Ships — Wind and Sea] 

I ask whether the wave — or rather whether the ship travels as fast 
as the wave that bears it or as fast as the wind that drives, it or whether 
it shares in both the one movement and the other. 

And if the mariner has the current favourable and the wind contrary 
I know if they are of equal strength the ship remains in its first posi- 
tion. M 41 V. 


[Resistance of the air] 













The line c s represents the movement of the gravity b m having taken 
away the resistance of the air a n. 

The air when devoid of clouds and mist starts thick at its base and 
at each degree of its height it acquires in the form of a pyramid degrees 
of thinness, as is shown by the line n a. 

And the weight that descends through this air also in each stage of 
its movement acquires a degree of speed, although I ought first to say 
that with every degree of time it acquired a degree of movement more 
than the degree of the time immediately past. This is why I suppose 
the degrees of the movement to be ten times as powerful as the degrees 
of the air that resists; consequently we may say that the said ten de- 
grees if one be taken away for the air which offers partial resistance 
become nine, and the second twenty degrees entering into the denser 
air has two taken away so that the twenty becomes eighteen, as is 
shown by the line b m. m 43 r. 



[Of weights moving through the air] 

If two equal weights are situated vertically one below the other and 
allowed to fall at the same time these in long descent will consume 
their interval and will come to touch. 

When the air is without mist or clouds you will find that at every 
degree of height it will acquire a corresponding degree of thinness. 
So also inversely every degree lower down it will acquire a degree of 
density. And this is why if two equal bodies are placed one below the 
other a braccio's space apart, that is they are attached by a thread and 
let fall together in a long movement they will touch because the one 
below always finds itself in thicker air than the one above it; and be- 
sides this the first has the hard work of opening the air and making the 
waves in it; part of this escapes upwards and charging strikes with its 
reflex movement against the second body, but the rest of the air above 
runs to fill the vacuum which exists behind this body. m 43 v. 

Proof of the proportion of the time and movement together with the 
speed made in the descent of heavy bodies in the shape of a pyramid, 
because the aforesaid powers are all pyramidal seeing that they com- 
mence in nothing and proceed to increase in degrees of arithmetical 

If you cut the pyramid at any stage of its height by a line equidistant 
to its base, you will find that whatever proportion there be between 
the height of this section from its base and the whole height of the 
pyramid, there will be the same proportion between the breadth of this 
section and the breadth of the whole base. m 44 r. 

This happens in the air of uniform thickness. 

The heavy body which descends, at each degree of time acquires a 
degree of movement more than the degree of the time preceding, and 
similarly a degree of swiftness greater than the degree of the preced- 
ing movement. Therefore at each doubled quantity of time the length 
of the descent is doubled and also the swiftness of the movement. 

It is here shown that whatever the proportion that one quantity of 
time has with another, the one quantity of movement will have the 
same with the other and similarly one quantity of swiftness, m 44 v. 

[Weight in the air — increase of speed] 
The heavy body which descends freely with every degree of time 



acquires a degree of movement, and with every degree of movement it 
acquires a degree of speed. 

Although the equal division of the movement of time cannot be in- 
dicated by degrees as is the movement made by the bodies, nevertheless 
the necessity of the case constrains me to ma/{e degrees after the man- 
ner in which they are made among musicians} 

Let us say that in the first degree of time it [i.e. the heavy body] ac- 
quires a degree of movement and a degree of speed, in the second de- 
gree of time it will acquire two degrees of movement and two of 
speed and so it continues in succession as has been said above. 

m 45 r. 

[ Waves of the air] 

The wave of the air that is produced by means of a body which 
moves through this air will be considerably swifter than the body that 
moves it. 

What is set forth above happens because as the body of the air is very 
volatile and moves very easily when a body moves through it it comes 
to make the first wave in its first movement, and at the same time that 
wave cannot be produced without it causing another after it and that 
another. And so this body moving through the air creates beneath it in 
each stage of time multiplications of waves, which in their flight pre- 
pare the path of movement for the movement of their mover. 

The wave of the air forming and reforming itself prepares the way 
of movement for its mover. 

The air which is shut up by force becomes heavier than that which 
is at liberty. m 45 v. 

[Weight moving through the air] 

The heavy body which has a free descent with every degree of move- 
ment acquires a degree of weight. 

This arises out of the second of the first which says that 'that body 
will be heavier which has less resistance'. In this case of free descent of 
heavy bodies one sees clearly by the example already cited of the wave 
of the water, that the air makes the same wave beneath the thing which 
descends, because it finds itself pushed and drawn from the opposite 
side, that is that it makes a turning wave which helps to drive it down. 

1 Words crossed out in MS. 



Now for these reasons the air which flies in front of the weight that 
drives it shows clearly that it does not resist it and in consequence does 
not impede this movement; therefore the greater the descent of the 
wave which travels more rapidly than the heavy body that moves it, 
the longer the movement of this heavy body continues; and as the last 
wave becomes more remote the more it prepares the air which touches 
the weight to a more facile flight. m 46 r. 

[Density of waves. Weight in the air] 

When waves become divided in minute particles that quantity which 
above is united and powerful comes to descend to the ground. 

Things that fall may be continuous quantities such as staffs, beams 
and suchlike things, and liquid bodies, although these cease to be con- 
tinuous when their descent is long. 

Others are discontinuous such as stones and other bodies separated 
the one from the other. Others are neuter as would be the grain from 
the hopper that turns the mill, sand, and similar quantities of minute 
bodies of which you may make proof at a great height. And mark 
what a difference there is between the unity of their exit from the hop- 
per and their density when they arrive at the place of their percussion. 

If the air were of uniform thickness at each part of its height the 
bodies which descend would acquire at each stage of their movement 
equal degrees of speed. m 46 v. 

[Weight. Movement. Waves of water] 

Now we have found that the discontinuous quantity when moving 
acquires at each stage of its movement a degree of speed; and so in 
each stage of harmonic time they acquire a length of space from each 
other, and this acquisition is in arithmetical proportion. 

How then are we to account for the continuous quantity of liquid 
bodies in their descent, since in each interval of harmonic time it pours 
out the same weight and at each stage of movement it becomes longer 
and thinner, so that in a long course it shows itself ending in a point 
as does the pyramid; consequently such liquid body would not fall to 
earth, but it would rather be that each great mass of this body would 
remain in the air even though it should be a very great river which was 
continually rolling away; and experience shows the contrary; for as 
much as departs above strikes at the same time below. And if the same 



weight of this liquid body makes itself thinner it meets with less re- 
sistance from the air and consequently acquires speed; and if by being 
thinner it has acquired speed this same weight would also for this 
second reason come to make itself longer and in consequence still thin- 
ner and so would descend more rapidly; and this would go on in suc- 
cession to infinity. Therefore either nature or necessity has brought 
it about that in whatever manner the descent comes to assume the 
form of a pyramid, it makes intersection by changing its extremities 
from right to left and commences to divide itself; and the more it 
descends the more it divides; and thus with many ramifications it 
comes to lighten itself and to check its irregular movement. 

m 47 r. and v. 
[Weights falling in succession] 

If two bodies of equal weight and the same shape fall one after the 
other from the same height in each degree of time the one will be a 
degree more distant than the other. m 48 r. 

[ Weight. Increase of speed] 

The heavy thing descending freely gains a degree of speed with 
every stage of movement. 

And the part of the movement which is made in each degree of time 
is always longer successively, the new one than that which preceded it. 

It may be clearly shown that what is set forth above is true, for dur- 
ing the same time that the weight a descends at c, b which finds itself 
fifteen times swifter than a has covered fifteen times as much space in 
its descent. m 49 r. 


I wish to make a balance with arms of equal length, of which I wish 
to make one hanging downwards as is shown in b c and which weigh? 
at least four ounces; now I ask how much the arm ought to weigh 
which is straight and how much larger the one ought to be than the 
other for it to resist it in a position of equilibrium? m 51 r. 

[Bodies falling in succession] 

If the descent is made by two bodies equal in shape and weight of 
which one has commenced its movement before the other the propor- 
tion of their percussions will be as that of the length of their move- 
ments, m 52 r. 



[Proportions and projectiles} 

There will be the same proportion of base with base as there is of 
side with side and height with height. 

You will proceed by the rule of three and you will say if the height 
of the pyramid which I know for certain to be s c gives me a braccio 
for its base; or if a base of a braccio comes from a pyramid of ten 
braccia, what will another base of sixty five braccia come from ? 

There arises here an exception, namely that if a ball first goes up a 
hundred braccia with an ounce of powder it goes through air of 
greater thickness than would that which rose three thousand braccia> 
and consequently as that of the said three thousand braccia has been 
at each hundred braccia in a region of air that is thinner than that 
before it, it has always acquired more speed. 

