Who knows what weight pulls a hundred pounds upwards over this slope knows the capacity of the screw. In order to know accurately the quantity of the weight required to move a hundred pounds over the sloping road one must know the nature of the contact which this weight has with the surface on which it rubs in its movement, because different bodies have different frictions; thus let there be two bodies with different surfaces, one soft and polished, well greased or soaped, and let it be moved over a surface similar in kind, it will move much more easily than one that is roughened by lime or a rasping file; therefore, always when you wish to know the quantity of force required to drag the same weight over streets 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. . . .
Different slopes make different degrees of resistance at their contact; because, if the weight which must be moved is upon level ground and has to be dragged, it undoubtedly will be in the first strength of resistance, because everything rests on the earth and nothing on the cord which must move it. But if you wish to pull it up a very steep road all the weight which it gives of itself to the cord which sustains it is subtracted from the contact of its friction; but there is another more evident reason: you know that if one were to draw it straight up slightly grazing and touching a perpendicular wall, the weight is almost entirely on the cord that draws it, and only very little rests upon the wall where it rubs.
141
A thing which is consumed entirely by friction in its long movement will be consumed in part at the start of this movement.
This proves that it is impossible to give or make anything with absolute exactness; for if you want to make a perfect circle by moving one of the points of the compasses, and you admit what is set forth above, that this point tends to be worn away in the course of long movement, then the whole point will necessarily be worn away in a certain space of time and the part will be consumed in part of this time; and the beginning of such consumption will be indivisible in indivisible time.
And likewise the opposite point of these compasses as it rotates over the centre of the circle is being consumed at every stage of the movement while also consuming the place where it rests; we may therefore say that the end of the circumference does not join its beginning, but is imperceptibly nearer the centre of the circle.
142
2. WEIGHING INSTRUMENTS
Take a beam of uniform weight and thickness, and let it be suspended from a pole at its centre, or at a certain part of its length; and this beam thus suspended we shall call balance or scales, and the parts that protrude from the pole we shall call arms of the balance. . . .
143
The Commencement of this Book concerning Weights
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 then the arms of the balance will be 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 slant. Fourth. If the weights of 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.
144
The arrangement of the book will be as follows: First the simple poles, then supported from below, then partly suspended, then entirely, then let these poles support other weights.
145
Of weight proportioned to the power that moves it. One has to consider the resistance of the medium wherein such a weight is being moved; and a treatise will have to be written on this subject.
146
In order to test a man and see whether he has true judgement on the nature of weights, ask him where you should divide one of the two equal arms of a balance so that the cut part if added to the end of its remainder, may form an exact counterweight to the arm opposite it. Which thing is not possible; and if he indicates a place he is a sorry mathematician.
147
Where the Science of weights is led into error by the practice
The science of weights is led into error by its practice, which in many instances 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, because necessity demands this for the support of the weight of these balances together with the weights which are gauged 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, such as exist in the mind and are without substance; which errors I set down here below.
148
The centre of the length of each arm of a 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 of the opposite weight which the one arm gives of itself in counterpoise to the other with its own arm which is the counterpoise.
149
The intercentric line is said to be that which starts from the centre of the world and which rising therefrom in one continuous straight line passes through the centre of the heavy substance suspended in an infinite quantity of space.
150
The centre of any heavy body whatsoever will stand in a perpendicular 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.
151
3. WHEELS AND WEIGHT
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, when it did not move away from its position.
152
In a circular balance no heavy body will raise more than its own weight by the force of its simple weight.
I call circular balance the wheel or pulley by which water is pulled from the wells, to which one never applies more weight than that which is attained by the water that is lifted.
153
If you wish with certainty to understand well the function and the force of the tackle it is necessary for you to know the weight of the thing that moves or the weight of the thing moved; and if you would know that of the thing that moves multiply it by the number of the wheels of the tackle, and the total that results will be the complete weight which will be able to be moved by the moving thing.
