As we embark on what we propose, therefore, we wish first to make plain what constitutes the tranquillity and the intranquillity of a realm or city; and of these, first tranquillity: for if this is not clear we cannot know what constitutes intranquillity. And since both of these seem to be dispositions of a city or realm (as we suppose from Cassiodorus) we shall without further delay make plain what needs to be clarified, i.e. what a realm or city is and what it is for, so that the description of tranquillity and its opposite will also become clearer.

2

So, since we wish (following the order we have set down) to describe the tranquillity of a city or realm, we should be aware – so as to avoid any ambiguity that may arise from the multiplicity of terms – that this term ‘realm’ in one of its significations implies a plurality of cities or provinces contained under one regime. In this sense a realm does not differ from a city in terms of the form of polity, but rather in terms of size. On another understanding of the word, this term ‘realm’ signifies a particular type of polity or temperate regime, which Aristotle calls ‘temperate monarchy’. In this sense there can be a realm in a single city just as there can be in several – as was the case around the beginnings of civil communities, when in most cases there was one single king in each single city. The third signification of the term, and the most familiar, is a mixture of the first and the second. The fourth sense is something common to every type of temperate regime, whether in a single city or in several cities; Cassiodorus took it in this sense in the words we placed at the beginning of this book, and it is in this same sense that we too shall use the term in determining the answers to our questions.

Now with regard to what we said in chapter 15, and likewise in other chapters subsequently, someone might quite well raise doubts and show, first of all, that the dignity of a bishop is greater and different in type from that which belongs solely or simply speaking to a priest; and that a bishop's dignity does not exist purely as a result of the human institution that we called ‘separable’, but rather by divine ordination as well, which we earlier called ‘essential’. It seems that this can be convincingly established from Luke 10, where we find this passage: ‘After these things the Lord appointed other seventy also, and sent them two and two before his face’. On this Bede writes: ‘Just as no one doubts that the twelve apostles prefigure the bishops, so likewise these seventy prefigured the priests of the second order of priests.’ The same thing can be shown from I Timothy 5, when the Apostle said: ‘Against a priest receive not an accusation, but before two or three witnesses.’ Therefore Timothy was superior in dignity to the other priests, but not by an election on the part of priests or the multitude of the faithful; therefore by divine ordination. Again, the same thing is apparent from the Letter of Pope Clement, headed ‘to James the brother of the Lord’. This seems also to have been the opinion of almost all bishops who are said to have succeeded Peter or Paul in the episcopal see of Rome, as is clear from the abovementioned Codex of Isidore.

2

Secondly, it seems that it can be shown that Saint Peter was superior to the rest of the apostles by a power or authority that was given him directly by Christ and not by another man or men; and in consequence that his successors are superior to the successors of the others.

Of the conclusions to be drawn, we shall set down as the first:

1

That in order to gain eternal beatitude, it is necessary to believe only in the truth of divine or canonic Scripture, what follows from it with any kind of necessity, and the interpretation of it that has been made by a common council of the faithful, if this is put to an individual in due fashion. The certainty of this was given in and can be taken from chapter 19 of the second discourse, sections 2–5.

2

That only a general council of the faithful or its prevailing multitude or part should determine the senses of divine law where there is doubt over the definition, especially those matters which are called the articles of the Christian faith, and anything else that must be believed of necessity of salvation; and that no other partial collective body or individual person, of whatever condition they may be, has the authority for the determination just mentioned. The certainty of this is given in chapter 20 of the second discourse, sections 4–13.

3

That no one is commanded in evangelical scripture to be compelled to observe the commands of divine law by temporal penalty or punishment: chapter 9 of the second discourse, sections 3–10.

4

That for eternal salvation it is necessary to keep only the commands of the evangelical law or those that follow from them of necessity, and those things which it is appropriate to do or to omit according to right reason; and not all the commands of the old law: chapter 9 of the second discourse, section 10 to the end.

5

That no mortal can give a dispensation from things that are commanded or prohibited by God or in the evangelical law; and that only a general council or the faithful human legislator can prohibit things that are permitted, binding people to fault or penalty for the status of this present world or of that to come, and no other partial college or individual person whatever their condition: chapter 12 of the first discourse; chapter 9 of the second, section 1, and 21 of the second, section 8.

Since, then, we have said that tranquillity is the good condition of a city in respect of the action of its parts, we must in consequence consider what the city is in itself, and what it is for; what and how many are its principal parts; the action appropriate to each of them, their causes, and their ordering in respect of each other. For these considerations are fundamental to the complete delineation of tranquillity and its opposite.

2

However, before we discuss the city – which is the perfect community – and its species or modes, we ought first to introduce the origin of civil communities and their regimes and ways of living. From these, as from the imperfect, men have progressed to perfect communities and regimes and ways of living in them. For nature's path, and that of art which imitates her, is always from the less to the more perfect. And men are not judged to know any particular thing unless they know it together with its primary causes and its first principles right down to its elements.

3

So, proceeding in this fashion, we should be aware that civil communities (according to their different times and places) have begun small, and by gradually receiving an increase have in the end been brought to completion – as we have just said happens in every action of nature or art. For the first and minimum human combination, from which all others have arisen, is that of male and female, as the best of philosophers says in Politics I, chapter 1, and is further apparent from his Economics. This combination produced more human beings, who first of all filled one household; and then as further combinations of this type occurred, the multiplication of human beings was so great that one household was insufficient for them and it was necessary to set up several households. A plurality of these is called a village or neighbourhood, and this (as Aristotle also writes, as above) was the first community.