I wish to know how much higher one small cannon or carbine 
throws than another, and to do this I train my instrument according to 
the line b c in a firm manner so that it will not alter its angle of eleva- 
tion. This done, I shall insert so small a quantity of powder that the 
ball will only be projected two braccia away from the carbine as b s, 
and I shall note where the ball falls, at n, then I shall double the charge 
of powder and see where it falls at m ; and if I shall find that the base 
m c is double the base n c I shall know that the height of the pyramid 
h c is double that of s c. m 53 r. 

[How to lift great weights} 

Great weights ought always to be supported by levers as you see done 
in order to draw the column out of the barge. 

a wedge which of itself supports at a. 

Wedge. Barge. Windlass. 

You will give as many turns with the cord to the column as you wish 
it to have turns to unroll as you draw it. m 56 v. 

[Fall of heavy bodies} 

If many bodies of similar weight and shape are allowed to fall one 
after the other at equal spaces of time the excesses of their intervals 
will be equal to each other. 




By the fifth of the first which says how the thing that descends at 
every stage of movement acquires equal degrees of speed. 

Therefore for this reason the movement of the last downward be- 
comes much more rapid than that of the first from the beginning. 

And by the eighth of the first which says that: the upper pair will 
have in their interval the same proportion with the interval of the 
lower pair as the speed of the lower pair has to that of the upper pair, 
and so conversely the speed with the distances will be as the distances 
with the speed. 

The experiment of the aforesaid conclusion as to movement ought 
to be made in this way, that is that one takes two balls of similar 
weight and shape and causes them to drop from a considerable height 
in such a way that at the inception of their movement they touch one 
another and that whoever is making the experiment stations himself 
on the ground in order to watch whether at the time of their fall they 
have remained touching each other or no. And this experiment should 
be made many times so that no accident may occur to hinder or falsify 
this proof — for the experiment might be false whether it deceived the 
investigator or no. m 57 r. and v. 


That air will become denser which is pressed upon by a greater 

Although b is as thick as a yet as it is twice as heavy it makes the air 
which flies below it twice as dense, and as it becomes denser below it 
becomes proportionately thinner above. m 58 r. 

[Weight, Percussion, Spring] 

That weight will show itself lighter which has a greater volume. 

That weight will make a less percussion which strikes in its descent 
with a part more distant from the central line of its gravity, or with a 
part that yields against the object as though it were pressing upon a 
spring, or with a part which yields against the thing that strikes the 
object with a spring or which jumps upon the point of its feet. 

m 59 r. 



[Falls of heavy bodies] 

Explanation of the movement of the separated quantity. 

Why the natural movement of heavy things at each stage of its 
descent acquires a degree of speed. 

And for this reason such movement as it acquires power shows itself 
of pyramidal shape, because the pyramid acquires similarly in each de- 
gree of its length a degree of breadth, and so such proportion of ac- 
quisition is found in arithmetical proportion because the parts that 
exceed are always equal. m 59 v. 

[Drawing: — two balls] 

These two figures are double in diameter the one of the other and I 
wish to know how much the one descends more rapidly than the other. 

[Drawing: — two cubes p q\ 

Although p is eight times q nevertheless it is not swifter in its descent 
than about the double of q, which descent will be spoken of here. 

Let us say therefore that q is three pounds and that the resistance of 
the air is a pound; consequently the weight which was three becomes 
two and consequently of the weight p there remains two pounds. So 
of the four dice below which are three pounds each there remain eight 
pounds and those above are twelve pounds and this makes twenty con- 
taining two ten times and so becoming ten times swifter. m 60 r. 

If two balls of equal weight and size are placed at a distance of one 
braccio one above the other and commence their descent at the same 
moment, always at each stage of movement the interval between them 
will be of the same size and will remain as shown at a b. 

If after the descent of a braccio made by one ball you allow another 
similar one to fall you will find that at each stage of movement there 
will be a proportionate change in their speed and force. 

It is clearly shown how when the ball has fallen from a to c it has 
traversed twice the distance that that ball has which has fallen only 
from a to b, and therefore it will be twice as swift and powerful and 
will bury itself twice as deeply as at b, and if when descending or in- 
deed when a has descended at d, b shall find itself at c, and the power 
of d will not be more than double that of c but will be half as much 
again. And so when a is at e, b will be at d aad the power of d will be 



three quarters that of e, then four fifths, then six sevenths and so 00 
to infinity. m 60 v. and 61 r. 



I wish to know what weight ought to be that of lead which will drive 
a ball of a pound of lead a greater distance from itself than any other 
weight which is also of lead, the said movers having also the same 

And I wish to know how far a weight equal to that of lead being of 
wood will drive in the same movement from itself the above-mentioned 
ball of lead. 

Among weights of similar shape that will be driven to a greater dis- 
tance by the same power which finds itself smaller in shape. 


Among the weights of like shape which are driven by the same 
power that which is of greater bulk will be of less movement. 

Aristotle says that if a power moves a body a certain distance in a 
certain time the same power will move half this body twice the distance 
in the same time. Therefore the millionth part of this weight will be 
driven by the same power a million times this distance in the same 
time; or if this weight was an ounce and it was transported a mile in 
a period of time the millionth part would be transported a million 
miles in the same period. And if you were to say that the air would 
make resistance I maintain that in proportion as this body was less in 
weight than an ounce the quantity of air would be less that withstood 
its course. m 61 v. and 62 r. 

[Movements and proportions] 

Of the movement made by things proportionately to the power 
which drives them. 

One ought to make the experiment with a cross-bow or other power 
which does not grow weaker, and also with balls of the same shape and 
of different substances and weights to test which goes farthest away 



from its motive power, and then to test with various shapes of various 
sizes breadths and lengths and to make a general rule. 



I wish to know what weight the power will have which shall drive 
to a greatest distance from itself a weight of a pound spherical in shape. 

m 62 v. 


The weight that charges the cross-bow has the same proportion to 
the weight of the arrow as the movement of the arrow of this cross- 
bow has to the movement of its cord. 

Here one ought to deduct three resistances made by the air, that is 
the percussion of the bow of the cross-bow made upon the air, and 
that of the cord; the third is that made against the arrow. And as the 
cord is thicker so its arrow encompasses it the less. m 63 r. 

[Of wedges] 

Ordinary wedge, immovable because of the mother (madre) . 

Wedge in sheath of iron for splitting stones. 

The force of the wedge is very great because of the percussion, and 
it acts with marvellous power in dividing the things united and in 
uniting those divided, in stamping sculptures of metals in bas-relief, in 
squeezing out the liquids from the places where they are produced, and 
in drying things that are moist, as well as in many other things as will 
be shown when treating of them; although it has the same nature as 
the screw it surpasses it altogether. m 63 v. 


Wedges are of two kinds of which one is called 'permanent' and the 
other 'transitive'. The permanent is that in which when the wedge has 
entered it cannot turn back, and the other enters and departs accord- 
ing to the necessity of the case. 

The axe and the hammer which comes to be mother (madre) of its 
handle wishes to be large above and narrow below, and their handles 



ought to be thick below and narrow above, and then with the perma- 
nent wedge one ouiiht to widen it above. m 64 r. 

[Movements of water and of sand] 

I ask where the water leaves the sand ribbed and where smooth, 
where thick and where thin, where pure and where mingled with 
various particles of straw-wood and leaves. 

Water falls from its heights and where it makes the greatest per- 
cussion it removes the things which are heaviest and fitted for resist- 
ance; and after this percussion it carries the heaviest things by the cur- 
rent that is greater and swifter, and so conversely it carries the lighter 
things in the part of the water that is slower and has less power. 

m 64 v. 

I ask where the water leaves its muddy banks, where mixed with 
sand and where thin and fine so that it slips away and where mixed 
with roots and wisps of straw and leaves. 

We shall speak of the cause of the movement that the sand makes 
upon its bed and what carries it off and how it moves and where and 
how it stops, and also of the small and large stones and how they group 
themselves when they stop all together and also of every other thing 
that goes rolling upon its bed. m 65 r. 

{Relations of surfaces] 

If two surfaces of different shapes and equal circumferences touch 
as one is placed upon the other, if that which touches is of like shape 
and circumference that which does not touch will be of varying shape 
and of equal circumference. m 65 v. 

[Movements of water and of sand] 

These waves of the sand are changeable together according to the 
direction of the river. 

