One of the important actions that is in nearly all of the foregoing chapters is measurement. We measure time, coordinates, proper motions, parallaxes, magnitudes, the positions of lines in spectra, and shifts in positions of spectral lines, for example. After a measurement has been made we want to know how good the measurement really is, and in order to evaluate our measurements, we must turn to statistics. We would also like to use our measured data to make predictions either within or beyond the range of the measurements. Here we shall describe some of the principles that permit us to achieve these two goals in practical situations.

As an example, let us assume that a student is asked to determine the position of a spectrum line by measuring the distance from some reference position to the line center with a ruler. The smallest divisions on the ruler are one mm apart. We should be able to read the position of the line to within one tenth of a millimeter (0.1 mm). Just as a check the student makes a second setting and reads a new value with the ruler. She tries again and again until she has made fifty tries, and she never makes quite the same reading twice. Which of these many readings should be adopted as the correct one?

In the night sky the stars appear as bright points on a dark spherical surface (Figure 1.1). No such surface really exists, of course, but the concept of a celestial sphere is a useful one that goes back thousands of years. Ptolemy described it and so did Pythagoras and many others. Today we no longer have to worry about the reality of that sphere, and so we eliminate the need for speculation on its composition, radius, thickness and so forth. On the other hand, even though the celestial sphere is not a physical entity, we have many practical uses for the concept. The observer is always at the center of it, and the direction from the observer to any star may be considered to be a radius of the celestial sphere.

The stars are so far very away that we can consider the celestial sphere to be very large and the Earth very small. From the perspective of an observer on the celestial sphere looking back, the entire Earth would appear as a single point. And on the surface of the Earth, when we point to objects in the sky, we don't need to know how far away they are for the purposes of positional astronomy. We need only be concerned with the angles between points on the celestial sphere. That's why a good planetarium fools us into thinking that we are looking at the real sky.

Astronomers, both professional and amateur, have cataloged thousands of variable stars across the celestial sphere. Their periods range from a few minutes to hundreds of days. Some are visible to the naked eye, but most can only be detected with large telescopes. Some behave in an erratic fashion, while others are as predictable as the sunrise. Within this large group there is something for the interests and equipment of every observer, and serious contributions to the overall body of astronomical data can be made by anyone who is willing to exercise care in all phases of the collection of data.

Astronomers have found the study of variable stars to be both pleasant and rewarding. We shall begin this chapter with some information on nomenclature, reference materials and the various classes of variable star. We shall then proceed to discuss methods of analysis of variable star light curves and the determination of precise periods.

Naming variable stars

Variable stars are named in accordance with a scheme that was introduced in the middle of the nineteenth century when variability was first being recognized as a common phenomenon in stars. The originator of the current practice was the German astronomer, Friedrich Argelander, who was mentioned in Chapter 3 as the force behind the BD charts and catalog. In each constellation the first variable to be discovered was identified with the letter “R” followed by the possessive form of the Latin name.

It is safe to say that the spectrograph, a relatively simple instrument, brought about a virtual revolution in astronomy. Although Newton had examined the spectrum of sunlight and Fraunhofer had seen the spectra of a few stars, the spectroscope was not extensively used on telescopes until the latter half of the nineteenth century. Beginning in about 1860, Sir William Huggins in England and Fr. Angelo Secchi in Rome performed their first experiments on the light from the Moon, the planets and the brighter stars. The spectrograph was slowly refined and improved, and eventually it made possible a series of new understandings of the nature of the Sun and stars. First came the identification of a few absorption lines in solar and stellar spectra; then came recognition of several distinct classes of spectra. By the end of the century the construction of spectrographs had been refined to the point that radial velocities could be measured with confidence. Today we have a remarkable understanding of the physical processes that occur in nebulae and the atmospheres of the Sun and stars.

Since the nineteenth century several technological developments have increased the efficiency of the spectrograph, and there have been changes in the means by which the light is dispersed. The general principles of the spectrograph are not complicated, however, and we will outline them below. We shall discuss first the prism and then the grating as dispersive elements. Then we will describe the practical considerations in the design and use of a spectrograph.

