Writings of Charles S. Peirce: A Chronological Edition, Vol. I, 1857-1866

ly considered, a system of like parabolas similarly placed, or any one of an infinity of systems of curves, is as simple as the system of straight lines. Again, motions and forces are combined according to the principle of the parallelogram, and a parallelogram appears to us a very simple figure. Yet the whole system of parallelograms is no more simple than any relief-perspective of them, or than any one of an infinity of other systems. As Sir Isaac Newton well said, geometry is but a branch of mechanics. No definition of the straight line is possible except that it is the path of a particle undisturbed by any force; and no definitions of parallels, etc. are possible which do not depend upon the definition of equal distances as measured by a rigid body, or other mechanical means. Thus, in dynamics, the natural ideas of the human mind tend to approximate to the truth of nature, because the mind has been formed under the influence of dynamical laws. Now, logical considerations show that if there is no tendency for natural ideas to be true, there can be no hope of ever reaching true inductions and hypotheses. So that philosophy is committed to the postulate,—without which it has no chance of success,—[. . .] Psychology has only lately become a positive science, and in my humble opinion the new views are now carried too far. I cannot see, for example, why psychologists should make such a bugbear of “faculties.” If in dynamics it has proved safe to rely upon our natural ideas, checked, controlled, and corrected by experience, why should not our natural ideas about mind, formed as they certainly have been under the influence of the true laws of mental action, be likely to approximate to the truth as much as natural ideas of space, force, and the like have been found to do? Upon this point, I must confess to entertaining somewhat heterodox opinions. The Herbartian philosophy, with the mode of reasoning which leads to it, seems to me thoroughly unsound and illusory,—though I fully admit the value and profundity of some of the suggestions of that philosophy. But to trust to such reasoning in the slightest degree seems to me ever so much less safe than trusting to one’s native or natural notions about mind, though these no doubt need to be modified by observation and experiment. For my part it seems to me that the elementary phenomena of mind fall into three categories. First, we have feelings, comprising all that is immediately present, such as pleasure and pain, blue, cheerfulness, and the feeling which arises upon the contemplation of a complete theory. It is hard to define what I mean by feeling. If I say it is what is present, I W r i t i n g s o f C . S . P e i r c e 1 8 9 0 – 1 8 9 2 96 shall be asked what I mean by present, and must confess I mean nothing but feeling again. The only way is to state how any state of consciousness is to be modified so as to render it a feeling, although feeling does not essentially involve consciousness proper. But imagine a state of consciousness reduced to perfect simplicity, so that its object is entirely unanalyzed, then that consciousness reduced to that rudimentary condition, unattainable by us, would be a pure feeling, and not properly consciousness at all. Let the quality of blue, for example, override all other ideas, of form, of contrast, of commencement or cessation, and there would be pure feeling. When I say that such impossible states exist as elements of all consciousness, I mean that there are ideas which might conceivably thus exist alone and monopolize the whole mind. Besides feelings, we have in our minds sensations of reaction, as when a person blindfold suddenly runs against a post, when we make a muscular effort, or when any feeling gives way to another feeling. Suppose I had nothing in my mind but a feeling of blue, which were suddenly to give place to a feeling of red; then, at the instant of transition there would be a sense of reaction, my blue life being transmuted to red life. If I were now also endowed with a memory, that sense would continue for some time. This state of mind would be more than pure feeling, since in addition to the feeling of red a feeling analogous to blue would be present, and not only that but a sense of reaction between the two. This sense of reaction would itself carry along with it a peculiar feeling which might conceivably monopolize the mind to the exclusion of the feelings of blue and red. But were this to happen, though the feeling associated with a sense of reaction would be there, the sense of reaction as such would be quite gone; for a sense of reaction cannot conceivably exist independent of at least two feelings between which the reaction takes place. A feeling, then, is a state of mind having its own living quality, independent of any other. A sense of reaction, or say for short a sensation, is a state of mind containing two states of mind between which we are aware of a connection, even if that connection is no more than a contrast. No analysis can reduce such sensations to feelings. Looking at the matter from a physiological point of view, a feeling only calls for an excited nerve-cell,—or indeed a mere mass of excited nerve-matter without any cell, or shut up in any number of cells. But sensation supposes the discharge or excitation of a nerve-cell, or a transfer of excitement from one part of a mass of nerve-matter to another, or the spontaneous production or cessation of an excited condition. 22. Architecture of Theories. Initial Version, 1890 97 Besides feelings and sensations, we have general conceptions; that is, we are conscious that a connection between feelings is determined by a general rule; or, looking at the matter from another point of view, a general conception is the being aware of being governed by a habit. Intellectual power is simply facility in taking habits and in following them in cases essentially analogous to, but in non-essentials widely remote from, the normal cases of sensation, or connection of feelings, under which those habits were formed. The one primordial law of mental action is a tendency to generalization; that is, every connection between feelings tends to spread to neighboring feelings. If you ask what are neighboring feelings, it is like the question that was answered by the parable of the Good Samaritan. A neighboring feeling is simply a connected feeling. These connections are of two kinds, internal or manifest, and external or occult. A feeling is manifestly connected with feelings which it resembles or contrasts with; the connection is merely an identity of feeling. A feeling is occultly connected with feelings bound to it by some external power, as the roll of thunder with the flash of lightning. The mental law belongs to a widely different category of law from physical laws. A physical law determines that a certain component motion must take place, otherwise the law is violated. But such absolute conformity is not required by the mental law. It does not call for any definite amount of assimilation in any case. Indeed such a precise regulation would be in downright conflict with the law. For it would instantly crystallize thought and prevent all further formation of habit. The law of mind makes something the more likely to happen. It thus resembles the “non-conservative” forces of physics, such as viscosity and the like, which are due to chance encounters of molecules.