The water that is swifter is that which wears away the bed of the 
rivers most. Hence it comes about how when the sand forms those 
shells, or after the manner of certain undulations it is seen on the sur- 
face of the water, how the sand moved by the greater current of the 
water becomes more sifted. 

How water can flow above by one line and below by another, cross- 
wise; as is shown at a b and c d. m 66 r. 



Conception. Of everything that moves the space that it acquires is 
as great as that which it leaves. 

Conception. If one thing is removed from contact with another, the 
extent of the movement which the part opposite to the said contact 
makes will be as great as the space that is interposed between the parts 
that at first were touching. m 66 v. 


The way in which the tail of the fish moves in order to drive the 
fish forward and so also with the eel the snake and the leech. 

Conception. If parts of two surfaces touch one another the part of 
the one that is touched by the other will correspond to that of the other 
that is touched by the one, or rather there is as much of the first that is 
touched by the second as there is of the second that touches the first. 

m 67 r. 
[With drawing] 

Here contend two forces of water and now one conquers and now 
the other. m 67 v. 

[Cross-bow — relation of weight and movement] 

If the cord of the cross-bow draws four hundred pounds weight 
upon its notch with the movement of a third of a braccio, as it dis- 
charges itself it will draw two hundred pounds with two thirds of a 
braccio distance from its notch, and a hundred pounds will be removed 
from its position by such power for a space of one braccio and a third. 

And so as much as you shall diminish the weight of the movable 
thing so the power will cause it to make a greater movement, in such a 
way that you will always find that the movement of the cord and the 
movement of the thing moved will be in the same proportion as the 
weight that drew the cord to the notch was to the weight that was 
driven by the cord (if the air did not restrain it). m 71 v. 

A bow bends half a braccio and has power of one hundred and pulls 
two hundred braccia; and another has power of two hundred and 
opens a quarter of a braccio. 

Which will discharge its arrow proportionately farther, the one or 
the other? 

And if another opened a hundred times less and was a hundred 



times stronger which would carry farthest in proportion with the said 
arrows? m 72 r. 



If you twist the cord n m twenty times it will uncoil and make 
thirty nine revolutions in the contrary direction, and will then roll the 
opposite way and will make thirty-eight; and so in succession it will 
go on diminishing in arithmetical proportion until its movement 

The length of the first turn will bear the same proportion to the 
second as the first right twist has to that of the left. m 72 v. 

[ Risings of water] 

If the reservoir be of uniform breadth and in all the parts of its 
height so that it touches the water, and from the bottom water issues 
forth which may be guided upwards somewhat by the edges of its 
tube, it will spout up in as great force issuing by a narrow tube as by 
a thick one, because the water as it issues forth in a thin stream is 
driven by the thin part that is in the reservoir, and the thick part will 
be guided and driven by the thick part which is likewise in the said 
reservoir. m 73 r. 

[Movement and weight. Cross-bow] 


The length of the movement made by the thing driven by the 
cross-bow will have the same proportion with the movement of its 
cord as that of the weight driven to the weight that loads the said 

A b is the movement made by the cord of the cross-bow when it is 
laden and acquires its force. 

B c is the contrary movement which the cord makes when it loses 
its force, and there is the same proportion between the movement that 
the cord makes in acquiring force and the movement made by the 
death of the force, as between the weight driven and the weight that is 
the creator of the force. m 73 v. 



If a weight moves the cord of the cross-bow as far as its notch, the 
half of this weight will pass the half of the movement of such cord, 
and five sevenths of the weight will create three quarters of the move- 
ment of the cord towards the notch. 

Now in order to give a uniform weight to the movement of the 
cord which as it slackens diminishes its degrees of force with each de- 
gree of movement, you take away the half of the force; thus if it was 
four hundred in the beginning and in the last stage nothing, take the 
half of the pyramidal force which was two hundred and calculate with 
the weight of the arrow. ' m 74 r. 


Although the force of the cross-bow is great at the beginning and 
nothing in the last stage, nevertheless the movement of the cord 
through the impetus acquired makes itself swifter towards the end 
than at the beginning of its movement: wherefore we conclude that 
the arrow goes to the end of the movement of the cord. m 74 v. 

[Of the power of percussion] 

Many small blows cause the nail to enter into the wood, but if you 
join these blows together in one single blow it will have much more 
power than it had separately in its parts. But if a power of percussion 
drives a nail entirely into a piece of wood this same power can be 
divided into ever so many parts, and though the percussion of these 
occur on the nail for a long time they can never penetrate to any extent 
in the said wood. 

If a ten-pound hammer drives a nail into a piece of wood with one 
blow, a hammer of one pound will not drive the nail altogether into 
the wood in ten blows. Nor will a nail that is less than the tenth part 
[of the first] be buried more deeply by the said hammer of a pound 
in a single blow although it may be in equal proportions to the first 
named, because what is lacking is that the hardness of the wood does 
not diminish the proportion of its resistance, that is that it is as hard 
as at first. 

If you wish to treat of the proportions of the movement of the 
things that have penetrated into the wood when driven by the power 




of the blow, you have to consider the nature of the weight that strikes 
and the place where the thing struck buries itself, m 83 v. and 84 r. 

[Weight of pyramid] 

If you wish to divide the pyramid in two equal weights divide it 
lengthwise into four parts and join together the quarter towards the 
apex and the quarter towards the base; these two parts joined together 
will be equal in weight and quantity to the two centre parts; that is 
that one common measure will measure them precisely, as is shown 
here below. m 85 v. 

[Movement — cross-bow] 

Of the movement made in such manner that the mover ends its 
course before the cord remains drawn: 

This movement will be of like power to that which is made by the 
cord which remains drawn when the arrow passes out of the cross- 
bow, provided that the mover finishes its course in pyramidal power, 
that is great at the beginning and finishes in nothing. The movement 
of the arrow however which is great at the beginning and also 
ends in nothing preserves the length of its power of movement more 
than the cord does, seeing that its movement was capable of being of 
the length of four hundred braccia and the movement of the cord 
which drove this arrow was capable only of a third of a braccio. 

Pyramid of the power and movement of the mover. 

Pyramid of the power and movement of the thing moved. 

If the cord of the cross-bow after the flight that it has given to the 
arrow remains curved, it is certain that its power at each degree of 
movement has acquired degrees of slowness and infinite weakness; 
consequently we may say such power to be pyramidal because it com- 
mences at the base and ends in a point. 

The arrow also being driven by the cord of the cross-bow is pyra- 
midal, because at each degree of movement it acquires degrees of slow- 
ness and feebleness; but because this pyramid is longer than that of its 
mover, the arrow has parted from the cord before this cord has stopped; 
even when its mover was in its greatest power. m 90 r. and v. 



| Weight and counterpoise. Cross-bow] 


Here one ought to experiment upon the same counterpoise and with 
the same fall of this counterpoise, with different weights of its mover 
and different shapes, and see first what weight being of spherical shape 
will be that which will be driven to a greater distance from its mover 
than any other. In addition to this when you have found what this 
weight will be, which as I have said is spherical in shape, you will then 
prove how the weight of this movable thing varies in length and is 
equipped with feathers after the fashion of an arrow. And further 
such shape of arrow while of the same weight may be made of differ- 
ent substances heavier or lighter in themselves. And where such a 
shape stops in proportion to the power of its mover, it may then be 
tried by experiment whether the movement of its mover is increased 
if the movement made by the thing moved becomes longer or shorter, 
although this experiment ought first to be made whilst the movable 
thing is of spherical shape. 

And remember the means which are made use of between the 
mover and the thing moved, that is the weight of the instrument and 
the other things. m 91 r. and v. 


[Movement — Water] 



Here the thing moved can never be less rapid than the movement 
made by its mover. In effect if the thing moved had movement equal 
to that of its mover the mover could not make percussion with it, and 
would only be able to move as much weight as was equal to as much 
of the water as follows the movement of the wave that drives it. 

m 92 r. 



1 Counterpoise and cross-bow] 


This movement is the contrary to that of the cross-bow seeing that 
the mover of this acquires at every stage of movement degrees of im- 
petus, and the cross-bow does the opposite because its cord commences 
in its force and ends in nothing, whereas the counterpoise as it falls 
commences in nothing and ends in great power. 

Now one understands here that with the great movement that the 
cord of the cross-bow makes at the commencement of its movement, 
the arrow which is moved by this impetus does not slacken its move- 
ment at the same time as does that of the cord; on the contrary it fol- 
lows the quality of the first speed and comes to separate itself from the 
cord before this cord has finished its movement. 

And the thing driven by the counterpoise does the opposite, for as it 
commences slowly and ends with great impetus it will never be able to 
separate itself as does the cross-bow, that is to say the thing moved, 
until such mover has finished its own course. m 92 v. and 93 r. 




