![2 Self-Organized Construction in...Conway's Game of Life
time step t is determined, according to a transition rule,1
by those of a set of
in-neighbors at step t — 1, and its own state at step t — 1 can affect the state of i
out-neighbors at t. In GoL, in-neighbors and out-neighbors coincide, and include
the cell itself. The neighborhood is a 3 x 3 square of cells. GoL's transition rule
specifies that a cell is in state 1 at step t if and only if either of the following
held at t - 1.
1. The cell and either two or three other cells in its neighborhood were in state
1.
2. The cell was in state 0, and exactly three other cells in its neighborhood were
in state 1.
By a random array, I mean one in which the initial probability p of each
cell being in state 1 is the same for all cells, and the initial state is determined
independently for each cell. Of course, we cannot actually construct such an
array, but we can reason about it. Toward the end of the chapter, large finite
random arrays will be considered, but it is simpler to start with the infinite
case. In fact, none of the reasoning used in the infinite case depends upon the
distribution of state 1 cells being strictly random, provided the frequency of all
finite arrangements of cell-states is as expected in a random array with the same
density of state 1 cells. A sparse random array is one in which p is very low (a
more precise definition is given below).
In a popular book on GoL, Poundstone [20] says:
Speculation about "living" Life patterns focuses on infinite, low-density
random fields.... If there are self-reproducing Life patterns, they would
have room to grow in such a field [pp. 175-176].
Poundstone may have drawn on material published in Berlekamp et al. [3], which
claims that self-replicating patterns can be shown to exist in GoL. (Such patterns
are finite arrangements of state 1 cells that produce multiple disjoint copies of
themselves in an otherwise empty—state 0—array.) Berlekamp et al. [3], then
say:
Inside any sufficiently large random broth, we expect just by chance, that
there will be some of these self-replicating creatures.... It's
probable.. .that after a long time, intelligent self-reproducing animals
will emerge and populate some parts of the space [p. 849] (emphasis in
original).
lr
The chapter uses a good deal of CA and Game of Life terminology, some of it novel.
Terms of this kind are italicized when first used, and explained unless their meaning is clear
from context.](https://image.slidesharecdn.com/87856-250327113811-50996978/85/New-Constructions-in-Cellular-Automata-David-Griffeath-20-320.jpg)
![CHAPTER XV.
THE MICROMETER.
It will have been gathered from the previous chapter that the perfect
circles nowadays turned out by our best opticians, and armed in
different parts by powerful reading microscopes, in conjunction with
a cross wire in the field of view of the telescope to determine the
exact axis of collimation, enable large arcs to be measured with an
accuracy comparable to that with which an astronomical clock
enables us to measure an interval of time.
We have next to see by what method small arcs are measured in the
field of view of the telescope itself. This is accomplished by what are
termed micrometers, which are of various forms. Thus we have the
wire micrometer, the heliometer, the double-image micrometer, and
so on. These we shall now consider in succession, entering into
further details of their use, and the arrangements they necessitate
when we come to consider the instrument in conjunction with which
they are generally employed.
The history of the micrometer is a very curious one. We have already
spoken of a pair of cross wires replacing the pinnules of the old
astronomers in the field of view of the telescope, so that it might be
pointed to any celestial object very much more accurately than it
could be without such cross wires. This kind of micrometer was first
applied to a telescope by Gascoigne in 1639. In a letter to Crabtree
he writes:[10]
“If here (in the focus of the telescope) you place the
scale that measures ... or if here a hair be set that it appear
perfectly through the glass ... you may use it in a quadrant for the
finding of the altitude of the least star visible by the perspective
wherein it is. If the night be so dark that the hair or the pointers of](https://image.slidesharecdn.com/87856-250327113811-50996978/85/New-Constructions-in-Cellular-Automata-David-Griffeath-34-320.jpg)
![In order to do this accurately we must determine how many
divisions of the screw correspond to the distance of the wires when
on two stars, say, one second apart. Here we must take advantage
of the rate at which a star travels across the field when the
telescope is fixed, and we separate the wires by a number of turns
of the screw, say twenty, and find what angle this corresponds to, by
letting a star on or near the equator[11]
traverse the field, and
noticing the time it requires to pass from one wire to the next.
Suppose it takes 26⅔ seconds, then, as fifteen seconds of arc pass
over in one second of time, we must multiply 26 by 15, which gives
400, so that the distance from wire to wire is 400 seconds of arc;
but this is due to twenty revolutions of the screw, so that each
revolution corresponds to 400
20
˝, or twenty seconds, and as each
revolution is divided into 100 parts, and 20
100
˝ = ⅕˝ therefore each
division corresponds to ⅕˝ of arc.
We shall return to the use of this most important instrument when
we have described the equatorial, of which it is the constant
companion.
THE HELIOMETER.](https://image.slidesharecdn.com/87856-250327113811-50996978/85/New-Constructions-in-Cellular-Automata-David-Griffeath-39-320.jpg)
![Fig. 106.—Object-glass
cut into two parts.
Fig. 105.—A B C. Images of Jupiter supposed to
be touching; B being produced by duplication, C
duplicate image on the other side of A.
A B, Double Star; A, A´ & B, B´, the
appearance when duplicate image is moved to
the right; A´, A & B´, B, the same when moved
to the left.
