# Mechanics and Natural Philosophy in History

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## Definition

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## 1. Introduction

Natural philosophy encompassed all natural phenomena of the physical world. It sought to discover the physical causes of all natural effects and was little concerned with mathematics. By contrast, the exact mathematical sciences—such as astronomy, optics, and mechanics—were narrowly confined to various computations that did not involve physical causes. Natural philosophy and the exact sciences functioned independently of each other. Although this began slowly to change in the late Middle Ages, a much more thoroughgoing union of natural philosophy and mathematics occurred in the seventeenth century and thereby made the Scientific Revolution possible. The title of Isaac Newton’s great work, The Mathematical Principles of Natural Philosophy, perfectly reflects the new relationship. Natural philosophy became the “Great Mother of the Sciences”, which by the nineteenth century had nourished the manifold chemical, physical, and biological sciences to maturity, thus enabling them to leave the “Great Mother” and emerge as the multiplicity of independent sciences we know today [2].(Backcover)

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- Traditional natural philosophers: people trained in the investigation of nature by using the concepts of matter, causation, ethics. Examples are Aristotle, Plato, Descartes, and Leibniz, but not Newton and many of his successors.
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- Mathematicians: people trained in theoretical mathematics and in practical activities.

## 2. Mechanics and Natural Philosophy in Ancient Greece

- Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium, but incline towards the weight that is at the greater distance.
- If, when weights at certain distances are in equilibrium, something is added to one of the weights, they are not in equilibrium, but incline towards the weight to which the addition was made [4] (p. 189).

Weights which balance at equal distances are equal. For if they are unequal, take away from the greater the difference between the two. The remainder will not then balance [supposition 3], which is absurd [supposition 1]. Therefore the weights cannot be unequal [4].(p. 190)

Further, the truth of what we assert is plain from the following considerations. We see the same weight or body moving faster than another for two reasons, either because there is a difference in what it moves through, as between water, air, and earth, or because, other things being equal, the moving body differs from the other owing to excess of weight or of lightness. Now the medium causes a difference because it impedes the moving thing, most of all if it is moving in the opposite direction, but in a secondary degree even if it is at rest; and especially a medium that is not easily divided, that is a medium that is somewhat dense. A, then, will move through B in time G, and through D, which is thinner, in time E (if the length of B is equal to D), in proportion to the density of the hindering body. For let B be water and D air; then by so much as air is thinner and more incorporeal than water, A will move through D faster than through B. Let the speed have the same ratio to the speed, then, that air has to water. Then if air is twice as thin, the body will traverse B in twice the time that it does D, and the time G will be twice the time E [5].IV, 8, 215b

## 3. New Sciences and Philosophies

Thus there are found some keen mathematicians of our time who assert that mechanics may be considered either mathematically, [removed from physical considerations], or else physically. As if, at any time, mechanics could be considered apart from either geometrical or actual motion! Surely when this distinction is made, it seems to me (to deal gently with them) that all they accomplish by putting themselves forth alternately as physicists and as mathematicians is simply that they fall between stools, as the saying goes. For mechanics can no longer be called mechanics when it is abstracted and separated from machines [8].(Preface. Translation into English in [9])

Albeit that since, for the time being, we here verge away from Geometry to a physical consideration, our discussion will accordingly be somewhat freer, and not everywhere assisted by diagrams and letters or bound by the chains of proofs, but, looser in its conjectures, will pursue a certain freedom in philosophizing. Despite this, I shall exert myself, if it can be done, to see that even this part be divided into propositions [10].(p. 5. Translation in [11])

## 4. Natural Philosophy of Newton and Afterwards

The Proof you sent me I like very well. I designed ye whole to consist of three books, the second was finished last summer being short & only wants transcribing & drawing the cuts fairly. Some new Propositions I have since thought on wch I can as well let alone. The third wants ye Theory of Comets. In Autumn last I spent two months in calculations to no purpose for want of a good method, wch made me afterwards return to ye first Book & enlarge it wth divers Propositions some relating to Comets others to other things found out last Winter. The third I now designe to suppress [emphasis added by us]. Philosophy is such an impertinently litigious Lady that a man had as good be engaged in Law suits as have to do with her. I found it so formerly & now I no sooner come near her again but she gives me warning. The two first books without the third will not so well beare ye title of Philosophiæ naturalis Principia Mathematica [Mathematical principles of natural philosophy] & therefore I had altered it to this De motu corporum libri duo [Two books on the motion of bodies]: but upon second thoughts I retain ye former title. Twill help ye sale of ye book wc I ought not to diminish now tis yours. The Articles are wth ye largest to be called by that name. If you please you may change ye word to sections, thô it be not material. In ye first page I have struck out ye words uti posthac docebitur [will be taught to use from now on] as referring to ye third book [13].(vol. 2, p. 437. Newton to Halley 26 June 1686)