Someone will raise doubts, however, about what we have said, objecting that the authority to pass or institute laws does not belong to the universal body of the citizens. Firstly because something that is mostly wicked and undiscerning ought not to establish the law; for these two faults, sc. malice and ignorance, must be excluded from the legislator. Indeed it was in order to avoid them in judgements, as well, that we understood the necessity of laws in chapter 11 of this discourse. But the people or the universal body of the citizens is of this nature; for men are visibly wicked and stupid for the most part, since ‘the number of the stupid is infinite’ as it says in Ecclesiastes 1. Again, because it is very hard or impossible to get the opinions of many wicked and foolish individuals to agree, whereas this is not the case with a few who are virtuous. It is therefore more expedient for law to be passed by a few men rather than by the universal body of the citizens or an unnecessary number of them. Again, in any civil community the wise and the learned are few in respect of the rest of the untaught multitude. Since, therefore, it is more expedient for law to be passed by the wise and learned than by the ignorant and the uneducated, it seems that the authority to pass them belongs to the few, and not to many or to all. Further still, it is in vain for something to be done by many if it can be done by fewer. Since, therefore, it is possible for law to be passed by the wise (who are few) – as said before – it would be in vain for the entire multitude or its greater part to be occupied in this business. The authority to legislate does not, therefore, belong to the universal body of the citizens or its prevailing part.

Until now, we have been concerned with the kinematics of particles where the objective has been to determine the motion of a particle or rigid body without regard to the cause of the motion. Clearly in any physical system motion cannot occur without the application of some kind of external stimulus. In particular, in order for a particle to accelerate, it is necessary to apply a force to the particle. In order to study the motion that results from the application of a force (or, in general, the application of multiple forces) to a particle, it is necessary to study the kinetics of the particle. The objective of kinetics is threefold: (1) to describe quantitatively the forces that act on a particle; (2) to determine the motion that results from the application of these forces using postulated laws of physics; and (3) to analyze the motion.

The first topic in this chapter is the development of models for forces that are commonly used in dynamics. In particular, models are developed for contact forces, spring forces, and gravitational forces. These models will be used throughout the remainder of this book when solving problems.

The next topic in this chapter covers Newton's laws, which are the fundamental postulates that govern the nonrelativistic motion of particles.

The first topic in the study of dynamics is kinematics. Kinematics is the study of the geometry of motion without regard to the forces the cause that motion. For any system (which may consist of a particle, a rigid body, or a system of particles and/or rigid bodies) the objectives of kinematics are fourfold: to determine (1) a set of reference frames in which to observe the motion of a system; (2) a set of coordinate systems fixed in the chosen reference frames; (3) the angular velocity and angular acceleration of each reference frame (and/or rigid body) resolved in the chosen coordinate systems; and (4) the position, velocity, and acceleration of each particle in the system. In order to develop a comprehensive and systematic approach, the study of kinematics given in this Chapter is divided into two parts: (1) the study of kinematics of particles and (2) the study of kinematics of rigid bodies.

This Chapter is organized as follows. First, both a qualitative and precise definition of a reference frame is given. In particular, it is discussed that a reference frame provides a perspective from which to observe the motion of a system.

In Chapter 3 we discussed the important principles and methods used in the formulation, solution, and analysis of the motion of a single particle. In this chapter we extend the results of particle kinetics to systems consisting of two or more particles.

The first topic covered in this chapter is the center of mass of a system of particles. Using the definition of the center of mass, the linear momentum of a system of particles is defined. Then, using the definition of linear momentum, the velocity and acceleration of the center of mass of the system are defined.

The second topic covered in this chapter is the angular momentum of a system of particles. In particular, expressions for the angular momentum are derived relative to an arbitrary point, an inertially fixed point, and the center of mass of the system. Then, relationships between these three different forms of angular momentum are derived.

The third and fourth topics covered in this chapter are Newton's 2nd law and the rate of change of angular momentum for a system of particles. In particular, it is shown that the center of mass of the system satisfies Newton's 2nd law. Furthermore, the key results relating the rate of change of angular momentum for a system of particles to moment applied to the system are derived.

Until now we have been concerned with the kinetics of particles, i.e., the kinetics of objects that have nonzero finite mass but do not occupy any physical space. Furthermore, in Section 2.15 of Chapter 2 we studied the kinematics of motion of a rigid body. In this chapter we turn our attention to the kinetics of rigid bodies. To this end, the objectives of this chapter are threefold: (1) to describe quantitatively the forces and moments that act on a rigid body; (2) to determine the motion that results from the application of these forces and moments using postulated laws of physics; and (3) to analyze the motion.

The key difference between a particle and rigid body is that a particle can undergo only translational motion whereas a rigid body can undergo both translational and rotational motion. In general, for motion in ℝ3 it is necessary to specify three variables for the translational motion and to specify another three variables for the rotational motion of the rigid body.

Mechanics is the study of the effect that physical forces have on objects. Dynamics is the particular branch of mechanics that deals with the study of the effect that forces have on the motion of objects. Dynamics is itself divided into two branches called Newtonian dynamics and relativistic dynamics. Newtonian dynamics is the study of the motion of objects that travel with speeds significantly less than the speed of light while relativistic dynamics is the study of the motion of objects that travel with speeds at or near the speed of light. This division in the subject of dynamics arises because the physics associated with the motion of objects that travel with speeds much less than the speed of light can be modeled much more simply than the physics associated with the motion of objects that travel with speeds at or near the speed of light. Moreover, nonrelativistic dynamics deals primarily with the motion of objects on a macroscopic scale while relativistic dynamics deals with the study of the motion of objects on a microscopic or submicroscopic scale.