There are other kinds of micrometers
which we must also briefly consider. In
the heliometer[12]
we get the power of
measuring distances by doubling the
images of the objects we see, by means
of dividing the object-glass. The two
circles, A and B, Fig. 105, represent the
two images of Jupiter formed, as we
shall show presently, and touching each
other; now, if by any means we can
make B travel over A till it has the
position C, also just touching A, it will
manifestly have travelled over a
distance equal to the diameters of A and
B, so that if we can measure the distance traversed and divide it by
2, we shall get the diameter of the circle A, or the planet. The same
principle applies to double stars, for if we double the stars A and B,
Fig. 105, so that the secondary images become A´ and B´, we can
move A´ over B, and then only three stars will be visible; we can then
move the secondary images back over A and B till B´ comes over A,
and the second image of A comes to A´. It is thus manifest that the
images A´ and B´ on being moved to A´ and B´ in the second position
have passed over double their distance apart. Now all double-image
micrometers depend on this principle, and first we will explain how
this duplication of images is made in the heliometer. It is clear that
we shall not alter the power of an object-glass to bring objects to](https://image.slidesharecdn.com/87856-250327113811-50996978/85/New-Constructions-in-Cellular-Automata-David-Griffeath-40-320.jpg)
New Constructions in Cellular Automata David Griffeath
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- 5. New Constructions in Cellular Automata David Griffeath Digital Instant Download Author(s): David Griffeath, Cristopher Moore ISBN(s): 9780195137170, 0195137183 Edition: illustrated edition File Details: PDF, 22.57 MB Year: 2003 Language: english
- 7. New Constructions in Cellular Automata
- 8. Santa Fe Institute Studies in the Sciences of Complexity Lecture Notes Volume Author Eric Bonabeau, Marco Dorigo, and Guy Theraulaz M. E. J. Newman and R. G. Palmer Title Swarm Intelligence: From Natural to Artificial Systems Modeling Extinction Proceedings Volumes Editor James H. Brown and Geoffrey B. West Timothy A. Kohler and George J. Gumerman Lee A. Segel and Irun Cohen H. Randy Gimblett James P. Crutchfield and Peter Schuster David Griffeath and Cristopher Moore Title Scaling in Biology Dynamics in Human and Primate Societies Design Principles for the Immune System and Other Distributed Autonomous Systems Integrating Geographic Information Systems and Agent-Based Modeling Techniques Evolutionary Dynamics: Exploring the Interplay of Selection, Accident, Neutrality, and Function New Constructions in Cellular Automata
- 9. New Constructions in Cellular Automata Editors David Griffeath University of Wisconsin Madison, WI Cristopher Moore Santa Fe Institute Santa Fe, NM and University of New Mexico Albuquerque, NM Santa Fe Institute Studies in the Sciences of Complexity OXFORD UNIVE
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- 11. About the Santa Fe Institute research and education center, founded in 1984. Since its founding, SFI has research and education center, founded in 1984. Since its founding, SFI has devoted itself to creating a new kind of scientific research community, pursuing emerging science. Operating as a small, visiting institution, SFI seeks to catalyze new collaborative, multidisciplinary projects that break down the barriers between the traditional disciplines, to spread its ideas and methodologies to other individuals, and to encourage the practical applications of its results. All titles from the Santa Fe Institute Studies in the Sciences of Complexity series will carry this imprint which is based on a Mimbres pottery design (circa A.D. 950-1150), drawn by Betsy Jones. The design was selected because the radiating feathers are evocative of the out-reach of the Santa Fe Institute Program to many disciplines and institutions.
- 12. Santa Fe Institute Editorial Board September 2000 Ronda K. Butler-Villa, Chair Director of Publications, Facilities, & Personnel, Santa Fe Institute Dr. David K. Campbell Department of Physics, Boston University Prof. Marcus W. Feldman Institute for Population & Resource Studies, Stanford University Prof. Murray Gell-Mann Division of Physics & Astronomy, California Institute of Technology Dr. Ellen Goldberg President, Santa Fe Institute Prof. George J. Gumerman Arizona State Museum, University of Arizona Dr. Thomas B. Kepler Vice President for Academic Affairs, Santa Fe Institute Prof. David Lane Dipartimento di Economia Politica, Modena University, Italy Prof. Simon Levin Department of Ecology & Evolutionary Biology, Princeton University Prof. John Miller Department of Social & Decision Sciences, Carnegie Mellon University Prof. David Pines Department of Physics, University of Illinois Dr. Charles F. Stevens Molecular Neurobiology, The Salk Institute
- 13. Contributors List Kellie M. Evans, California State University, 18111 Nordhoff Street, Northridge, CA 91330; E-mail: kellie.m.evans@csun.edu Nick M. Gotts, MLURI, Land Use Science Group, Aberdeen AB15 8QH, Scotland, United Kingdom; E-mail: n.gotts@mluri.sari.ac.uk Janko Gravner, University of California, Mathematics Department, Davis, CA 95616; E-mail: gravner@math.ucdavis.edu David Griffeath, University of Wisconsin, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706; E-mail: griffeat@math.wisc.edu Dean Hickerson, Mathematics Department, University of California, Davis, CA 95616; E-mail: dean@rnath.ucdavis.edu George E. Homsy, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139; E-mail: ghomsy@ai.mit.edu Joy V. Hughes, 3954 Jarvis Road, Scotts Valley, CA 95066; E-mail: hughes@scruznet. com Norman H. Margolus, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 545 Technology Square, Cambridge, MA 02139; E-mail: nhm@mit.edu Bernd Mayer, Institute for Theoretical Chemistry, and Radiation Chemistry, University of Vienna, UZAII, Althanstrafte 14, A-1090 Vienna, Austria; E-mail: bernd@asterix. msp.univie.ac. at Cristopher Moore, Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 and University of New Mexico, Department of Computer Science and Department of Physics and Astronomy, Albuquerque, NM 87131; E-mail: moore@santafe.edu Gadi Moran, Department of Mathematics, University of Haifa, Haifa 31905, Isreal; E-mail: gadi@mathcsS.haifa.ac.il Mark D. Niemiec, S260 Par Lane PH7, Willoughby Hills, OH 44094; E-mail: mniemiec@interserv. com Martin Nilsson, Los Alamos National Laboratory, EES-5 & T-CNLS, Mail Stop D450, Los Alamos, New Mexico 87545 ; E-mail: nilsson@lanl.gov Nienke A. (Domes, University of Wisconsin, Department of Economics, 1180 Observatory Drive, Madison, WI 53706; E-mail noomes@ssc.wisc.edu Steen Rasmussen, Los Alamos National Laboratory, EES-5 & T-CNLS, Mail Stop D450, Los Alamos, NM 87545; E-mail: steen@lanl.gov Rudy Rucker, Department of Mathematics and Computer Science, San Jose State University, San Jose, CA 95192; E-mail: rucker@mathcs.sjsu.edu Raissa D'Souza, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139; E-mail: raissa@im.lcs.mit.edu David Whitten, CST-1 MS J565, Los Alamos National Laboratory, Los Alamos, NM 87545; E-mail: whitten@lanl.gov
- 14. This page intentionally left blank