If a weight of four hundred pounds draws the cord of the cross- 
bow over the notch the cord has the force of four hundred pounds and 
as it slackens it ends in nothingness. 

And this diminution of force comes about by stages after the man- 
ner of a pyramid of which the projecting parts are equal; consequently 
we may say that the centre of this pyramid is that which may be called 
the centre of the force, as the nature of its weight acts in the simple staff, 
in which if one takes its centre one will find the weight of the whole 
taking in the same way the centre of the rise of the cord of the cross- 
bow. And measuring the weight which draws the said cord in this posi- 
tion, one will find that this weight is equal to the weight of all the ar- 
rows which could be stretched along the length of the movement that 
the arrow makes when drawn from the notch of the cross-bow on its 
last course. And if this arrow were long and thin or short and thick 



or if it was a ball of lead consider how you ought to measure it in the 
whole route of its course. Think about this and make a general rule 
for it because it is a matter that requires consideration. 

m 93 v. and 94 r. 

Excessive force against a like resistance profits the projectile noth- 
ing. But if the force of the mover should find itself in proportion to its 
projectile the movement made by the projectile will be in the first stage 
of its strength. It is as though I were to attempt to draw a bladder filled 
with wind against the air; for if this were moved by excessive force the 
air where it strikes would make such resistance through its becoming 
compressed that the bladder striking upon it there would leap back just 
as though it had been driven against a wall. But if this bladder were 
moved by a motive power proportionate in force and movement to the 
lightness of the said projectile, then this projectile will advance as far 
forward as its power enables it to drive slowly before it the air that 
withstands its course. b.m. 54 r. 

There are two different kinds of percussions, simple and complex. 
Simple is that made by the movable thing in its falling movement 
upon its object. Complex is the name given when this first percussion 
passes beyond the resistance of the object which it strikes first, as in the 
blow given to the sculptor's chisel which is afterwards transferred to 
the marble that he is carving. This blow also is divided into two others, 
namely a simple and a double blow. The simple blow has been suffi- 
ciently described: the double is that that occurs when the hammer 
descends with force in its natural movement and flies back rebounding 
from the greater blow and then creates an inferior blow and makes this 
percussion in two places with the two opposite sides of the hammer. 
And this blow grows less and less in proportion to the number of the 
obstacles which are interposed between it and the final resistance, just 
as if someone were to strike a book on its front page when they were 
all touching, the last page would feel the damage very slightly. 

b.m. 82 r. 

All movements are caused by abundance or dearth, and where there 
is the greater excess there the movement will be greater, b.m. 132 r. 

Movements are of two kinds, of which one is called simple and the 



cither composite. The simple is divided into two parts, and the one is 
when the body moves round its axis without change of position, as the 
wheel or millstone or things like these; the second is when the thing 
moves from its position without any revolution of itself. 1 The com- 
posite movement is that which in addition to moving from its position 
also moves round its axis, as the movement of the wheels of a waggon 
or other similar things. 

Circular movements are of two kinds of which one is called simple 
and the other composite. b.m. 140 v. 

The straightness of the transverse movement continues in the mov- 
able thing as long as the whole of the power given to it by its mover 

The straightness fails in the transverse movement because the power 
which the movable thing acquires from its mover becomes less. 

b.m. 147 v. 

Force is produced by dearth or profusion. It is the child of material 
movement, the grandchild of spiritual movement, the mother and 
source of gravity. This gravity is confined within the element of water 
and of earth, and this force is infinite, for by means of it infinite 
worlds could be set in motion if it were possible to make the instru- 
ments by which this force could be produced. 

Force with material movement and weight with percussion are the 
four accidental powers in which all the works of mortals have their 
being and their end. 

Force has its origin in spiritual movement which courses through the 
limbs of sentient animals thickening their muscles, and by this process 
of thickening the muscles become contracted and so draw back the 
tendons which are connected with them, and from this originates the 
force that exists in men's limbs. 

The quality and quantity of the force in a man will have the power 
of giving birth to other force, and this will be proportionately so much 
the greater according as the movement of the one is longer than that 
of the other. b.m. 151 r. 

Gravity and force together with material movement and percussion 
are the four accidental powers by which the human race in its marvel- 

1 Note in margin, 'progressive movement'. 



lous and varied works seems to reveal itself as a second nature in this 
world; seeing that by the use of such powers all the visible works of 
mortals have their existence and their death. 

Gravity is a power created by movement which transports one ele- 
ment into another by means of force, and this gravity has as much life 
as is the effort made by this element to regain its native place. 

Force and gravity have much in common in all their powers, differ- 
ing only in the movements of their birth and death. For simple gravity 
merely dies, that is as it approaches its centre. But force is born and 
dies in every movement. 

The spirit of the sentient animals moves through the limbs of their 
bodies, and when it finds that the muscles in those it has entered are 
responsive, it sets itself to enlarge them; and as soon as they enlarge 
they shorten and in shortening draw back the tendons which are 
joined to them. And from this arises the force and movement of hu- 
man limbs. Consequently material movement springs from spiritual. 

b.m. 151 v. 

No element possesses gravity or levity in its natural state. Gravity 
and levity are caused by one element being drawn into another. 

When an equal quantity of elements naturally contiguous have ex- 
changed places they will offer an equal amount of resistance one to 
another. b.m. 174 v. 

Weight descends for lack of resistance and that resistance which is 
weak gives way before it more speedily. b.m. 175 v. 

No movement can ever be so slow that a moment of stability is found 
in it. 

That movement is slower which covers less distance in the same time. 

That movement is swifter which covers a greater distance in the 
same time. 

Movement can extend to infinite degrees of slowness. 

And the power of the movement can extend to infinite degrees of 
slowness and likewise to infinite degrees of swiftness. b.m. 176 v. 

No element possesses weight within its sphere, and when by chance 
an element passes over into a lighter one it instantly creates gravity; 



aiu\ not being able to be supported there it falls back again into its own 
element, and there immediately this gravity dies. 

Gravity and force which are interchangeably daughters and mother 
of motion and sisters of impetus and percussion are always fighting 
against their cause; and after this has been subdued they conquer 
themselves and die. 

Gravity is a particular action which takes place when one element is 
drawn into another and not being able to be received there attempts 
with perpetual combat to return to its own place. 

Gravity is a particular fortuitous action of one element when drawn 
into another; it has as much life as there is desire in these elements to 
return to their own place. 

That which moves towards the centre is termed weight and that 
which flies from it is termed lightness; but each is of equal power and 
life and movement. b.m. 181 r. 

Every heavy body desires to lose its heaviness . . . 

Gravity, force, together with percussion, are to be spoken of as 
producers of movement as well as being produced by it. 

Of these three fortuitous powers two have in their birth, their desire 
and their end one and the same nature. b.m. 184 v. 

Of the things that support themselves without movement in the 
water no part that is above the water has weight of itself. This is 
proved as follows: if neither the still water has weight of itself nor 
the things supported by it, and it is proved by the passage that its 
weight is wanting and also the weight of the water, how then can we 
suppose that the motionless water which is without weight should 
support the weight? And if it supports the bodies it does not support 
their weight which has been consumed, for this weight is ended with 
the movement of the penetration which it made in this water. 

b.m. 267 v. 

When anyone wishes to make a bow carry a very long way he should 
draw himself right up on one foot raising the other so far from it as to 
create the necessary balance for his body which is thrown forward on 
the first foot, and he should not have his arm fully extended; and so 
that he may be better equipped for the hard work he should fit to the 



bow a piece of wood which as used in cross-bows should go from the 
hand to the breast, and when he wishes to discharge the arrow, instantly 
at the same time he should leap forward and extend the arm that holds 
the bow and release the cord. And if by dexterity he does all this at the 
same moment it will travel a very long way. 

The reason given for this is as follows: — know that as the leap for- 
ward is swift it lends a degree of fury to the arrow, and the extending 
of the arm because it is swifter lends a second; the driving of the cord 
being also more swift gives a third. If the other arrows therefore are 
driven by three degrees of fury and this by the dexterity shown is 
driven by six it ought to travel double the distance. And I would re- 
mind you that you leave the bow relaxed, so that it will spring forward 
and remain taut. Forster i 44 r. 


If there are two men who hold a sheet by its borders in which sheet 
there is a man who weighs two hundred pounds, and each pulls his 
end so much that the weight does not touch the ground, know that 
each of those who are pulling is holding up a weight as great as the 
man weighs who is in the middle, because he supports half the weight 
of the man in the centre and half that of the man opposite who is pull- 
ing, so it appears that the weight in the centre being two hundred 
pounds since each of those pulling has two hundred becomes four hun- 
dred pounds. Forster 1 48 v. 