154
If you multiply the number of the pounds that your body weighs by the number of the wheels that are situated in the tackle, you will find that the number of the total that results will be the complete quantity of pounds that it is possible to raise with your weight.
155
Cause an hour to be divided into three thousand parts, and this you will do by means of a clock by making the pendulum higher or heavier.
156
As the attachment of the heavy body is further from the centre of the wheel the revolving movement of the wheel round its pivot will become more difficult although the motive power may not vary. The same is seen with the time of clocks, for, if you place the two weights nearer or further away from the centre of the timepiece, you make the hours shorter or longer.
157
The fact that a thing may be either raised or pulled causes great difference of difficulty to its mover; for if it weighs a thousand pounds and one moves it by simply lifting it, it shows itself as a thousand pounds; whereas if it is pulled it becomes less by a third; and if it is pulled with wheels it is diminished by as many degrees in proportion to the sizes of the wheels, and also according to the multiplied number of the wheels. And with the same time and power it can make the same journey, with different degrees of time and power also in the same time and movement; and this is done merely by increasing the number of the wheels on which rest the axles which must be multiplied.
158
4. THE SCREW
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.
159
Vitruvius says* that small models are of no avail for ascertaining the effects of large ones; and I here propose to prove that this conclusion is a false one. And chiefly by bringing forward the very same argument which led him to this conclusion; that is, by an experiment with an auger. For he proves that if a man, by a certain exertion of strength, makes a hole of a given diameter, and afterwards another hole of double the diameter, this cannot be made with only double the exertion of the man’s strength, but needs much more. To this one may very well reply that an auger of double the diameter cannot be moved by double the exertion because the surface of a body similar in shape and of double the width has four times the quantity and power of the smaller one.
160
III
FLIGHT
Vasari reports that Leonardo used to buy caged birds in order to free them. Their flight was a continuous source of inspiration to him. He loved to watch them moving through the air, soaring, gliding, sailing, and flapping and compared their behaviour with that of flying insects and bats while drawing their instantaneous actions in his notebooks. Here also we find representations of the structure of wings and tails with accounts of their co-operation in steering the body.
This interest went hand in hand with the study of air. How did the air react? Here was an opportunity to study the powers of nature, i.e. weight, movement, and impetus at work within one of the four elements, and to compare and contrast its properties with those of another element that he had studied—water. How were bodies heavier than air able to sustain and propel themselves in it? Man should be able to do the same if suitably equipped. His designs show various machines with the aviator in horizontal and vertical positions, while using his limbs to operate fabricated wings; also an aerial screw heralding the helicopter of the present day, a parachute, a hydrometer to determine humidity, and an instrument to indicate the direction and to measure the force and velocity of the wind.
While absorbed in these studies, he once jotted down in his notebook that he must have been predestined to write on this because one of his earliest recollections was a dream of a kite paying a visit to his cradle and opening his lips with its tail (see pp. 269-70)
.
His booklet on the flight of birds, written and illustrated in 1505, is in the Biblioteca Reale, Turin.
I. MOVEMENT THROUGH WIND AND WATER
The instrumental or mechanical science is the noblest and most useful above all others, since by means of it all animated bodies which have movement perform all their actions; and these movements have their origin from the centre of their gravity which is placed in the middle beside unequal weights, and it has scarcity and abundance of muscles, and also lever and counter-lever.
1
Before you write about creatures which can fly make a book about insensible things which descend in the air without the wind and another on those which descend with the wind.
2
In order to give the true science of the flight of birds through the air you must first give the science of the winds, which we shall prove by the motion of the waters; and the understanding of this science, which can be studied through the senses, will serve as a ladder to arrive at the perception of flying things in the air and the wind.
3
Divide the treatise on birds into four books, of which the first deals with their flight by flapping their wings; the second of flight without flapping wings and with the help of the wind; the third of flight in general such as that of birds, bats, fishes, animals, insects; the last of the mechanism of motion.
4