- 15. Contents Preface Cristopher Moore and David Griffeath xi Self-Organized Construction in Sparse Random Arrays of Conway's Game of Life Nicholas M. Gotts 1 Synthesis of Complex Life Objects from Gliders Mark D. Niemiec 55 A Two-Dimensional Cellular Automaton Crystal with Irrational Density David Griffeath and Dean Hickerson 79 Still Life Theory Matthew Cook 93 Replicators and Larger-than-Life Examples Kellie Michele Evans 119 Growth Phenomena in Cellular Automata Janko Gravner 161 Constructive Molecular Dynamics Lattice Gases: Three-Dimensional Molecular Self-Assembly Martin Nilsson, Steen Rasmussen, Bernd Mayer, and David Whitten 183 Simulating Digital Logic with the Reversible Aggregation Model of Crystal Growth Raissa D'Souza, George E. Homsy, and Norman H. Margolus 211 Universal Cellular Automata Based on the Collisions of Soft Spheres Norman H. Margolus 231 IX
- 16. Contents Emerging Markets and Persistent Inequality in a Nonlinear Voting Model Nienke A. Oomes 261 Cellular Automata for Imaging, Art, and Video Joy V. Hughes 285 Continuous-Valued Cellular Automata in Two Dimensions Rudy Rucker 295 Phase Transition via Cellular Automata Gadi Moran 317 Index 323
- 17. Preface This book is the long-awaited proceedings of a conference, held at the Santa Fe Institute in December, 1998, and sponsored by the National Science Foundation. "New Constructions in Cellular Automata" brought people together to discuss topics ranging from modeling physics and economics, to reversible computation, to the latest discoveries of bugs, puffers, and all the flora and fauna of the cellular automaton world. The first part of the book focuses on the best-loved CA rule, Conway's Life, and its variants. In the first chapter, Nick Gotts answers the cosmological question of what happens in a random low-density initial condition, showing that a surprising amount can be learned about what structures self-organize in the early Life universe. In the next chapter, Mark Niemiec shows us the latest methods of constructing complex objects from collisions of gliders, an essential engineering skill for Life devotees. David Griffeath and Dean Hickerson answer one of Life's open questions: whether an initial seed exists that populates the universe with an irrational density. Matthew Cook shows that telling when a "still life," a configuration which is stable under the Life rule, can be divided into New Constructions in Cellular Automata, edited by David Griffeath and Cristopher Moore, Oxford University Press. Xi
- 18. XII Preface separate pieces is an NP-complete problem. Moving on to Life's generalizations, Kellie Evans introduces Larger than Life and HighLife, and finds many families of replicators in these rules. To bring the book alive and to help the reader explore the many open ques- tions remaining in the field, many of the Life patterns discussed in these chapters can be downloaded from the book's companion web page, (http://psoup.math.wisc.edu/NewConstructions). In the next chapters, we put cellular automata to work as platforms for simulating phenomena in physics and economics. Janko Gravner introduces us to the mathematics of growth phenomena and studies the asymptotic shapes of various rules. Martin Nilsson, Steen Rasmussen, Bernd Mayer, and David Whitten discuss how to use lattice gases to simulate hydrophobic and hydrophilic polymers. (In recent work, they have achieved the formation of micelles with this method, and shown that CAs can reach time-scales several orders of magnitude longer than standard molecular dynamics (MD) simulations.) Raissa D'Souza, George Homsy and Norman Margolus then use reversible CAs to model how an aggregating cluster reaches equilibrium with its environ- ment, and show that their reversible aggregation (RA) rule can simulate universal reversible logic. Margolus shows that a soft-sphere model also has this degree of computational power, and Nienke Oomes rounds out this section by using CAs to model how economic inequality can persist in emerging markets. In the concluding chapters, Joy Hughes gives us beautiful examples of how CAs can be used in art and video, Rudy Rucker extols the virtues of CAs whose states are continuous rather than discrete, and Gadi Moran shows a phase tran- sition in majority-voting rules on graphs. We are deeply indebted to the Santa Fe Institute and Oxford University Press for making this book possible, and especially to Delia Ulibarri and Ronda K. Butler-Villa for their tireless work and extraordinary patience. We also thank the University of Wisconsin, Madison, for hosting the Primordial Soup web page and the book's companion page, <http://psoup.math.wisc.edu/NewConstructions), where many patterns and simulations relevant to these chapters can be down- loaded. Finally, we dedicate this book to Oscar, Rascal, Scurry, and Spootie the Cat. Cristopher Moore Santa Fe Institute and University of New Mexico David Griffeath University of Wisconsin
- 19. Self-Organized Construction in Sparse Random Arrays of Conway's Game of Life Nicholas M. Gotts 1 INTRODUCTION The construction problems and techniques described in this chapter arose out of a single problem: What happens in very low density infinite random arrays of Conway's Game of Life? However, the work reported has wider implications, briefly discussed in the final section. Conway's Game of Life (henceforth GoL) is a deterministic cellular automa- ton (CA), which is binary (a cell has two possible states: 0 and 1) and runs on an infinite two-dimensional grid of cells. A deterministic CA cell's state at New Constructions in Cellular Automata, edited by David Griffeath and Cristopher Moore, Oxford University Press. 1
- 20. 2 Self-Organized Construction in...Conway's Game of Life time step t is determined, according to a transition rule,1 by those of a set of in-neighbors at step t — 1, and its own state at step t — 1 can affect the state of i out-neighbors at t. In GoL, in-neighbors and out-neighbors coincide, and include the cell itself. The neighborhood is a 3 x 3 square of cells. GoL's transition rule specifies that a cell is in state 1 at step t if and only if either of the following held at t - 1. 1. The cell and either two or three other cells in its neighborhood were in state 1. 2. The cell was in state 0, and exactly three other cells in its neighborhood were in state 1. By a random array, I mean one in which the initial probability p of each cell being in state 1 is the same for all cells, and the initial state is determined independently for each cell. Of course, we cannot actually construct such an array, but we can reason about it. Toward the end of the chapter, large finite random arrays will be considered, but it is simpler to start with the infinite case. In fact, none of the reasoning used in the infinite case depends upon the distribution of state 1 cells being strictly random, provided the frequency of all finite arrangements of cell-states is as expected in a random array with the same density of state 1 cells. A sparse random array is one in which p is very low (a more precise definition is given below). In a popular book on GoL, Poundstone [20] says: Speculation about "living" Life patterns focuses on infinite, low-density random fields.... If there are self-reproducing Life patterns, they would have room to grow in such a field [pp. 175-176]. Poundstone may have drawn on material published in Berlekamp et al. [3], which claims that self-replicating patterns can be shown to exist in GoL. (Such patterns are finite arrangements of state 1 cells that produce multiple disjoint copies of themselves in an otherwise empty—state 0—array.) Berlekamp et al. [3], then say: Inside any sufficiently large random broth, we expect just by chance, that there will be some of these self-replicating creatures.... It's probable.. .that after a long time, intelligent self-reproducing animals will emerge and populate some parts of the space [p. 849] (emphasis in original). lr The chapter uses a good deal of CA and Game of Life terminology, some of it novel. Terms of this kind are italicized when first used, and explained unless their meaning is clear from context.