Every movement born of movement which is free either divagates or 
preserves the line of the movement which produces it, except the 
thunderbolt which descends from the clouds. 

For what reason does the club give a greater blow and moves more 
than the stone? Forster 11 32 r. 

If a man with his whole strength throws a stone of four pounds 
twenty braccia, one of one pound would he throw it eighty (MS. eight) 
braccia or no? One of half a pound would he throw it one hundred 
and sixty braccia (MS. pounds) or no? And if he does not throw these 
to such a distance what is the cause? Forster 11 33 v. 





should be with your body in a state of precisely 

equal bal 

59 1 


with the opposite counterpoise and you throw your arms up furiously. 
holding two weights in your hands, I am in doubt whether your weight 
would become light or heavy: light I said; by the movement made by 
its extremities it would wish to follow the impetus commenced, where- 
fore it uproots the weight and seems to lighten the man; also one may 
say that the air where the arms strike in its resistance may make heavi- 
ness after the manner of the jumper who presses down the ground at 
the beginning of his jump. Forster n 45 r. 

[S\etch — man mounting pair of steps] 

I ask : this weight of the man at every stage of movement upon this 
pair of steps, what weight does it give a b and a c? Observe its per- 
pendicular under the centre of the gravity of the man. 

Forster 11 45 v. 


Diagram let a be sixteen b one. 

I say that the resistance of the air will not allow the movement to be 
in the sixteenth proportion, and this experiment may be made upon 
fine mud of uniform fluidity by dropping two pieces of draw-plate iron 
into it of sixteen and of one. Forster 11 48 v. 

A man in running throws less of his weight on his legs than if he is 
standing still. And in like manner the horse when running is less con- 
scious of the weight of the man whom he is carrying; consequently 
many look upon it as marvellous that when a horse is in a race it can 
support itself upon one foot only. Therefore we may say as regards 
weight in transverse movement that the swifter this is the less it weighs 
perpendicularly towards the centre [of the earth]. Forster 11 50 v. 

The wheel as it turns upon its axle causes part of the axle to become 
lighter and the other heavier even more than double of what it was at 
first, not being able to move away from its position. Forster 11 51 r. 

By the law of the balance mathematically an infinite weight is raised. 

Forster 11 53 v. 



If the cross-bow or other engine drives a hundred braccia from itself 
a movable thing of a pound which has one degree of size, how far will 
it drive a pound of half a degree of size? And then of a quarter and 
then of an eighth. Forster n 57 v. 


The centre of the world is indivisible, therefore nothing alone being 
indivisible the centre is equal to nothing. And if one should make a 
hole which was with its diameter or indeed its centre the diameter of 
the world, and there were thrown there a weight, the more it were to 
move the greater would its weight become. 

So having arrived at the centre of the earth which has only the name 
and it being itself equal to nothing, the weight thrown would not find 
any resistance at this centre but would rather pass and then return. 

Forster 11 59 v. 

Every heavy thing which descends freely directs its course to the 
centre of the world; and that which has most weight descends most 
rapidly and the more it descends the more it becomes swift. 

[Sketch — ship in water] 

The water that is moved from its place by reason of the ship weighs 
as much as the actual weight of this ship exactly. Forster 11 65 v. 


If two cords support the same weight and are not equal in per- 
pendicular or slant they will not be equally burdened by this weight, 
but the one will receive so much more of the weight than the other as 
the one is shorter than the other and as the angles made by the line of 
the cords and by the beam above to which they are fixed are greater 
the one than the other. Forster 11 67 v. 



In this circle I wish to experiment in circular movement, that is to 
place there within things large and small of the same substance, things 
of equal size of different substances, and keep them mixed or as chance 
will have it, and see at the end of the movement what position each has 



And I wish to do the same with dust and a blow. Forstcr 11 68 v. 

The ball of the bombard shot through the mist makes a much 
shorter course and less percussion than that which is shot through the 
pure thin air; but it will make a considerably louder report. 

I believe also that the arrow shot slanting into water twists as does 
the line of sight; and of this I will make a proof by fixing the bow and 
shooting in a frame upon which a sheet of paper is stretched, this 
paper being over the water; and after you have shot on this paper 
without moving the bow or the sheet of paper take away the water 
And you will discover the arrow, and by means of a thin line you will 
be able to discern if the shaft of the cross-bow and the centre of the 
hole made in the paper and the length of the arrow are in the same 
line or no; and by this means you will make your general rule. 

Forster 11 69 v. 

If many bodies of equal weight and shape are allowed to drop one 
after another at equal intervals of time from the same altitude in such a 
way that there may always be one quantity in the air, I say that the 
spaces between them will be equal. 

If each thing that descends at every stage of movement acquires a 
stage of speed, we may say: a to descend to b in six intervals of time, 
and b to c in five intervals, and c to d in four, and d to e in three, and 
e to / in two, and / to g\ and thus the excesses are equal. It is necessary 
therefore that as many touch the ground as start above with equal time. 

Forster n 70 v. 

Whatever the proportion of the number of the cords placed in the 
pulley-blocks which draw the weight to those which sustain this 
weight, such is that of the weight that moves to that which is moved. 

Whatever the proportion of the number of the cords placed in the 
pulley-blocks, which pass through the pulley-block of the weight, to 
those which sustain this weight, such is that of the gravity suspended 
to the weight which sustains it. 

As many as are the wheels of the pulley-block, so many times the 
mover offers resistance of itself and this on one side only. 

Forster n 72 v. 

When the two ends of the cord which go out of the pulley are 



situated equally the power of the mover will be as that of its re- 

In proportion as the nature of the positions which the cords assume 
as they issue from the pulleys displays greater variety of shape, so the 
power of the mover varies to that of its resistance. Forster n 73 v. 

Ascertain always the proportion of the blow in company with the 
object which has to receive it. 

Since one hundred pounds applied at a single blow makes a greater 
percussion than a million applied one by one, I wish that when you 
train the battering-ram on the castle you cause the blow to be raised in 
the air by the simple weight of the men, and then you pull it back after 
the manner of a catapult or cross-bow, and you will have a good result. 

Forster 11 74 r. 

Prove what the difference is in giving to the arrow blow and move- 
ment, or merely blow alone or movement alone as is the custom. 

The blow and movement you will give to the middle of the move- 
ment usually made by the cord of the cross-bow. 

The blow alone you will give to the arrow at the end of the move- 
ment of the cord. The movement alone you will give when in all the 
movement of the cord you always find the arrow. Forster 11 75 r. 

Of the screws of equal thickness that will be most difficult which has 
most grooves upon it. 

And among those screws of equal length, thickness and number of 
ridges, you will find that the easiest to move which has the greatest 
number of curves of its ridges. 

That screw will be strongest to sustain weights of which the ridges 
have the less number of curves, but it will be most difficult to move. 

Forster 11 77 v. 

The screw will keep a straighter course which is rather drawn than 
driven; that is it keeps its direction better if you pull than if you press. 

If you drive or press with the screw, which touches the thing pressed 
with the extremity of its curves, this screw being forced will bend on 
the side opposite to this extremity of its curve which presses. 

Forster 11 78 r. 



I have ten measures of time and ten measures of force and ten 
incisures of movement and ten of weight, and I wish to raise up this 

If I double the weight and not the force in the movement it becomes 
necessary to double the time. 

If I double the weight and not the time or the force it becomes 
necessary to halve the movement. 

If I double the weight and not the movement or the time it becomes 
necessary to double the force. 

If I halve the weight and not the movement or the time, the force is 
halved. Forster 11 78 v. 


If you wish to know the weight of the cord that supports the last 
pulley, multiply always cubically the weight attached to the foot by 
the number of the pulleys, and the result of this multiplication will be 
the number of the pounds which this last cord receives from the afore- 
said weight attached to its foot. 

Let us suppose therefore that this weight attached to the foot is 
four, then you will say : four pounds multiplied by four the number of 
the pulleys makes sixteen; and then four times sixteen makes sixty- 
four; and it is multiplied cubically, and this cord above supports sixty- 
four pounds by the four attached to the feet; and if there were six 
such pulleys y^u would say: four times six are twenty-four, and four 
times twenty-four are ninety-eight (sic!) ; and this great weight is sup- 
ported by the last cord with the four pounds attached to the foot. 

Here it is shown how the four pounds proceeds to double con- 
tinually; with the addition of each wheel the previous weight is 
doubled. Forster 11 82 v. 