Writings of Charles S. Peirce: A Chronological Edition, Volume 5: 1884-1886

For a number of reasons the lunar theory of Jeremiah Horrocks is an important theory. Sketched out in a pre-final form in a letter of 20 December 1638 from Horrocks to William Crabtree,' and reported in more nearly final form in a letter of 21 June or July 1642 from Crabtree to William Gascoigne.i it reached the public only in the Opera posthuma of Horrocks, distributed privately in 1672 to Newton, Flamsteed, Wallis and others, and at length published in 1673. Horrocks himself regarded his lunar theory as his greatest discovery.' At the time of its publication it was the most promising lunar theory available: the most accurate in its predictions oflongitude and radius vector, and the one that Newton would put his hopes on as best lending itself to dynamic explanation. In describing the Moon's chief vagaries as due to a semi-annual wobble ofthe major axis of its elliptical orbit, coupled with a concomitant variation in eccentricity. Horrocks so formulated the basic facts as to reveal to Isaac Newton years later what needed to be accounted for; and it was these Horrocksian facts that Newton indeed attempted to explain, at least qualitatively, in Corollaries 7-9 of Proposition 66 of Book I of his Principia (all editions). Beginning in 1694, Newton undertook to develop a more accurate predictive theory of the Moon, and it was once more the Horrocksian theory that he took as point of departure, seeking to refine it by adding small sinusoidal terms. The approach failed,' and as a predictive model, Horrocks's own theory remained as good as any available down to the publication of Tobias Mayer's first lunar tables in 1753. Euler, who was acquainted with the Horrocksian theory, appears to have found it suggestive in leading to his invention of the method ofvariation of orbital parameters, so fruitful in the celestial mechanics of Lagrange and Laplace. 5 How did Horrocks arrive at this theory? Out of what considerations did it emerge? According to Robert Small, in his Account of the astronomical discoveries of Kepler (1804), Horrocks discovered the unequal motion of the lunar apsides

On the Origin of Horrocks's Lunar Theory

Johannes Kepler published his Astronomia nova in 1609, based upon a huge amount of computations. The aim of this paper is to show that Kepler's new astronomy was grounded on methods from numerical analysis. In his research he applied and improved methods that required iterative calculations, and he developed precompiled mathematical tables to solve the problems, including a transcendental equation. Kepler was aware of the shortcomings of his novel methods, and called for a new Apollonius to offer a formal mathematical deduction. He was also in great need of computational power, and his friend and colleague, Wilhelm Schickard, constructed the first prototype of a true mechanical calculator, although it never came into regular use. The article concludes that Kepler's new astronomy was clearly backed up by numerical methods and embedded concepts and challenges of great importance for the future development of numerical analysis.

Early Numerical Analysis in Kepler's New Astronomy

The Jesuit astronomer Giovanni Battista Riccioli (1598-1671) proposed an original astronomical system as an intermediate solution between the Copernican and the Ptolemaic systems. Many scholars describe his system as “geocentric”, and so does Riccioli. He uses the Tychonic solution by adding a lot of differences, for instance, the order of the orbits and the movement of Jupiter and Mars. In his astronomical work, Riccioli is rigorous and serious and offers a great quantity of observational data. The analysis of these results is crucial to understand the evolution of the Copernican question during the first half of the XVII century. The article will look at some general lines of Riccioli’s system, to understand how an astronomer of the XVII century really worked and to understand which problems the Copernican system could not resolve. Finally, I will try to consider some philosophical implications of this historical case.

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Robert Small

Papers

"An Account of the Astronomical Discoveries of Kepler", Robert Small, Madison 1963 : [recenzja] / J. D.