- 21. Exploring the Variety of Random Documents with Different Content
- 22. The details of the escapement may be seen in Fig. 96, which gives a general view of a portion of the back plate of the clock movement, supposing the pendulum removed; a and b are the front and back plates respectively of the clock train; c is a cock supporting one end of the crutch axis; d is the crutch rod carrying the pallets, and e an arm carried by the crutch axis and fixed at f to the left-hand pallet arm; g is a cock supporting a detent projecting towards the left and curved at its extreme end; at a point near the top of the escape wheel this detent carries a pin (jewel) for locking the wheel, and at its extreme end there is a very light “passing spring.” The action of the escapement is as follows:—Suppose the pendulum to be swinging from the right hand. It swings quite freely until a pin at the end of the arm e lifts the detent; the wheel escapes from the jewel before mentioned, and the tooth next above the left-hand pallet drops on the face of the pallet (the state shown in the figure), and gives impulse to the pendulum; the wheel is immediately locked again by the jewel, and the pendulum, now detached, passes on to the left; in returning to the right, the light passing spring, before spoken of, allows the pendulum to pass without disturbing the detent; on going again to the left, the pendulum again receives impulse as already described. The right-hand pallet forms no essential part of the escapement, but is simply a safety pallet, designed to catch the wheel in case of accident to the locking-stone during the time that the left-hand pallet is beyond the range of the wheel. The escape wheel carrying the seconds hand thus moves once only in each complete or double vibration of the pendulum, or every two seconds. IV. The Chronometer. We have now given a description of the astronomical clock—the modern astronomical instrument which it was our duty to consider. There is another time-keeper—the chronometer—which we have to dwell upon. In the chronometer, instead of using the pendulum, we
- 23. have a balance, the vibration of which is governed by a spiral spring, instead of by gravity, as the pendulum is. By such means we keep almost as accurate time as we do by employing a pendulum, the balance being corrected for temperature on principles, one of which we shall describe. We must premise by saying that fully four-fifths of the compensation required by a chronometer or watch-balance is owing to the change in elasticity of the governing spiral spring, the remainder, comparatively insignificant, being due to the balance’s own expansion or contraction. The segments R1, R2 of the balance (see Fig. 97) are composed of two metals, say copper and steel, the copper being exterior; then as the governing spiral spring loses its elasticity by heat, the segments R1, R2 curve round and take up positions nearer the axis of motion, the curvature being produced by the greater expansion of copper over steel; and thus the loss of time due to the loss of elasticity of the spiral spring is compensated for. This balance may be adjustable by placing on the arms small weights, W W, which may be moved along the arms, and so increase or diminish the effect of temperature at pleasure.
- 24. Fig. 97.—Compensating Balance. Of the number of watch and chronometer escapements we may mention the detached lever—the one most generally used for the best watches, the form is shown in Fig. 98. P P are the pallets working on a pin at S as in the dead-beat clock escapement; the pallets carry a lever L which can vibrate between two pins B B. R is a disc carried on the same axis with the balance, and it carries a pin I, which as the disc goes round in the direction of the arrow, falls into the fork of the lever, and moves it on and withdraws the pallet from the tooth D, which at once moves onwards and gives the lever an impulse as it passes the face of the pallet. This impulse is communicated to the balance through the pin I, the balance is kept vibrating in contrary directions under the influence of the hair-spring, gaining an impulse at each swing. On the same axis as R is a second disc O with a notch cut in it into which a tongue on the lever enters; this acts as a safety lock, as the lever can only move while the pin I is in the fork of the lever.
- 25. Fig. 98.—Detached Lever Escapement. The escapement we next describe is that most generally used in chronometers. S S, Fig. 99, is the escape wheel which is kept from revolving by the detent D. On the axis of the balance are two discs, R1, R2, placed one under the other. As the balance revolves in the direction of the arrow, the pin P2 will come round and catch against the point of the detent, lifting it and releasing the escape-wheel, which will revolve, and the tooth T will hit against the stud P1, giving the balance an impulse. The balance then swings on to the end of its course and returns, and the stud P2 passes the detent as follows: a light spring Y Y is fastened to the detent, projecting a little beyond it, and it is this spring, and not the detent itself, that the pin P2 touches: on the return of P2 it simply lifts the spring away from the detent and passes it, whereas in advancing the spring was supported by the point of the detent, and both were lifted together.
- 26. Fig. 99.—Chronometer Escapement. Fig. 100.—The Fusee. In watches and chronometers and in small clocks a coiled spring is used instead of a weight, but its action is irregular, since when it is fully wound up it exercises greater force than when nearly down. In order to compensate for this the cord or chain which is wound round the barrel containing the spring passes round a conical barrel called a fusee (Fig. 100): B is the barrel containing the spring and A A the fusee. One end of the spring is fixed to the axis of the barrel, which is prevented from turning round, and the other end to the barrel, so
- 27. that on winding up the clock by turning the fusee the cord becomes coiled on the latter, and the more the spring is wound the nearer the cord approaches the small end of the fusee, and has therefore less power over it; while as the clock goes and the spring becomes unwound, its power over the axis becomes greater. The power, therefore, acting to turn the fusee remains pretty constant. 9. By Messrs. E. Dent and Co. of the Strand.
