{142}

[Sidenote: Certainty of Mathematics]

5.

He who blames the supreme certainty of mathematics feeds on confusion and will never be able to silence the contradictions or sophistical sciences which lead to an everlasting clamour.

[Sidenote: Of Science]

6.

There is no certainty [in science] where one of the mathematical sciences cannot be applied, or in those [sciences] which are not in harmony with mathematics.

[Sidenote: From Leonardo's Dictionary]

7.

Syllogism: to speak doubtfully.

Sophism: to speak confusedly; falsehood for truth.

Theory: knowledge without practice.

[Sidenote: Definition of Science]

8.

Science is that discourse of the mind which derives its origin from ultimate principles beyond which nothing in nature can be found which forms a part of that science: as in the continued quant.i.ty, that is to say, the science of geometry, which, starting from the surfaces of bodies, has its origin in the line, which is the end of the superficies; and we are not satisfied by this, because we know that the line terminates in the point, and the point is that which is the least of things. Therefore the point is the first principle of geometry, and nothing else can exist either {143} in nature or in the human mind from which the point can issue. Because if you say that the contact between a surface and the extreme point of an iron instrument is the creation of the point, it is not true; but let us say that this point of contact is a superficies which surrounds its centre, and in the centre the point dwells. And such a point is not a part of the substance of the superficies, neither it nor all the points of the universe can, even if combined,--it being granted that they could be combined,--compose any part of a superficies. And granted, as you imagined, a whole composed of a thousand points, if we divide any part of this quant.i.ty of a thousand, we can very well say that this part shall equal its whole; and this we can prove by zero, or naught, that is, the tenth figure of arithmetic, which is represented by a cipher as being nothing, and placed after unity it will signify 10, and if two ciphers are placed after unity it will signify 100, and thus the number will go on increasing by ten to infinity whenever a cipher is added, and the cipher in itself is worth nothing more than naught, and all the naughts in the universe are equal to one naught alone, in regard to their substance and value.

[Sidenote: True Science based on the Testimony of the Senses]

9.

Knowledge which is the issue of experience is termed mechanical; that which is born and ends {144} in the mind is termed scientific; that which issues from science and ends in manual work is termed semi-mechanical. But I consider vain and full of error that science which is not the offspring of experience, mother of all cert.i.tude, and which does not result in established experience, that is to say, whose origin, middle and end do not pa.s.s through any of the five senses. And if we doubt of everything we perceive by the senses, should we not doubt much more of what is contrary to the senses, such as the existence of G.o.d and of the soul, and similar matters constantly under dispute and contention?

And it is truly the case that where reason is lacking it is supplemented by noise, which never happens in matters of certainty. On account of this we will say that where there is noise there is no true science, because truth has one end only, which, when it is made known, eternally silences controversy, and should controversy come to life again, it is lying and confused knowledge which is reborn, and not certainty. But true science is that which has penetrated into the senses through experience and silenced the tongue of the disputers, and which does not feed those who investigate it with dreams, but proceeds from the basis of primary truths and established principles successively and by true sequence to the end; as, for instance, what comes under the heading of elementary mathematics, {145} that is, numeration and measurement, termed arithmetic and geometry, which treat with the highest truth of the discontinued and continued quant.i.ty.

Here there will be no dispute as to whether twice three make more or less than six, nor whether two angles of a triangle are less than two right angles, but eternal silence shall ignore all controversy, and the devotees of the true science will finish their studies in peace, which the lying mental sciences cannot do. And if thou sayest that true and established science of this kind is a species of mechanics, because they can only be completed by the hand, I will say the same of all the arts, such as that which pa.s.ses through the hand of the sculptor, which is a kind of drawing, a part of painting; and astrology and the other sciences pa.s.s through manual operation, but they are mental in the first place, as painting, which first of all exists in the mind of the composer, and cannot attain to fulfilment without manual labour. With regard to painting, its true and scientific principles must be established: what const.i.tutes a shaded body, what const.i.tutes a primary shade, a derivative shade, what const.i.tutes light: that is, darkness, light, colour, size, shape, position, distance, propinquity, motion, rest, which are comprehended by the mind only, and without manual labour. And this is the science of painting which remains in the mind of those who meditate on it, from which {146} issues the work in due time, and is infinitely superior to the aforesaid contemplation or science.

[Sidenote: Mechanics]

10.

Mechanics are the paradise of scientific mathematics, because with them we arrive at the fruits of mathematics.

[Sidenote: Mechanics and Experience]

11.

Experience is indispensable for the making of any instrument.

12.

Proportion is not only to be found in figures and measurements, but also in sound, weight, time and position, and in whatever power which exists.

[Sidenote: Reason and Experience]

13.

The power of the projecting force increases in proportion as the object projected is smaller; the acceleration of the motion increases to infinity proportionately to this diminution. It would follow that an atom would be almost as rapid as the imagination or the eye, which in a moment attains to the height of the stars, and consequently its voyage would be infinite, because the thing which can be infinitely diminished would have an infinite velocity and would travel on an infinite course (because every continuous quant.i.ty is divisible to infinity). And this opinion is {147} condemned by reason and consequently by experience.

Thus, you who observe rely not on authors who have merely by their imagination wished to be interpreters between nature and man, but on those alone who have applied their minds not to the hints of nature but to the results of their experience. And you must realize the deceptiveness of experiments; because those which often appear to be one and the same are often different, as is shown here.

[Sidenote: Effects correspond to the Force of their Cause]

14.

A spherical body which possesses a dense and resisting superficies will move as much in the rebound resulting from the resistance of a smooth and solid plane as it would if you threw it freely through the air, if the force applied be equal in both cases.

Oh, admirable justice of thine, thou first mover! thou hast not permitted that any tone should fail to produce its necessary effects, either as regards order or quant.i.ty. Seeing that a force impels an object which it overcomes a distance of one hundred arms' length, and if in obeying this law it meets with resistance, thou hast ordained that the force of the shock will cause afresh a further movement, which in its various bounds recuperates the whole sum of the distance it should have travelled. And if you measure the distance {148} accomplished by the aforesaid bounds you will find that they equal the length of distance through which a similar object set in motion by an equal force would travel freely through the air.

15.

Every action must be caused by motion.

16.

Motion is the cause of all life.

[Sidenote: Of Force]

17.

What is force? Force, I say, is a spiritual virtue, an invisible power, which by accidental external violence is caused by motion, and communicated and infused into bodies which are inert by nature, giving them an active life of marvellous power.

18.