The cord doubles its natural strength as many times as it is suspended 
in different parts of its length. Forster 11 83 r. 

Gravity with suspended cord at every degree of movement makes 
degrees of weight. 

The force that moves gravity on suspended cord will be as great as 
that which moves this gravity over rollers or balls which are placed 



upon a surface that is quite smooth — because each is supported exactly. 

But at this point a doubt seems to arise, that is by the fifteenth of the 
first, where it says that the centre of the gravity of a suspended cord is 
beneath the centre of this cord, and this centre of the gravity desires as 
far as possible to approach to the centre of the earth; and if you draw 
this weight cross-wise, fixed weight makes a revolving movement and 
raises itself up and goes away from the centre of the earth, and so in- 
treases weight in its mover. The weight which is resting on balls on a 
smooth surface always has its centre at an equal distance from the 
centre of the world, and consequently it does not increase the resistance 
in its mover. 

The resistance created by friction for the movement of weights is 
separate and remote from this weight. 


This is shown by the things said before, that is that it is clearly seen 
that the movement made by the weights along the horizontal line does 
not of itself offer any other resistance to its mover than its natural 
friction which it makes with a smooth surface where it touches it; 
which movement becomes more difficult in proportion as the smooth 
surface becomes more scoured and rough. And in order to see the 
truth of this move the said weight upon balls on an absolutely smooth 
surface: you will then see that it will move without effort. 

The weight the movement of which is rendered difficult by the 
friction which it makes with the smooth surface where it moves, will 
increase in gravity as it lacks effort in the friction which it has with the 
smooth surface where it moves. 

This is shown as it raises itself on a line that has a considerable slant, 
for as it were its simple weight is in the force of the mover, and the 
friction is small. Forster n 86 r. and 85 v. 


Whoever knows how great a weight raises the hundred pounds up- 
wards by this slope knows the capacity of the screw. 

If you desire true knowledge of the quantity of the weight required 
to move the hundred pounds over the sloping road, it is necessary to 
know the nature of the contact which this weight has with the smooth 



surface where it produces friction by its movement, because different 
bodies have different kinds of friction; because if there shall be two 
bodies with different surfaces, that is that one is soft and polished and 
well greased or soaped, and it is moved upon a smooth surface of a 
similar kind, it will move much more easily than that which has been 
made rough by the use of lime or a rasping-file. Therefore always 
when you wish to know the quantity of the force that is required in 
order to drag the same weight over beds of different slope, you have to 
make the experiment and ascertain what amount of force is required to 
move the weight along a level road, that is to ascertain the nature of its 
friction. And if you neither know this nor wish to make trial of it, set 
up an obstacle in your way, and that goes changing according to the 
slope of the road whence this weight ought to be drawn. Seeing that 
different slopes make different degrees of resistance at their contact; 
and the reason is that if the weight which ought to move is placed 
upon level ground and for this reason has to be dragged, undoubtedly 
this weight will be in the first strength of resistance, because everything 
rests upon the earth and nothing upon the cord which ought to move 
it. But if you wish to draw it along a very steep road all the weight 
which it gives of itself to the cord which sustains it is substracted from 
the contact of its friction; but as it is necessary to show another more 
palpable reason: — you know that if one were to draw it upright graz- 
ing and touching a wall somewhat, that this weight is almost all upon 
the cord which draws it and only a minute part rests upon the wall 
where it rubs. Forster 11 87 r. and 86 v. 

If the centre of the weight be outside the perpendicular of the centre 
of the screw which moves it: 

This weight will show itself heavier to its mover, and the teeth of the 
screw together with those of the screw-box which encloses them will 
be oppositely weighed down by two contrary forces. Forster 11 97 r. 

Weights work in balances along the line of their perpendicular. 

You have in the ninth of my theory that when the weight is attached 
to the transverse cord within equal angles each extremity of this cord 
is equally burdened by this weight; moreover the fact of these ex- 
tremities being at varying distances from this weight does not make 
any difference. Forster 11 99 r. 



Why the small gimlet makes its hole without anything to guide it 
and the large one requires two or three turns for this hole to be made 
larger. Forster n ioo v. 

Let the weight be affixed with as many cords as you wish to the arms 
of the balances, so that you have only to seek, if it is not the perpen- 
dicular of the centre of the weight, in what part it intersects with the 
arm of the balance which is above it. Forster n 105 r. 

I Diagrams] 

I have affixed three different weights somehow to one of the arms 
of the balance at three different places chosen by chance, and I would 
wish on the opposite arm to give a counterpoise to the said weights; 
which counterpoises are two, that is the one of four pounds and the 
other of two, and finally I should wish to attach them separately at 
such a place that they would be equal in weight to three other weights 
opposite. Forster 11 105 v. 


To one of the arms of the balance I have attached three different 
weights, that is one of one pound, another of two and a third of three 
pounds, and these said weights are at varying distances from each other 
as chance may have it; now I have a weight of eight pounds and I 
would wish to set it upon the opposite balance as counterpoise to these 
three such weights; I ask in what position it is to be placed to make 
itself equal to those opposite to it: you will do as you see here below. 

Forster 11 106 r. 

The centre of any heavy body whatsoever will stand in a perpen- 
dicular line beneath the centre of the cord on which it is suspended. 

I ask if you were to suspend a pole outside the centre of its length 
what degree of slant it will assume. 

The pole which is suspended outside the centre of its length by a 
single cord will assume such a slant as will make with its opposite 
sides together with the perpendicular of the centre of the cord that 
supports it, two equal acute angles or two equal obtuse angles. 

Forster 11 115 r. 

If the wheels are of equal height the waggon will move with a sure 



degree of force. But if you change the two back wheels for wheels of 
greater height it will move with greater ease. If however in the case of 
the first wheels you were to change the wheels in front for some less 
in height, in such a way that in the same manner the wheels in front 
were low and those behind high, the first movement will have been 
made more difficult and harder. Forster 11 124 r. 

The balance with three equal arms will remain stable of itself in 
whatever position you may turn it, and the weights will always stand 
in double proportion, except when one of the arms is in a perpen- 
dicular line, because then the proportion will be that of the equality. 
Take away the perpendicular of the centres of the weights of each arm 
of this triangular balance, and observe how they stand there with the 
centre of the balance; and if you find two on one side, take their centre 
against the centre of the opposite arm and you will see a double pro- 
portion of spaces and weights. Forster 11 126 r. 

If centres of weights are equidistant from their common centre these 
weights will be equal in equilibrium. 

If perpendiculars of centres of weights are equidistant from the per- 
pendicular of their common centre these weights will stand equal in 
equilibrium if they are equal. 

For this reason the centre of the world is always movable through 
the change in the overflowing of the ocean. Forster 11 126 v. 

Gravity is all in all the length of its support and all in every part 
of it. 

Why has it been found by experiment that when the pole stands in a 
slanting line and remains with its parts equidistant to the central line 
it does not remain slanting but rather becomes horizontal, forming 
four right angles with the above-mentioned central line ? 

The answer is that it proceeds from the imperfection of the pole. 

Forster 11 128 r. 

A weight of one pound falls one braccio and gives a blow of a cer- 
tain force; the question is asked if a weight of half a pound were to fall 
double the height, or twice from the first height, or twice the weight 
from half the height or four times from a quarter of the height, if it 
would produce the same result. Forster 11 130 r. 



Every heavy body weighs in the line of its movement. 

Forster n 130 v. 
[ Diagram ] 

Although the time in which the movement of heavy bodies occurs 
together with the length of this movement is divisible, it does not fol- 
low that the act of percussion because made on the surface of these 
bodies can itself be divided. 

Though the figure s strikes on the slant a n and that it will appear 
that being fleeting it may not be powerful, it will not fail to be the case 
that the act of percussion will be much more powerful than if it was a 
round body and that its rebound will be along the line. 

Forster 11 131 r. 


The action of friction is divided into parts of which one is simple 
and all the others are compound. Simple is when the object is dragged 
along a plain smooth surface without anything intervening; this alone 
is the form that creates fire when it is powerful, that is it produces 
fire, as is seen with water-wheels when the water between the sharp- 
ened iron and this wheel is taken away. 

The others are compound and are divided into two parts; and the 
first is when any greasiness of any thin substance is interposed be- 
tween the bodies which rub together; and the second is when other 
friction is interposed between this as would be the friction of the poles 
of the wheels. The first of these is also divided into two parts, namely 
the greasiness which is interposed in the aforesaid second form of fric- 
tion and the balls and things like these. Forster 11 131 v. 

All things and everything whatsoever however thin it be which is 
interposed in the middle between objects that rub together lighten the 
difficulty of this friction. 