- 28. CHAPTER XIV. CIRCLE READING. One of the great advantages which astronomy has received from the invention of the telescope is the improved method of measuring space and determining positions by the use of the telescope in the place of pointers on the old instruments. The addition of modern appliances to the telescope to enable it to be used as an accurate pointer, has played a conspicuous part in the accurate measurement of space, and the results are of such importance, and they have increased so absolutely pari passu with the telescope, that we must now say something of the means by which they have been brought about. For astronomy of position, in other words for the measurement of space, we want to point the telescope accurately at an object. That is to say, in the first instance we want circles, and then we want the power of not only making perfect circles, but of reading them with perfect accuracy; and where the arc is so small that the circle, however finely divided, would help us but little, we want some means of measuring small arcs in the eyepiece of the telescope itself, where the object appears to us, as it is called, in the field of view; we want to measure and inspect that object in the field of view of the telescope, independently of circles or anything extraneous to the field. We shall then have circles and micrometers to deal with divisions of space, and clocks and chronographs to deal with divisions of time. We require to have in the telescope something, say two wires crossed, placed in the field of view—in the round disc of light we see in a telescope owing to the construction of the diaphragm—so as to
- 29. be seen together with any object. In the chapter on eyepieces it was shown that we get at the focus the image of the object; and as that is also the focus of the eyepiece, it is obvious that not only the image in the air, as it were, but anything material we like to put in that focus, is equally visible. By the simple contrivance of inserting in this common focus two or more wires crossed and carried on a small circular frame, we can mark any part of the field, and are enabled to direct the telescope to any object. In the Huyghenian eyepiece, Fig. 60, the cross should be between the two convex lenses, for if we have an eyepiece of this kind the focus will be at F, and so here we must have our cross wires; but, if instead of this eyepiece we have one of the kind called Ramsden’s eyepiece, Fig. 62, with the two convex surfaces placed inwards, then the focus will be outside, at F, and nearer to the object-glass: therefore we shall be able to change these eyepieces without interfering with the system of wires in the focus of the telescope. We hence see at once that the introduction of this contrivance, which is due to Mr. Gascoigne, at once enormously increases the possibility of making accurate observations by means of the telescope. Fig. 101.—Diggs’ Diagonal Scale.
- 30. Hipparchus was content to ascertain the position of the celestial bodies to within a third of a degree, and we are informed that Tycho Brahe, by a diagonal scale, was able to bring it down to something like ten seconds. Fig. 101 will show what is meant by this. Suppose this to be part of the arc of Tycho’s circle, having on it the different divisions and degrees. Now it is clear that when the bar which carried the pointer swept over this arc, divided simply into degrees, it would require a considerable amount of skill in estimating to get very close to the truth, unless some other method were introduced; and the method suggested by Diggs, and adopted by Tycho, was to have a series of diagonal lines for the divisions of degrees; and it is clear that the height of the diagonal line measured from the edge of the circle could give, as it were, a longer base than the direct distance between each division for determining the subdivisions of the degree, and a slight motion of the pointer would make a great difference in the point where it cuts the diagonal line. For instance, it would not be easy to say exactly the fraction of division on the inner circle at which the pointer in Fig. 101 rests, but it is evident that the leading edge of the pointer cuts the diagonal line at three-fourths of its length, as shown by the third circle; so the reading in this case is seven and three-quarters; but that is, after all, a very rough method, although it was all the astronomer had to depend upon in some important observations.
- 31. Fig. 102.—The Vernier. The next arrangement we get is one which has held its own to the present day, and which is beautifully simple. It is due to a Frenchman named Vernier, and was invented about 1631. We may illustrate the principle in this way. Suppose for instance we want to subdivide the divisions marked on the arc of a circle, Fig. 102 a b, and say we wish to divide them into tenths, what we have to do is this—First, take a length equal to nine of these divisions on a piece of metal, c, called the vernier, carried on an arm from the centre of the circle, and then, on a separate scale altogether, divide that distance not into nine, as it is divided on the circle, but into ten portions. Now mark what happens as the vernier sweeps along the circle, instead of having Tycho’s pointer sweeping across the diagonal scale. Let us suppose that the vernier moves with the telescope and the circle is fixed; then when division 0 of the vernier is opposite division 6 on the circle we know that the telescope is pointing at 6° from zero measured by the degrees on this scale; but suppose, for instance, it moves along a little more, we find that line 1 of the vernier is in contact with and opposite to another on the circle, then the reading is 6° and ⅒°; it moves a little further, and we find that the next line 2, is opposite to another, reading 6° and 2 10 °, a little further still, and we find the next opposite. It is clear that in this way we have a readier means of dividing all those spaces into tenths, because if the length of the vernier is nine circle divisions the length of each division on the vernier must be as nine is to ten, so that each division is one-tenth less than that on the circle. We must therefore move the vernier one-tenth of a circle division, in order to make the next line correspond. That is to say, when the division of the vernier marked 0 is opposite to any line, as in the diagram, the reading is an exact number of degrees; and when the division 1 is opposite, we have then the number of degrees given by the division 0 plus one-tenth; when 2 is in contact, plus two-tenths;
- 32. when 3 is in contact, plus three-tenths; when 4 is in contact, plus four-tenths, and so on, till we get a perfect contact all through by the 0 of the vernier coming to the next division on the circle, and then we get the next degree. It is obvious that we may take any other fraction than to for the vernier to read to, say 1 60 , then we take a length of 59 circle divisions on the vernier and divide it into 60, so that each vernier division is less than a circle division by 1 60 . This is a method which holds its own on most instruments, and is a most useful arrangement. But most of us know that the division of the vernier has been objected to as coarse and imperfect; and Sharp, Graham, Bird, Ramsden, Troughton, and others found that it is easy to graduate a circle of four or five feet in diameter, or more, so accurately and minutely that five minutes of arc shall be absolutely represented on every part of the circle. We can take a small microscope and place in its field of view two cross wires, something like those we have already mentioned, so as to be seen together with the divisions on the circle, and then, by means of a screw with a divided head, we can move the cross wires from division to division, and so, by noting the number of turns of the screw required to bring the cross wires from a certain fixed position, corresponding to the pointer in the older instruments, to the nearest division, we can measure the distance of that division from the fixed point or pointer, as it were, just as well as if the circle itself were much more closely divided. We can have matters so arranged that we may have to make, if we like, ten turns of the screw in order to move the cross wires from one graduation to the next, and we may have the milled head of the screw itself divided into 100 divisions, so that we shall be able to divide each of the ten turns into 100, or the whole division into 1,000 parts. It is then simply a question of dividing a portion of arc equal to five minutes into a thousand, or, if one likes, ten thousand parts by a delicate screw motion. We are now speaking of instruments of precision, in which large telescopes are not so necessary as large circles. With reference to
- 33. instruments for physical and other observations, large circles are not so necessary as large telescopes, as absolute positions can be determined by instruments of precision, and small arcs can, as we shall see in the next chapter, be determined by a micrometer in the eyepiece of the telescope.