The text

Nietzsche published each of the first three parts of Thus Spoke Zarathustra (TSZ hereafter) separately between 1883 and 1885, during one of his most productive and interesting periods, in between the appearance of The Gay Science (which he noted had itself marked a new beginning of his thought) and Beyond Good and Evil. As with the rest of his books, very few copies were sold. He later wrote a fourth part (called “Fourth and Final Part”) which was not published until 1892, and then privately, only for a few friends, by which time Nietzsche had slipped into the insanity that marked the last decade of his life. Not long afterwards an edition with all four parts published together appeared, and most editions and translations have followed suit, treating the four parts as somehow belonging in one book, although many scholars see a natural ending of sorts after Part iii and regard Part iv as more of an appendix than a central element in the drama narrated by the work. Nietzsche, who was trained as a classicist, may have been thinking of the traditional tragedy competitions in ancient Greece, where entrants submitted three tragedies and a fourth play, a comic and somewhat bawdy satyr play. At any event, he thought of this final section as in some sense the “Fourth Part” and any interpretation must come to terms with it.

The Honey Sacrifice

– And again moons and years passed over Zarathustra's soul and he took no notice of it; but his hair had turned white. One day as he sat on a stone before his cave and gazed outward – there where one looks out upon the sea and beyond twisting abysses – his animals walked around him pensively until finally they stood before him.

“Oh Zarathustra,” they said. “Are you perhaps on the lookout for your happiness?” – “What does happiness matter!” he answered. “I haven't strived for happiness for a long time, I strive for my work.” – “Oh Zarathustra,” said the animals again. “You say that as one who has had overly much of the good. Do you not lie in a sky-blue lake of happiness?” – “You foolish rascals,” answered Zarathustra, smiling.

The Child with the Mirror

At this time Zarathustra returned again to the mountains and to the solitude of his cave and withdrew from mankind, waiting like a sower who has cast his seeds. But his soul grew full of impatience and desire for those whom he loved, because he still had much to give them. For this is the hardest thing: to close the open hand out of love, and to preserve a sense of shame as a bestower.

Thus moons and years passed for the lonely one; but his wisdom grew and its fullness caused him pain.

But one morning he woke already before dawn, reflected for a long time on his bed and at last spoke to his heart:

What frightened me so in my dream that it waked me? Did not a child approach me carrying a mirror?

“Oh Zarathustra” – spoke the child to me – “look at yourself in the mirror!”

But when I looked into the mirror I cried out, and my heart was shaken; for I did not see myself there, but a devil's grimace and scornful laughter.

Zarathustra's Prologue

When Zarathustra was thirty years old he left his home and the lake of his home and went into the mountains. Here he enjoyed his spirit and his solitude and for ten years he did not tire of it. But at last his heart transformed, – one morning he arose with the dawn, stepped before the sun and spoke thus to it:

“You great star! What would your happiness be if you had not those for whom you shine?

For ten years you have come up here to my cave: you would have tired of your light and of this route without me, my eagle and my snake.

But we awaited you every morning, took your overflow from you and blessed you for it.

Behold! I am weary of my wisdom, like a bee that has gathered too much honey. I need hands that reach out.

I want to bestow and distribute until the wise among human beings have once again enjoyed their folly, and the poor once again their wealth.

For this I must descend into the depths, as you do evenings when you go behind the sea and bring light even to the underworld, you super–rich star!

Like you, I must go down as the human beings say, to whom I want to descend.

So bless me now, you quiet eye that can look upon even an all too great happiness without envy!

Bless the cup that wants to flow over, such that water flows golden from it and everywhere carries the reflection of your bliss!

The Wanderer

It was around midnight that Zarathustra started his route over the ridge of the island, in order to arrive at the other coast by early morning; for there he intended to board a ship. At that location there was safe harborage where even foreign ships liked to anchor; these would take the occasional passenger who wanted to cross the sea from the blessed isles. Now as Zarathustra climbed up the mountain he thought as he traveled about his many lonely wanderings since the time of his youth, and about how many mountains and ridges and peaks he had already climbed.