Sr I must now again beg you, not to let your resentments run so high, as to deprive us of your third book, wherin the application of your Mathematicall doctrine to the Theory of Comets, and severall curious Experiments, which, as I guess by what you write, ought to compose it, will undoubtedly render it acceptable to those that will call themselves philosophers [13].(vol. 2, p. 443. Halley to Newton 29 June 1686)

Being engaged in this business, not only have I fallen upon many questions not to be found in previous tracts, to which I have been happy to provide solutions, but also I have increased our knowledge of the science by providing it with many unusual methods, by which it must be admitted that both mechanics and analysis are evidently augmented more than just a little.(Preface. Translation into English by I. Bruce)

I propose to condense the theory of this science and the method of solving the related problems to general formulas whose simple application produces all the necessary equations for the solution of each problem. I hope that my presentation achieves this purpose and leaves nothing lacking. In addition, this work will have another use. The various principles presently available will be assembled and presented from a single point of view in order to facilitate the solution of the problems of mechanics. [...] No figures will be found in this work. The methods I present require neither constructions nor geometrical or mechanical arguments, but solely algebraic operations subject to a regular and uniform procedure. Those who appreciate mathematical analysis will see with pleasure mechanics becoming a new branch of it [emphasis added by us] and hence, will recognize that I have enlarged its domain [15].(Preface. English translation in [16])

## 5. Natural Philosophy and the New Sciences Relying on Mechanics in the 19th Century

[Clapeyron] agrees with Mr. Carnot in referring power to vis viva developed by caloric contained in the vapour, in its passage from the temperature of the boiler to that of the condenser. I conceive that this theory, however ingenious, is opposed to the recognised principles of philosophy, because it leads to the conclusion that vis viva may be destroyed by an improper disposition of the apparatus: thus Mr. Clapeyron draws the inference, that “the temperature of the fire being from 1000${}^{\circ}$ (C.) to 2000${}^{\circ}$ (C.) higher than that of the boiler, there is an enormous loss of vis viva in the passage of the heat from the furnace into the boiler”. Believing that the power to destroy belongs to the Creator alone, I entirely coincide with Roget and Faraday in the opinion, that any theory which, when carried out, demands the annihilation of [living] force, is necessarily erroneous [19].(pp. 382–383)

## 6. Coming Back to the Past

To picture the relations among bodies, it may help to consider $\mathsf{\Omega}$ as being the collection of all open sets in the Euclidean plane and to take ∨ as begin the sign of inclusion, ⊂, so that the suggestive sketches often called “Venn diagram” are easy to draw [20].(p. 8)

The bodies of interest in mechanics have mass, as we say they are massy [20].(p. 16)

A system of forces on a universe $\mathsf{\Omega}$ is an assignment of vectors in some inner-product space $\mathcal{F}$ to all pairs of separate bodies of $\mathsf{\Omega}$ [20]:(pp. 19–20)

Axiom F1.$\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\mathbf{f}:{(\mathsf{\Omega}\times \mathsf{\Omega})}_{0}\to \mathcal{F}$

**Definition**

**1.**

**Theorem**

**1.**

Axiom I1. There is a frame such that if $\mathbf{m}(\mathcal{B},\mathsf{\xd8})$ is constant over an open interval of time, then in that interval $\mathbf{f}(\mathcal{B},{\mathsf{\Sigma}}^{e})=\mathbf{0}$, and conversely [20].(p. 65)

Axiom I2. Newton, Euler, and others. In an inertial frame

## 7. Final Remarks

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## Conflicts of Interest

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**MDPI and ACS Style**

Capecchi, D.; Ruta, G.
Mechanics and Natural Philosophy in History. *Encyclopedia* **2022**, *2*, 1333-1343.
https://doi.org/10.3390/encyclopedia2030089

**AMA Style**

Capecchi D, Ruta G.
Mechanics and Natural Philosophy in History. *Encyclopedia*. 2022; 2(3):1333-1343.
https://doi.org/10.3390/encyclopedia2030089

**Chicago/Turabian Style**

Capecchi, Danilo, and Giuseppe Ruta.
2022. "Mechanics and Natural Philosophy in History" *Encyclopedia* 2, no. 3: 1333-1343.
https://doi.org/10.3390/encyclopedia2030089