- 34. CHAPTER XV. THE MICROMETER. It will have been gathered from the previous chapter that the perfect circles nowadays turned out by our best opticians, and armed in different parts by powerful reading microscopes, in conjunction with a cross wire in the field of view of the telescope to determine the exact axis of collimation, enable large arcs to be measured with an accuracy comparable to that with which an astronomical clock enables us to measure an interval of time. We have next to see by what method small arcs are measured in the field of view of the telescope itself. This is accomplished by what are termed micrometers, which are of various forms. Thus we have the wire micrometer, the heliometer, the double-image micrometer, and so on. These we shall now consider in succession, entering into further details of their use, and the arrangements they necessitate when we come to consider the instrument in conjunction with which they are generally employed. The history of the micrometer is a very curious one. We have already spoken of a pair of cross wires replacing the pinnules of the old astronomers in the field of view of the telescope, so that it might be pointed to any celestial object very much more accurately than it could be without such cross wires. This kind of micrometer was first applied to a telescope by Gascoigne in 1639. In a letter to Crabtree he writes:[10] “If here (in the focus of the telescope) you place the scale that measures ... or if here a hair be set that it appear perfectly through the glass ... you may use it in a quadrant for the finding of the altitude of the least star visible by the perspective wherein it is. If the night be so dark that the hair or the pointers of
- 35. the scale be not to be seen, I place a candle in a lanthorn, so as to cast light sufficient into the glass, which I find very helpful when the moon appeareth not, or it is not otherwise light enough.” This then was the first “telescopic sight,” as these arrangements at the common focus of the object-glass and eyepiece were at first called. It is certain that we may date the micrometer from the middle of the seventeenth century; but it is rather difficult to say who it was who invented it. It is frequently attributed to a Frenchman named Auzout, who is stated to have invented it in 1666; but we have reason to know that Gascoigne had invented an instrument for measuring small distances several years before. Though first employed by Gascoigne, however, they were certainly independently introduced on the Continent, and took various forms, one of them being a reticule, or network of small silver threads, suggested by the Marquis Malvasia, the arc interval of which was determined by the aid of a clock. Huyghens had before this proposed, as specially applicable to the measures of the diameters of planets and the like, the introduction of a tapering slip of metal. The part of the slip which exactly eclipsed the planet was noted; it was next measured by a pair of compasses, and having the focal length of the telescope, the apparent diameter was ascertained.
- 36. Fig. 103.—System of Wires in a Transit Eyepiece. Malvasia’s suggestion was soon seized upon for determinations of position. Römer introduced into the first transit instrument a horizontal and a number of vertical wires. The interval between the three he generally used was thirty-four seconds in the equator, and the time was noted to half seconds. The field was illuminated by means of a polished ring placed outside of the object-glass. The simple system of cross wires, then, though it has done its work, is not to be found in the telescope now, either to mark the axis of collimation, or roughly to measure small distances. For the first purpose a much more elaborate system than that introduced by Römer is used. We have a large number of vertical wires, the principal object of which is, in such telescopes as the transit, to determine the absolute time of the passage of either a star or planet, or the sun or moon, over the meridian; and one or more
- 37. horizontal ones. These constitute the modern transit eyepiece, a very simple form of which is shown in the above woodcut. THE WIRE MICROMETER. The wire micrometer is due to suggestions made independently by Hooke and Auzout, who pointed out how valuable the reticule of Malvasia would be if one of the wires were movable. Fig. 104.—Wire Micrometer. x and y are thicker wires for measuring positions on a separate plate to be laid over the fine wires. The first micrometer in which motion was provided consisted of two plates of tin placed in the eyepiece, being so arranged and connected by screws that the distances between the two edges of the tin plates could be determined with considerable accuracy. A planet could then be, as it were, grasped between the two plates, and its diameter measured; it is very obvious that what would do as well as these plates of tin would be two wires or hairs representing the edges of these tin plates; and this soon after was carried out by Hooke, who left his mark in a very decided way on very many astronomical arrangements of that time. He suggested that all that was necessary to determine the diameter of Saturn’s rings was to have a fixed wire in the eyepiece, and a second wire travelling in the field of view, so that the planet or the ring could be grasped between those two wires.
- 38. The wire-micrometer. Fig. 104, differs little from the one Hooke and Auzout suggested, A A is the frame, which carries two slides, C and D, across the ends of each of which fine wires, E and B, are stretched; then, by means of screws, F and G, threaded through these movable slides and passing through the frame A A, the wires can be moved near to, or away from, each other. Care must be taken that the threads of the screw are accurate from one end to the other, so that one turn of the screw when in one position would move the wire the same distance as a turn when in another position. In this micrometer both wires are movable, so as to get a wide separation if needful, but in practice only one is so, the other remaining a fixture in the middle of the field of view. There is a large head to the screw, which is called the micrometer screw, marked into divisions, so that the motion of the wire due to each turn of the screw may be divided, say into 100 parts, by actual division against a fixed pointer, and further into 1,000 parts by estimation of the parts of each division. Hooke suggested that, if we had a screw with 100 turns to an inch, and could divide these into 1,000 parts, we should obviously get the means of dividing an inch into 100,000 parts; and so, if we had a screw which would give 100 turns from one side of the field of view of the telescope to the other, we should have an opportunity of dividing the field of view of any telescope into something like 100,000 parts in any direction we chose. The thick wires, x, y, are fixed to the plate in front of, but almost touching, the fine wires, and in measuring, for instance, the distance of two stars the whole instrument is turned round until these wires are parallel to the direction of the imaginary line joining them. This was the way in which Huyghens made many important measures of the diameters of different objects and the distances of different stars. Thus far we are enabled to find the number of divisions on the micrometer screw that corresponds to the distance from one star to another, or across a planet, but we want to know the number of seconds of arc in the distance measured.
