*Events* in history can either extend over an interval of time (*extended events*), or be instantaneous (*point events*). A good example of a point event is the achievement of Indian independence at midnight in New Delhi, 14/15 August 1947.

Another type of point event is the midnight common to two consecutive days. At least, this is the common usage, and our analysis ought to accept it. In spite of having an instant in common, it is always considered that the first day is earlier than the second. This, too, should be taken into account.

Whatever definition is adopted for one event to be 'earlier than' another, if two events have a stretch of time in common, then neither of them can be said to be earlier than the other. In this case we shall not obtain a linear sequence of events ordered by 'earlier than', however much we know about their dating. By breaking the events down into shorter events it may be possible to obtain a linear sequence, but we do not always have sufficient information to do this. Hence a set of events will often be only partially ordered by the relation 'earlier than'.

Underlying our concept of historical events is the linear sequence of time. Time is measured by clocks, whose different settings in different time zones are well understood. Behind these different settings there is only the one time line, corresponding to times represented by real numbers (see also a technical note on the real time line).

One final preliminary point is that extended historical events seem to require both end points to be included. A single day is a good example of this. So although it would be possible to consider events with one or both end points omitted, this will not be done here.

The logical structure we require, then, for a practical system of relative dating, is an ordering by the an appropriate relation 'earlier than', involving both point events and extended events, where extended events include both end points. This structure is provided by a mathematical structure giving a suitable partial ordering of any set of such events.

An appropriate definition of an event *a* being earlier than an event*b* is that *either* *a* ends before *b* begins,*or* *a, b* are both extended events and *a* ends at the same time that *b* begins.

The two basic results of this definition are, (1) If event *a* is earlier than event *b*, and *b* is earlier than event *c*, then *a* is earlier than *c*. (2) No event is earlier than itself. (So things are as they should be.)

(Properties (1), (2) are precisely those needed to give a partial order in the technical mathematical sense.)

Now imagine that we have two events *a, b* such that *a* is earlier than *b*, and *b* is earlier than *a*. This is certainly not possible in reality, but is it possible in our system? The answer is No, because (1) now entails that *a* is earlier than itself, which is forbidden by (2).

In exactly the same way, we find that the system does not allow us to have a closed sequence of events *a, b, c, ... , a*, such that each event in the sequence is earlier than the next one. Such a sequence is called a 'circuit'.

Suppose three events *a, b, c* are such that *a* is earlier than*b* and *c*, and *b* is earlier than *c*. This is what we mean by saying that the three events are 'linearly ordered' (or, that they form a 'chain'). The events can then conveniently be arranged in a diagram:

Fig.1 o--->---o--->---o.abc

It is not necessary to include a line for the ordering '*a* is earlier than *c*', and such lines are always omitted.

If we do not know anything about the relative ordering of the events*b, c*, then the diagram becomes something like:

ao /\ Fig.2 / \bo oc

where the arrows have been omitted (although we need to know that earlier is at the top in this particular diagram).

If we start with a diagram instead of with the orderings of events, then provided the diagram has no circuits, it will represent correctly the ordering of the events in it.

With events as in Fig.2, if we later discover that *b* is earlier than *c*, then we have the ordering given by the diagram in Fig.1. It might turn out, however, that *c* is earlier than *b*, which gives a different chain of events.

It is always the case that a partially ordered set can become a chain by adding orderings. In our case, this is meaningful only if the events do not overlap.

The Gospels of Matthew (chap.2) and Luke (2:1-39) describe different events connected with the birth of Jesus, some of which we shall consider. It is possible to get very close to a linear ordering. Let events be labelled as follows:

=========================================================== b: birth of Jesus Matthew Luke ----------------------------------------------------------- m: visit of the Magi s: visit of the shepherds f: flight to Egypt p: presentation in Jerusalem r: return to Israel n': return to Nazareth n: settlement in Nazareth ============================================================

(where n' might be the same as n). From this data the events are immediately ordered as:

m f r n o------o------o------o / b o ---> time \ o------o------o s p n'

If we decide that s is earlier than m, p earlier than f, and that n' is the same as n (each of them possible decisions), we obtain the diagram:

m o /\ / \ / \ o------o---o--o------o------o b s p f r n

There are then two possibilities for the relative dating of the Magi's visit (between s,p or between p,f). (This example is modified from my book,*"We Have Seen His Star"*, TRACI, New Delhi, 1983.

Three different opinions will be noted on the relative dating of the periods of composition, or periods of activity, of the following works and authors:

================================================= g : Govindasvâmin (9th cent AD) j : astrological Jaimini Sûtras jv: vedic Jaimini pv: vedic Parâsara p : Brihat Parâsara Horâ Sâstra [BHPS] (p1: BHPS, Part I) (p2: BHPS, Part II) r : Rig Veda s : Sârâvali (7th or 8th cent AD) v : Varâhamihira (6th cent AD) =================================================

(The letters will also be used to refer to the works and authors themselves.)

The traditional view of Indian astrology gives the following order:

r o \ v o----o----o ---> time / s g o-----o pv,p jv,j (Strictly speaking, pv o - and similarly for jv,j, the events pv, p _/ \_ but the diagram has should appear as - \ / been simplified.) p o

Professor David Pingree (*The Yavana Jâtaka of Sphujidhvaja*, Harvard Oriental Series 48, 1978, Vol.II, p. 448) argues for a different ordering. He points out that p1 uses v and contains a verse of s in a different order, while p2 is commented on by g and mentions that p1 exists. His relative dating scheme for these particular events is, in simple terms:

s o / v o---o---o---o p1 p2 g

My own view (*Hindu Astrology: Myths, Symbols and Realities*, Select Books, New Delhi, 1981, p.80 [with a diagram]) is different from both:

s r --o-- o / \ \ / \ v o---o--o---o / \ p2 g /p1 o-----o \ / pv jv --o-- j

This is mainly because j contains late elements, and the extant p1 is a compilation of much of the horoscopic astrology existing before recent times.

(As an alternative to using diagrams, the relative dating of a set of events may be expressed using an algebraic method.)

Last updated: 07 January 2005