Observe the friction of great weights, which make rubbing move- 
ments, how I have shown in the fourth of the seventh that the greater 
the wheel that is interposed the easier this movement becomes; and so 
also conversely the less easy in proportion as the intervening thing is 
thinner as would be any thin greasy substance; and so increasing tiny 
grains such as millet make it better and easier, and even more the balls 



of wood or rollers, that is wheels shaped like cylinders, and as these 
rollers become greater so the movements become easier. 

Forster 11 132 r. 

That thing which is entirely consumed by the long movement of its 
friction will have part of it consumed at the beginning of this move- 

This shows us that it is impossible to give or make anything of any 
absolute exactness, for if you desire to make a perfect circle of the 
movement of one of the points of the compasses, and you admit or con- 
firm what is set forth above, namely that in the course of long move- 
ment this point tends to become worn away, it is necessary to concede 
that if the whole be consumed in the whole of a certain space of time, 
the part will be consumed in the part of this time, and that the indi- 
visible in the indivisible time may give a beginning to such con- 

And thus the opposite point of these compasses which turns in itself 
over the centre of this circle, at every stage of movement is in process of 
being itself consumed and of consuming the place on which it rests; 
whence we may say that the end of the circle is not joined with its be- 
ginning, rather the end of such line is some imperceptible part nearer 
towards the centre of such circle. 

The friction made by the same weight will be of equal resistance at 
the beginning of its movement although the contact may be of different 
breadths or lengths. 

The greatness of the contact made by compact bodies in their fric- 
tion will have so much more permanence as it is of greater bulk; and 
so also conversely it will be so much less enduring as it is of less size. 

That which is said is shown in the case of the friction made by the 
head of the handle of the knife, for in equal time it is more perceptible 
than that which is made by its point. Forster 11 133 r. and 132 v. 

Impetus transports the movable thing beyond its natural position. 

Every movement has terminated length, according to the power 
which moves it, and upon this one forms the rule. 

Every movable thing which acquires velocity in the act of move- 
ment is moved under its natural movement, and so conversely when it 
loses it moves with accidental movement. Forster 11 141 v, 




Proof how these cords have equal weight. 

If the center of the weight is located in the middle of the equal 
number of cords which support it this weight is equally distributed 
between each cord. 

Here one is supposing the pole to be unbendable, and not taking 
count in cases like these of the weight of the instrument but only of the 
weight attached. Forster n 142 r. 

The gravity which is moved in conformity with its natural position 
with every degree of movement acquires a degree of speed. 

And if the gravity shall move in opposition to its natural position 
with every degree of movement it loses a degree of speed. 

In transverse movements the degrees of diminution are in the case 
of that which goes upward. Forster 11 144 r. 

The pole which is suspended at its extremities by two cords divides 
its weight equally between these cords. 

But if one of the cords remains fixed and the other moves towards it, 
weight moves from this fixed cord and joins itself to the weight of the 
movable one. 

The more a cord is moved towards the centre of the pole the more 
weight is taken from the other cord. 

The weight which is moved within the cords is in the same propor- 
tion to the first weights as is the movement made by the cord to the 
remainder of the pole. Forster 11 150 r. 

But if the one cord is fixed and the other moves towards it, weight 
moves from this fixed cord and is united with that of the movable one. 

Forster n 150 v. 

The pole which at its extremities is suspended to two cords divides its 
weight equally between these cords. 

But if one of the cords is moved towards the other every degree of 
movement corresponds to this change of weight. 

The weight that moves between the cords has such proportion to the 
first weights as the movement made by the cord has to the remainder 
of the pole. 

The remainders of the weights which will be left to these cords will 



have together such proportion as the opposite spaces which are enclosed 
between the two cords and the centre of the pole have between them. 

Forster 11 151 r. 

The variety of the weight which this cord acquires by its movement 
will have such proportion with that of the first weight as its movement 
has with the remainder of the pole. 

And the weights that are changed on the said cords will have to- 
gether such proportion as the spaces which intervene between the two 
cords and the centre of the pole have between themselves. 

And weights which have remained on the said cords will have to- 
gether such proportion as oppositely have the spaces that are enclosed 
between the centre of the pole and the two cords. 

The weight which moves between the cords has such proportion to 
the first weights as the movement made by the cord has to the re- 
mainder of the pole. Forster 11 151 v. 

The pole which is suspended by two cords at its extremities divides 
its weight equally between these cords; and although these cords may 
be moved equally towards the centre of the pole they do not vary their 
first weight. 

But should one of the cords towards the centre of the pole be moved 
and the other remain fixed at its extremity every degree of movement 
occasions among them variation of weight; and the remainder of the 
weights of the cords will have such proportion one with another as have 
the spaces opposite to them which are enclosed between the centre of 
the pole and the two cords. Forster 11 152 r. 

Such proportion as there is between those spaces which are enclosed 
between the centre of the length of the suspended pole and the two 
cords which sustain this pole, such will there be one with another the 
opposite weights which this pole gives of itself and the cords which 
support it. 

The thing which moves by natural movement at every degree of 
movement acquires degrees of speed, which degrees will bear the same 
proportion, the last to the last but one, as the second has to the first. 

Forster 11 152 v. 



The pole which is suspended by the extremities of its length to two 
cords divides its weight equally between these cords. 

But if the one of the cords towards the middle of the length of the 
pole be moved there will be the same proportion between the weight 
separated from the stationary cord and joined to that which moved, as 
between the movement made by the cord and the remainder of the 
pole which is supported between the two cords. 

But if the one of the cords is stationary and the other is moved 
towards it, weight departs from this stationary cord and unites itself 
to that which is moved, which has the same proportion to the remain- 
ing part of the first weight as the movement made by the cord has to 
the first space of these cords. Forster n 153 r. 

Whoever speaks of arms of a balance means them to be of equal 
thickness and weight if they are of equal length. 

The spaces which are interposed between the centre of the arms and 
the pole of the balance have between them such proportion as the 
opposite weights have with them as the one arm serves as counterpoise 
to the other. 

The spaces which are enclosed between the centre of the two arms of 
the balance and the pole of this balance have between them such pro- 
portion as is that which the weights of these arms have between them 
together with their length. Forster 11 154 v. 

The centre of the length of each arm of the balance is the true 
centre of its gravity. 

The arms of the balance make of themselves a counterpoise the one 
to the other; which counterpoise will have with these arms as many 
varieties as the proportions of these arms will be varied. 

That proportion which the one arm of the balance has to its opposite 
arm such the weight will have with it as this lightens the opposite arm. 

Forster 11 155 v. 

The centre of the length of each arm of the balance is the true centre 
of its gravity. 

Arm of balance is said to be that space which is found between the 
weight attached to this balance and its pole. 

That proportion which exists between the spaces that come between 



the centres of the arms and the pole of the balance, is as that oi the 
opposite weights which the one arm gives of itself in counterpoise to 
the other with its own arm which is the counterpoise. 

Arms of the balance are said to be those which are found between the 
centres of the weights affixed to it and the pole of the said balance. 

Forster 11 157 r. 

Where the support makes less resistance there the weight supported 
by it shows itself heavier; and that part of the support makes less re- 
sistance which is more remote from its foundation. Forster 11 157 v. 

Of the pyramids of equal height the proportion of the weight will be 
as that of the bases. 

Pyramids of varying lengths upon equal bases will be of as many 
different proportionate weights as their lengths are varying; the 
pyramids of equal bases with different lengths enclosed in a parallelo- 
gram will be of equal weight. Forster 11 158 r. 


If a chimney-sweeper weighs two hundred pounds how much force 
does he exert with his feet and back in the chimney? 

Forster in 19 v. 
I ask why the blow of the hammer causes the nail to jump out. 

Forster in 20 v. 

The air which closes itself up with fury behind the bodies which 
move through it offers more resistance than that which remains sta- 
tionary, consequently the ball when struck covers a greater distance 
than the jump or the leap can serve as the occasion of. 

Forster in 27 r. 

Why it is first the blow rather than the movement caused by it; the 
blow has performed its function before the object has started on its 
course. Forster 111 28 r. 


It will be impossible to break that support which is the centre of the 
gravity placed upon it; and the centre of this itself will be the perpen- 
dicular upon the centre of its base. Forster in 29 r. 



A blow is an end of movement created in an indivisible period of 
time, because it is caused at the point which is the end of the line of 
the movement made by the weight which is the cause of the blow. 

Forster m 32 r. 

No animal can simply move more weight than is the load that finds 
itself outside the centre of its support. Forster in 34 r. 

The movement made by the arrow in ordinary simple flight will 
increase as much as the power of the composite movement of a second 
tethered (?) [apicata] flight. Forster in 38 v. 