- 39. In order to do this accurately we must determine how many divisions of the screw correspond to the distance of the wires when on two stars, say, one second apart. Here we must take advantage of the rate at which a star travels across the field when the telescope is fixed, and we separate the wires by a number of turns of the screw, say twenty, and find what angle this corresponds to, by letting a star on or near the equator[11] traverse the field, and noticing the time it requires to pass from one wire to the next. Suppose it takes 26⅔ seconds, then, as fifteen seconds of arc pass over in one second of time, we must multiply 26 by 15, which gives 400, so that the distance from wire to wire is 400 seconds of arc; but this is due to twenty revolutions of the screw, so that each revolution corresponds to 400 20 ˝, or twenty seconds, and as each revolution is divided into 100 parts, and 20 100 ˝ = ⅕˝ therefore each division corresponds to ⅕˝ of arc. We shall return to the use of this most important instrument when we have described the equatorial, of which it is the constant companion. THE HELIOMETER.
- 40. Fig. 106.—Object-glass cut into two parts. Fig. 105.—A B C. Images of Jupiter supposed to be touching; B being produced by duplication, C duplicate image on the other side of A. A B, Double Star; A, A´ & B, B´, the appearance when duplicate image is moved to the right; A´, A & B´, B, the same when moved to the left. There are other kinds of micrometers which we must also briefly consider. In the heliometer[12] we get the power of measuring distances by doubling the images of the objects we see, by means of dividing the object-glass. The two circles, A and B, Fig. 105, represent the two images of Jupiter formed, as we shall show presently, and touching each other; now, if by any means we can make B travel over A till it has the position C, also just touching A, it will manifestly have travelled over a distance equal to the diameters of A and B, so that if we can measure the distance traversed and divide it by 2, we shall get the diameter of the circle A, or the planet. The same principle applies to double stars, for if we double the stars A and B, Fig. 105, so that the secondary images become A´ and B´, we can move A´ over B, and then only three stars will be visible; we can then move the secondary images back over A and B till B´ comes over A, and the second image of A comes to A´. It is thus manifest that the images A´ and B´ on being moved to A´ and B´ in the second position have passed over double their distance apart. Now all double-image micrometers depend on this principle, and first we will explain how this duplication of images is made in the heliometer. It is clear that we shall not alter the power of an object-glass to bring objects to
- 41. Fig. 107.—The parts separated, and giving two images of any object. focus if we cut the object- glass in two, for if we put any dark line across the object-glass, which optically cuts it in two, we shall get an image, say of Jupiter, unaltered. But suppose instead of having the parts of the object- glass in their original position after we have cut the object-glass in two, we make one half of the object-glass travel over the other in the manner represented in Fig. 107. Each of these halves of the object-glass will be competent to give us a different image, and the light forming each image will be half the light we got from the two halves of the object-glass combined; but when one half is moved we shall get two images in two different places in the field of view. We can so alter the position of the images of objects by sliding one half of the object-glass over the other, that we shall, as in the case of the planet Jupiter, get the two images exactly to touch each other, as is represented in Fig. 105; and further still, we can cause one image to travel over to the other side. If we are viewing a double star, then the two halves will give four stars, and we can slide one half, until the central image formed by the object-glasses will consist of two images of two different stars, and on either side there will be an image of each star, so that there would appear to be three stars in the field of view instead of two. We have thus the means of determining absolutely the distance of any two celestial objects from each other, in terms of the separation of the centres of the two halves of the object-glass. But as in the case of the wire micrometer we must know the value of the screw, so in the case of the heliometer we must know how much arc is moved over by a certain motion of one half of the object-glass.
- 42. Fig. 108.—Double images seen through Iceland spar. Fig. 109.—Diagram showing the path of the ordinary and extraordinary rays in a crystal of Iceland spar, producing two images apparently at E and O. THE DOUBLE-IMAGE MICROMETER.
- 43. Now there is another kind of double-image micrometer which merits attention. In this case the double image is derived from a different physical fact altogether, namely, double refraction. Those who have looked through a crystal of Iceland spar, Fig. 108, have seen two images of everything looked at when the crystal is held in certain positions, but the surfaces of the crystal can be cut in a certain plane such that when looked through, the images are single. For the micrometer therefore we have doubly refracting prisms, cut in such a way as to vary the distance of the images. Generally speaking, whenever a ray of light falls on a crystal of Iceland spar or other double refracting substance, it is divided up into two portions, one of which is refracted more than the other. If we trace the rays proceeding from a point S, Fig. 109, we find one portion of the light reaching the eye is more refracted at the surfaces than the other, and consequently one appears to come from E and the other from O, so that if we insert such a crystal in the path of rays from any object, that object appears doubled. There is, however, a certain direction in the crystal, along which, if the light travel, it is not divided into two rays, and this direction is that of the optic axis of the crystal, A A, Fig. 110; if therefore two prisms of this spar are made so that in one the light shall travel parallel to the axis, and in the other at right angles to it, and if these be fastened together so that their outer sides are parallel, as shown in Fig. 111, light will pass through the first one without being split up, since it passes parallel to the axis, but on reaching the second one it is divided into two rays, one of which proceeds on in the original course, since the two prisms counteract each other for this ray, while the other ray diverges from the first one, and gives a second image of the object in front of the telescope, as shown in Fig. b. The separation of the image depends on the distance of the prisms from the eyepiece, so that we can pass the rays from a star or planet through one of these compound crystals and measure the position of the crystal and so the separation of the stars, and then we shall have the means of doing the same that we did by dividing our object-glass, and in a less expensive way, for to take a large object-glass of eight or ten
- 44. inches in diameter and cut it in two is a brutal operation, and has generally been repented of when it has been done. Fig. 110.—Crystals of Iceland Spar showing, A A ´, the optic axis. It is obvious that a Barlow lens, cut in the same manner as the object-glass of the heliometer, will answer the same purpose; the two halves are of course moved in just the same manner as the halves of the divided object-glass. Mr. Browning has constructed micrometers on this principle. Fig. 111.—Double Image Micrometer. Fig. a, p q, single image formed by object-glass. Fig. b,
- 45. p1 q1, p2 q2, images separated by the double refracting prism. Fig. c, same, separated less, by the motion of the prism. There is yet another double-image micrometer depending on the power of a prism to alter the direction of rays of light. It is constructed by making two very weak prisms, i.e., having their sides very nearly parallel, and cutting them to a circular shape; these are mounted in a frame one over the other with power to turn one round, so that in one position they both act in the same direction, and in the opposite one they neutralise each other; these are carried by radial arms, and are placed either in front of the object-glass or at such a distance from it inside the telescope that they intercept one half of the light, and the remaining portion goes to form the usual image, while the other is altered in its course by the prism and forms another image, and this alteration depends on the position of the movable prism. 10. Grant’s History of Physical Astronomy, p. 454. 11. More accurately the time of transit is to be multiplied by the cosine of the star’s declination. 12. So called because the contrivance was first used to measure the diameter of the sun.
