Rule I. All sciences are nothing but human wisdom, which always remains one and same, and is no more altered by the different subjects it is applied to, than is the light of the sun by the variety of the objects it illuminates. All sciences are interconnected and interdependent, therefore it is not necessary for the mind to be confined within any limits.
Before delving into some particular science, we should inquire about the nature and limits of human knowledge, and improve the natural light of his reason. (Between things of equal difficulty, investigate things more useful for the discovering of truth.)
Rule II. Only concern ourselves with those objects regarding which our minds are capable of obtaining certain and indubitable knowledge, with a certainty equal to that of the demonstrations of arithmetic and geometry.
Rule III. Knowledge can be obtained in two ways only: intuition or in/deduction.
Deduction is a continuous and uninterrupted process of thought, in which each part of the process is clearly intuited. In deduction we perceive a movement or a certain succession of thought, while we do not in intuition; and because present evidence is not necessary to deduction as it is to intuition, but rather, in a certain measure, it derives its certainty from memory.
Rule IV. Universal mathematics is the study of order and measure, under which all other branches of sciences are subsumed.
The human mind has a certain touch of divinity in which the first seeds of useful thought already lie, so that often, no matter how much they are neglected and suffocated, they spontaneously produce fruit.
Rule V. Method consists entirely in the order and arrangement of things upon which the power (or eye) of the mind must be concentrated. Reduce complex and obscure propositions step by step to simpler ones, and then try to advance by the same gradual process from the intuition of the simplest to knowledge of all the rest.
Rule VI. Observe in every series of propositions previously deduced, which one is the simplest of all, and how far from this one each of the others is removed, either more, or less, or equally.
Rule VII. Examine each and every item which pertains to our design in a continuous and uninterrupted process of thought and include all of these in an adequate and orderly enumeration.
Rule IX. Focus our minds long and hard on the most minute and simple details so that we become accustomed to perceive the truth clearly and distinctly.
Exercise 1. The nature of force. whether it applies instantaneously to distant points
a) enumeration: magnetism; the influence of stars; the velocity of light
b) intuition: A stone cannot travel instantaneously from one place to another, because it is a body, but that a power, such as that which moves the stone, cannot be communicated otherwise than instantaneously if it goes by itself from one object to another, because it is communicated by itself, and does not exist in some body by which it is carried.
Exercise 2: How contrary effects can be produced simultaneously
a) enumeration: drugs that expel certain humours and retain others, a balance; in which the same weight causes one pan to rise and the other to drop simultaneously.
RULE X. Practice in seeking things which have been previously discovered by others, and in studying methodically even the most trivial human skills, but, above all, those which explain or presuppose order.
Whenever some book gave promise, in its title, of a new discovery. I tried, before reading further, whether perchance I could deduce something similar by some inborn sagacity, and I was very careful not to deprive myself of this innocent pleasure by precipitate reading.
We should especially investigate those arts in which there is more order, such as those of the artisans who weave fabrics and tapestries or those of women who embroider or weave threads in an infinite variety of patterns. They offer us very distinct examples of innumerable orders, all different from each other and nevertheless quite regular; and the scrupulous observation of these orders constitute almost all of human sagacity.
Rules XII. Use the intellect, the imagination, the senses, and the memory, not only for understanding simple propositions distinctly, but also for correctly comparing what is being sought with what is known, in order that they may be recognized; and finding those things which ought to be compared with each other.
Application in Geometry
The measurable proportions or relations between the whole and its parts, or any two objects, mediated through a unit in which both participates.
The line, the surface and the solid are not three distinct species of quantity. The line and the surface are not distinct from the solid/body, or from each other in reality; but if they are considered simply as abstractions of the intellect, then they are no more distinct species of quantity than animal and living are different species of substance in man. A line can be conceived as a rectangle, one side of which is the unity commensurate with the line.
The root, the square, the cube, and so forth, are nothing other than magnitudes in geometrical progression which we always assume are preceded by an arbitrary unit. To this unit the first proportional refers immediately and by a simple relationship. but the second by the mediation of the first and therefore through two relationships: the third by the mediation of the first and second, and by three relationships, and so on.