There will be such proportion between the amount of movement of 
a stone that is moved and that of the thing that is moving, one time 
more than another, as the time of the moving thing is swifter on the 
one occasion than on the other. Forster in 39 r. 

The infinite movements of the varieties of the instruments which 
may be constructed for drawing weights will be of equal power in the 
completed movement of the thing moving and of that moved. 

Forster in 40 v. 

Define to me why one who slides on the ice does not fall. 

Forster in 46 r. 

Prove which keeps its movement more, a wheel that revolves on the 
flat or on an edge. 

Prove whether the impetus of the revolving of the wheels acquires 
force from its mover. Forster in 48 r. 

Every free heavy body when falling directs its course to the centre, 
and that part which weighs most will be nearest to the centre of the 
world. Forster in 51 r. 

As is the proportion one to another of the spaces that are enclosed 
between the perpendicular of the weight attached to the slanting beam 
and the perpendiculars of the extremities of this beam, so will be that of 
the weights of the opposite extremities of the beam. Forster in 51 v. 

If two opaque bodies are moved one against the other with inter- 
secting movement the two bodies will seem three; and in like manner 




one thing will seem two, and the two without intersecting movement 
will appear to be four as the wings of birds when flying. Intersected 
movement. Simple movement. Forster in 55 r. 

The centre of gravity of any heavy suspended body will always fall 
below the centre of its support. 

The counterpoise divides with its weight if the beam divides it in 
haJf Forster m 60 r. 

If the sea bears down with its weight upon its bed, a man who lay 
on this bed and had a thousand braccia of water on his back would 
have enough to crush him. Forster m 66 r. 

The desire of every heavy body is that its centre may be the centre 
of the earth. Forster in 66 v. 

Friction produces double the amount of effort if the weight be 
doubled. Forster in 72 r. 


I ask how great a weight ought to be placed at m in order to draw 
one hundred pounds at n; and by degrees to ascertain what weight it 
will be which will overcome the other, giving the cord that goes from 
the one weight to the other sometimes one twist round the beam, 
sometimes two or three or four; and similarly if the beam be triangular 
or square or of a greater number of angles. Forster in 73 v. 

I ask which is swifter — a spark going upwards and living or turning 
downwards in death. Forster in 75 v. 

Everything attached or united to bodies .that have been struck will 
move against the place of the blow. 

That part of the cord that is twisted over the beam that lies equally, 
will press the part of the beam more which is nearer to the greater of 
the two weights that are fastened to the extremities of the cord. 

Forster in 77 v. 
[Duration of movement of liquid] 

The movement of the liquid made in any direction, proceeds as far 
in the revolution it has commenced as there lives in it the impetus 
given to it by its first mover. Fogli b 28 r. 



In the same space the arrow carries farther straight upwards than it 
does obliquely, and this arises from the fact that when the direction is 
upwards the arrow or bullet falls by the line in which it rose and 
obliquely it forms an arch. Quaderni n 15 r. 

[Law of inertia] 

The thing which moves will be so much the more difficult to stop as 
it is of greater weight. Quaderni iv 10 v. 

The thing which moves itself acquires as much space as it loses. 

Quaderni iv 15 v. 
[Of long and short steps] 

When one is descending one takes short steps because the weight 
rests on the hinder foot but when one is ascending one takes long 
steps because the weight is thrown on to the foot in front. 

Quaderni vi 18 r. 

That wheel will revolve more easily which has its axis of less 
thickness. Quaderni vi 21 r. 

It is proved how the air does not push the movable thing since it is 
separated by the power of its mover. 

If to the movable thing which separates itself from its mover there 
was given the perception of the movement of the air which pushed it 
behind it would happen that the bullet of the carbine in penetrating a 
leathern bottle full of water would immediately lose its movement at 
the beginning of its penetration, because instantly the water would 
close the entrance and separate it from the air which drives it; as to 
which experience shows to the contrary, seeing that this ball after the 
said penetration of the water moves for a long time. And if you were to 
say that the fury of the movement of the air or of the water, through 
which this bullet passes, which turns to fill up the vacuum from which 
the bullet departs point by point, is that which forms a wedge between 
the back of the bullet and the rest of the air which stays behind it; 
here the reply is that the air is more powerful and more compressed 
in front of the bullet than that on the opposite side, because this oppo- 
site side is the air reflected by the percussion of the bullet. 'The reflec- 
tion of anything is always of less power than its incidence'; and if you 
should gainsay me as to this by urging that this power cannot be in- 



fus^d in the body that is moved, because 'no movable thing moves <>l 
itself, unless its members exert force in other bodies outside it', as when 
a man in the centre of a boat pulls the rope attached to the stern of it, 
in order to give movement to the ship, which work is useless unless this 
rope is fastened to the bank where he wishes to move, or unless he 
pushes the oars in the water or the pole on the bottom; therefore the 
power not being in the air which drives the said bullet it is necessary 
that it is poured into the bullet; and if it is thus poured what has been 
said above serves as an example of the result; and in addition to this, 
this power so poured in would be of equal force through all its sides, 
because it would be spread equally in equal quantities through all that 
bullet; this however is not so, and the other premise you do not grant 
me; let us therefore seek for a third to which no exception can be 
taken. 'The potency of the mover is separated from it entirely and ap- 
plied to the body moved by it, and it goes on to consume itself in course 
of time in penetrating the air which is always compressed before the 
movable thing'. And this happens because 'every impression is pre- 
served for a long time in the object on which it is impressed', as is seen 
in the circles created by its percussion within the surface of the water, 
which move within the water for a long distance, and in the eddies and 
waves, formed in one spot, and carried by the impetus of the water to 
another, without their destruction; and radiance creates the same effect 
in the eye, and sound in the ear. But if you would also say that the 
air preserves the power of the mover which accompanies it and pushes 
its movable thing, how are we to reconcile to this the case of the wheel 
which in a storm of wind turns for a long time, although its mover is 
separated from it? 

It is not air that moves it for as it is equally distributed round its axis 
as regards its outline and its weight, the wind which embraces it on one 
side only, if it caresses the half of the wheel which flies from it, op- 
poses and resists the other half of the wheel which moves against it, 
and consequently the wind which stops the movement as much as it 
aids it does not render this wheel any service or disservice; therefore the 
potency of the mover was left imprinted on the outside of the wheel 
and was not poured into it or into the air that lay about it. If you wish 
to see the movement the air makes when it is penetrated by a movable 
thing take an example in the water, that is, underneath its surface, for 



it may have mingling with it thin millet or other minute seed which 
floats at every stage of height of the water; and afterwards place some 
movable thing within it which floats in the water and you will see 
the revolution of the water, which ought to be in a square glass vessel 
shaped like a box. 'Every natural act is communicated from the doer 
to the object in the shortest possible time'; and the air beaten and com- 
pressed by the movable thing that moves within it need not therefore 
be that which restores the vacuum, for the movable thing makes a 
succession of vacuums as it flies from it; but it is that which is nearer 
the opposite side of the movable thing, that is that by which it leaves 
the path, that continually rarefies the condensation already made; and 
by means of this rarefaction the before mentioned vacuum is restored. 
'Never, in the same time will the greater power be subdued by the 
lesser power' : therefore, the swift movement of the rarefied air in order 
to fill up the place in the vacuum, caused by the movable thing depart- 
ing from it, is much weaker than that which is continually being 
compressed before the movable thing; of which compression the air 
that is thinner than it will never be the cause. Therefore we have con- 
cluded that the movable thing does not move on account of the wave 
of the air created by the impetus of the mover. And if you wish to say 
that the flooding of the air which escapes before the movable thing is 
that which prepares the movement of the movable thing, together with 
the air, and runs after it in order to restore the rarefaction of the air, 
to this one replies that this air is here flooded by the movable thing and 
not by itself, and 'it is impossible that at one and the same time the 
mover should move the movable thing and the movable thing move its 
mover': therefore, your reason does not hold, because if the aforesaid 
flooding were that which had to draw itself after the cause of its 
movement, 'it is impossible that any thing of itself alone can be the 
cause of its creation; and those things which are of themselves are 
eternal'. Leic. 29 v. 

Gravity comes into being when an element is placed above another 
element thinner than itself. 
Gravity is caused by one element having been drawn within another. 

Sul Volo (f.m.) 1 r. 



Gravity is caused by one element being situated in another; and it 
moves by the shortest line towards its centre, not by its own choice, not 
because the centre draws it to itself; but because the mean in which it 
Ends itself cannot withstand it. Sul Volo (f.m.) 2 v.