- 46. BOOK IV. MODERN MERIDIONAL OBSERVATIONS.
- 47. CHAPTER XVI. THE TRANSIT CIRCLE. We are now, then, in full possession of the stock-in-trade of the modern astronomer—the telescope, the clock, and the circle,—and we have first to deal with what is termed astronomy of position, that branch of the subject which enables us to determine the exact position of the heavenly bodies in the celestial sphere at any instant of time. Before, however, we proceed with modern methods, it will be well, on the principle of reculer pour mieux sauter, to refer back to the old ones in order that we can the better see how the modern instruments are arranged for doing the work which Tycho, for instance, had to do, and which he accomplished by means of the instruments of which we have already spoken. First of all let us refer to the Mural Quadrant, in which we have the germ of a great deal of modern work, its direct descendant being the Transit Circle of the present time. We begin then by referring to the hole in the wall at which Tycho is pointing (see Fig. 112), and the circle, of which the hole was the centre, opposite to it, on which the position of the body was observed, and its declination and right ascension determined. This then was Tycho’s arrangement for determining the two co-ordinates, right ascension and declination, measured from the meridian and equator. It is to be hoped that the meaning of right ascension and declination is already clear to our readers, because these terms refer to the fundamental planes, and distances as measured from them are the very A B C of anything that one has to say about astronomical instruments.
- 48. We know that Tycho had two things to do. In the first place he had to note when a star was seen through the slit in the wall, which was Tycho’s arrangement for determining the southing of a star, the sun, or the moon; and then to give the instant when the object crossed the sight to the other observer, who noted the time by the clocks. Secondly, he had to note at which particular portion of the arc the sight had to be placed, and so the altitude or the zenith distance of the star was determined; and then, knowing the latitude of the place, he got the two co-ordinates, the right ascension and declination. How does the modern astronomer do this? Here is an instrument which, without the circle to tell the altitude at the same time, will give some idea of the way in which the modern astronomer has to go to work. In this we have what is called the Transit Instrument, Fig. 113; it is simply used for determining the transit of stars over the meridian. It consists essentially of a telescope mounted on trunnions, like a cannon, having in the eyepiece, not simple cross wires, but a system of wires, to which reference has already been made, so that the mean of as many observations as there are wires can be taken; and in this way Tycho’s hole in the wall is completely superseded. The quadrant is represented by a circle on the instrument called the transit circle, of which for the present we defer consideration.
- 49. Fig. 112.—Tycho Brahe’s Mural Quadrant. Fig. 113.—Transit Instrument (Transit of Venus Expedition).
- 50. Fig. 114.—Transit Instalment in a fixed Observatory. Now there are three things to be done in order to adjust this instrument for observation. In the first place we must see that the line of sight is exactly at right angles to the axis on which the telescope turns, and when we have satisfied ourselves of that, we must, in the second place, take care, not only that the pivots on which the telescope rests are perfectly equal in size, but that the entire axis resting on these pivots is perfectly horizontal. Having made these two adjustments, we shall at all events be able, by swinging the telescope, to sweep through the zenith. Then, thirdly, if we take care that one end of this axis points to the east, and the other to the west, we shall know, not only that our transit instrument sweeps through the zenith, but sweeps through the pole which happens to be above the horizon—in England the north pole, in Australia the south pole. That is to say, by the first adjustment we
- 51. shall be able to describe a great circle; by the second, this circle will pass through the zenith; and by the third, from the south of the horizon to the north, through the pole. Of course, if the pole star were at the pole, all we should have to do would be to adjust the instrument (having determined the instrument to be otherwise correct) so as simply to point to the pole star, and then we should assure ourselves of the east and west positions of the axis. Some details may here be of interest. The first adjustment to be made is that the line of sight or collimation shall be at right angles to the axis on which the instrument moves: to find the error and correct it, bring the telescope into a horizontal position and place a small object at a distance away, in such a position that its image is bisected by the central wire of the transit, then lift the instrument from its bearings or Ys, as they are called, and reverse the pivots east for west, and again observe the object. If it is still bisected, the adjustment is correct, but if not, then half the angle between the new direction in which the telescope points and the first one as marked by the object is the collimation error, which may be ascertained by measuring the distance from the object to the central wire, by a micrometer in the field of view, and converting the distance into arc. To correct it, bring the central wire half way up to the object by motion of the wire, and complete the other half by moving the object itself, or by moving the Ys of the instrument. This of course must be again repeated until the adjustment is sensibly correct. The second adjustment is to make the pivots horizontal. Place a striding level on the pivots and bring the bubble to zero by the set screws of the level, or note the position of it; then reverse the level east for west, and then if the bubble remains at the same place the axis of motion is horizontal, but, if not, raise or lower the movable Y sufficiently to bring the bubble half way to its original position, and complete the motion of the bubble, if necessary, by the level screw until there is no alteration in the position of the bubble on reversing the level.
- 52. Fig. 115.—Diagram explaining third adjustment, H, R, plane of the horizon; H, Z, A, P, B, R, meridian; A and B places of circumpolar star at transit above and below pole P. The third adjustment is to place the pivots east and west. Note by the clock the time of transit of a circumpolar star, when above the pole, over the central wire, and then half a day later when below it, and again when above it; if the times from upper to lower transit, and from lower to upper are equal, then the line of collimation swings so as to bisect the circle of the star round the pole, and therefore it passes through the pole, and further it describes a meridian which passes through the zenith by reason of the second adjustment. This is therefore the meridian of the place, and therefore the pivots are east and west. If the periods between the
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