# De mundi systemate


by @ulaulaman about #IsaacNewton #physics #gravity

I published this post some years ago ([archived version](https://web.archive.org/web/20101129141519/http://sciencebackstage.blogosfere.it/2010/11/de-mundi-systemate.html)), but for unilateral decision of the online publisher, it is deleted, so I decide to recover it.

![De mundi systemate](https://cdn.hashnode.com/res/hashnode/image/upload/v1743071361312/499b059e-e1d1-4c11-aa4f-c4172165e89f.jpeg)

It's the third pard o [Newton](http://scienceworld.wolfram.com/biography/Newton.html "Isaac Newton")'s [_Philosophiae Naturalis Principia Mathematica_](http://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica "Philosophiae Naturalis Principia Mathematica Wikipedia"), one of the most famous physics _tractatus_ (latin world to _treatise_). In 2010 **Dave Richeson** proposes [An amazing paragraph from Euler's Introductio](http://divisbyzero.com/2010/11/09/an-amazing-paragraph-from-eulers-introductio/), a wonderful post in which he extraxts some Euler's quotes from _Introductio in analysin infinitorum_ (_Introduction to analysis of the infinite_). So, because I'm working on a learning project about gravity, I decide to propose an analouge post about _De mundi systemate_ (_On worlds' system_). For english version of the following Newton's quotes I'll ask help to [Andrew Motte](http://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846) "Philosophiae Naturalis Principia Mathematica Wikisource").  
We began with _Propositio II_ (Proposition II):

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1743071362878/ce0594f2-6831-47ff-a73d-4840ac2bb951.jpeg)

> Forces, by which the primary planets are continually drawn off from rectilinea motion, and retained in their orbits, come from the Sun, and they are reciprocally as the square of the distances from the centre.

In mathematical world, it means: \\\[F = \\frac{K}{r^2}\\\] Similarly the interaction between Moon and Earth:

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1743071364665/4685df6d-9f22-4a31-95ed-cdae18f7d9d5.jpeg)

> Force, by which the Moon is retained in its orbit, tends to the Earth, and are reciprocally as the square of the distances by centre.

The following quote (_Proposition VII_) is very interesting:

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1743071366365/a58c5747-851f-40b0-b9d1-eaa54cb0d788.jpeg)

> The weight is in all bodies, and it's proportional to matter which they contain

In this quote we can see the first use of _gravity_, but I think that Newton means the translation that I proposed.

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1743071367872/59dbb732-21ed-414e-8f74-9858abdb361e.jpeg)

That it means that _gravity under planete's surface decreases proportionally to the distance from the centre_: this is the classic problem of a tube through the earth passing to the centre. In order to obtain Newton's result we can think like the great mathematician: we consider the relation between mass, volume and density. Using modern mathematics we can write: \\\[M = \\rho V = k r^3\\\] Using it in the following expression: \\\[a= \\frac{GM}{r^2}\\\] we obtain \\\[a=kr\\\] _Proposition XIII_ consists in first and second [Kepler's law](http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion "Kepler's laws of planetary motion Wikipedia"):

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1743071369439/780ef5ca-dac5-41da-9f03-e50a3945fd72.jpeg)

P.S.: you can read [_De mundi systemate_ on Google Books](http://goo.gl/GjTEs).