I am a wanderer and a mountain climber, he said to his heart. I do not like the plains and it seems I cannot sit still for long.

And whatever may come to me now as destiny and experience – it will involve wandering and mountain climbing: ultimately one experiences only oneself.

The time has passed in which accidents could still befall me, and what could fall to me now that is not already my own?

Mankind has always been fascinated by fossils, by their beauty and their mystery, their charm and their strangeness, their mute testimony to lives and worlds lost unimaginably long ago. In prehistoric times, our forebears not only collected fossils, but evidently treated them as valued artefacts, as indicated, for example, by the discovery of an ammonite at an Upper Palaeolithic burial site in Aveline's Hole in Burrington in the West Country (Rahtz, 1993), and numerous different types of fossil at Cro Magnon sites in the V´ez`ere valley in the P´erigord region of France, truly the birthplace of European civilisation (many of which are now displayed in the magnificent Museum of Prehistory in Les Eyzies). The habit persisted both in so-called primitive and so-called advanced societies through historical times (Mayor, 2000).

Palaeontology, that is, the scientific study of fossils, may be said to have originated at least as long ago as the sixteenth century (Thackray, in Briggs and Crowther, 1990), and, obviously, continues to be practised to the present day. The earliest written observations on fossils were made by the German Bauer, or Agricola, in his book De natura fossilum, and the earliest illustrations by the Swiss Gesner in his book De rerum fossilium lapidum et gemmarum, both of which date from the sixteenth century. The usage by these and other early observers of the term ‘fossil’, from the Latin fodere, meaning ‘to dig’, pertained to literally anything dug up from the ground or mined, including what we would now classify as minerals, crystals and gemstones. The earliest interpretations as to the nature of what we would now accept as fossils were made by the Danish anatomist Stensen, or Steno, working in the Medici court in Florence, in his publications dating from the latter part of the seventeenth century (Cutler, 2003). Steno applied Descartes’ ‘method of doubt’ and his own deductive logic to demonstrate that the so-called glossopetrae or ‘tongue stones'much valued in medieval Europe for their supposed medicinal properties were in fact not the tongues of snakes turned to stone by St Paul, as was the superstition, but the fossilised equivalents of the shark's teeth he was familiar with from his dissection work.

Palaeobiology deals with the documentation, analysis and interpretation of the relationships of fossils to evolving earth and life processes and environments, and their application to the elucidation thereof. This chapter deals with applications of fossils in the interpretation of earth and life processes (excluding evolution, covered in Chapter 5), and environments. Readers interested in further details of the principles and practice of palaeobiology are referred to the works of Briggs and Crowther (1990), Dodd and Stanton (1990), Bosence and Allison (1995), Brenchley and Harper (1998), Erwin and Wing (2000), Gastaldo and DiMichele (2000), Briggs and Crowther (2001) and Cohen (2003).

Life strategy

There are a number of theoretical biological models of population dynamics, one of the most elegant and influential of which involves only two variables, r and K, at either end of a continuum (Pianka, 1970). Of these, r refers to the theoretically initially limitless rate of increase of the population of a species, assuming no competition, for example, on occupying a new environmental niche; K to the ultimately limiting, carrying capacity of that niche. Whether a species or a higher-level taxon selects in favour of a strategy that tends to maximise K, through the ability to compete, or r, through the ability to reproduce at a high rate, will depend on a number of factors, including population density etc. K-strategy selection typically involves slow development to reproductive maturity, at large size, sexual reproduction, small numbers of progeny, typically in more than one issue or confinement (iteroparity), long generation time, and a long lifespan; r-strategy selection, rapid development to reproductive maturity, at small size, asexual reproduction in some groups, large numbers of progeny, typically in one issue (semelparity), short generation time and a short lifespan.

In general, there appears to be a tendency towards conservative, or specialist, K-strategy selection in stable environments, and radical, or generalist opportunistic, r-strategy selection in unstable or otherwise stressed environments. Interestingly, environments that are both stable and stressed favour selections involving a combination of components of both strategies. For example, deserts favour plants characterised by a K-type tolerance of long periods of drought, and an r-type ability to react by reproducing rapidly whenever it rains.