The scientific method focuses on the regular and reproducible and is without any doubt very successful in this; however, this means that it leaves out a large part of our experience that is considered accidental, a random noise in no need of further attention. Nevertheless, we encounter meaningful coincidences that play as large a role in our lives as the dependably regular. These, then, from a scientific point of view, are just psychological facts that can be explained away as apophenia; but bearing in mind that our perceptions are not raw sense impressions, but already cast in the mould of archetypal and previously formed concepts that consequently shape our scientific laws and can also be seen as psychological facts, the dividing line appears less clearly drawn, suggesting that a study of meaningful coincidences, although by definition outside the reach of regular scientific law, may still be interesting and worthwhile. (This brings to mind the remarkable collection of observed morally meaningful connections we would tend to interpret as purely coincidental in Linné's astounding book Nemesis Divina.)
As a contribution to this study, C.G. Jung in the first part of the present book (Synchronicity: an Acausal Connecting Principle) expounds on his concept of synchronicity. In his first chapter he gives an introduction to the idea and adduces supporting evidence from experiments in extrasensory perception that he obviously found considerably more compelling than we (or certainly I) would today. In the second chapter, he describes an astrological experiment in which the birth horoscopes of married couples and, as a control group, unmarried people are compared. The first analysis indicated a higher incidence in the married couples of a conjunction of one partner's moon with the other's sun, moon or ascendent, corroborating the status of such coincidence as indicative of marriage in traditional astrology, but this fairly weak signal completely collapsed on cross validation. Well, what was he thinking? It is clear (and, judging from what he wrote in other parts of the book, it was also clear to him) that statistics is a way of establishing the dependably regular and, as such, the mortal enemy of the singular meaningful coincidence, which it just subsumes and drowns out in the residual noise.
In the third chapter, in my view the most interesting of this part of the book, Jung gives an overview of precursors of the idea of synchronicity such as the ancient Chinese thought of the I Ching and taoism, Agrippa, Paracelsus, Leibnitz etc.
[My personal impression is that Jung here accepts the concept of causality all to eagerly and easily as an established underlying principle of the scientific world view, to the point of setting up a dichotomy between causality and synchronicity, which he conceives as acausal by definition. However, science, dealing in quantitative relationships rather than metaphysical substances, uses the concept of causality in a quite off-hand manner; e.g. in physics causality is taken to mean little more than a preservation of temporal order, and even this is questionable: as already observed by the ancient Greek sceptics (cf. Sextus Empiricus, Outlines of Pyrrhonism, Book 3, Chapter IV seq.), the view that there is first a cause, then the causation, then the effect as a temporal sequence is naive and untenable, and cause and effect need to be conceived as simultaneous, which blurs the distinction of which is which. Contemporary research in quantum entanglement also indicates that things are not quite as simple as one might have thought. In this light it would make more sense to see causality not as an exclusive opposite, but rather a special case of synchronicity distinguished by its regularity and reproducibility.]
The book is saved, in my opinion, by its second part (The Influence of Archetypal Ideas on the Scientific Theories of Kepler), in which W. Pauli describes Kepler's pivotal role as a scientist favouring a mathematical, quantitative description of the world over the older holistic, qualitative way of alchemy and hermeticism while still retaining a very symbolic approach, e.g. when he describes the soul as the circle that arises as the intersection of a sphere (his model for the divine trinity) with the plane of the material. This part of the book provides ample source material from Kepler's writings and in particular from the controversy between Kepler and Fludd as representatives of the new and the old ways of thought. It is well worth reading independently of what one may think of Jung's ideas.
My **** rating arises as *** for Jung and ***** for Pauli.