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21.2.3.56 mysql_thread_id()

unsigned long mysql_thread_id(MYSQL *mysql)

Description

Returns the thread ID of the current connection. This value can be used as an argument to mysql_kill() to kill the thread.

If the connection is lost and you reconnect with mysql_ping(), the thread ID will change. This means you should not get the thread ID and store it for later. You should get it when you need it.

Return Values

The thread ID of the current connection.

Errors

None.




ossible that with better data he might have made much more progress. He was in no hurry to publish anything, perhaps on account of possible opposition. Certainly Luther, with his obstinate conviction of the verbal accuracy of the Scriptures, rejected as mere folly the idea of a moving earth, and Melanchthon thought such opinions should be prohibited, but Rheticus, a professor at the Protestant University of Wittenberg and an enthusiastic pupil of Copernicus, urged publication, and undertook to see the work through the press. This, however, he was unable to complete and another Lutheran, Osiander, to whom he entrusted it, wrote a preface, with the apparent intention of disarming opposition, in which he stated that the principles laid down were only abstract hypotheses convenient for purposes of calculation. This unauthorised interpolation may have had its share in postponing the prohibition of the book by the Church of Rome.

According to Copernicus the earth is only a planet like the others, and not even the biggest one, while the sun is the most important body in the system, and the stars probably too far away for any motion of the earth to affect their apparent places. The earth in fact is very small in comparison with the distance of the stars, as evidenced by the fact that an observer anywhere on the earth appears to be in the middle of the universe. He shows that the revolution of the earth will account for the seasons, and for the stationary points and retrograde motions of the planets. He corrects definitely the order of the planets outwards from the sun, a matter which had been in dispute. A notable defect is due to the idea that a body can only revolve about another body or a point, as if rigidly connected with it, so that, in order to keep the earth’s axis in a constant direction in space, he has to invent a third motion. His discussion of precession, which he rightly attributes to a slow motion of the earth’s axis, is marred by the idea that the precession is variable. With all its defects, partly due to reliance on bad observations, the work showed a great advance in the interpretation of the motions of the planets; and his determinations of the periods both in relation to the earth and to the stars were adopted by Reinhold, Professor of Astronomy at Wittenberg, for the new Prutenic or Prussian Tables, which were to supersede the obsolete Alphonsine Tables of the thirteenth century.

In comparison with the question of the motion of the earth, no other astronomical detail of the time seems to be of much consequence. Comets, such as from time to time appeared, bright enough for naked eye observation, were still regarded as atmospheric phenomena, and their principal interest, as well as that of eclipses and planetary conjunctions, was in relation to astrology. Reform, however, was obviously in the air. The doctrine of Copernicus was destined very soon to divide others besides the Lutheran leaders. The leaven of inquiry was working, and not long after the death of Copernicus real advances were to come, first in the accuracy of observations, and, as a necessary result of these, in the planetary theory itself.

Chapter II.

Early Life of Kepler.

On 21st December, 1571, at Weil in the Duchy of Wurtemberg, was born a weak and sickly seven-months’ child, to whom his parents Henry and Catherine Kepler gave the name of John. Henry Kepler was a petty officer in the service of the reigning Duke, and in 1576 joined the army serving in the Netherlands. His wife followed him, leaving her young son in his grandfather’s care at Leonberg, where he barely recovered from a severe attack of smallpox. It was from this place that John derived the Latinised name of Leonmontanus, in accordance with the common practice of the time, but he was not known by it to any great extent. He was sent to school in 1577, but in the following year his father returned to Germany, almost ruined by the absconding of an acquaintance for whom he had become surety. Henry Kepler was obliged to sell his house and most of his belongings, and to keep a tavern at Elmendingen, withdrawing his son from school to help him with the rough work. In 1583 young Kepler was sent to the school at Elmendingen, and in 1584 had another narrow escape from death by a violent illness. In 1586 he was sent, at the charges of the Duke, to the monastic school of Maulbronn; from whence, in accordance with the school regulations, he passed at the end of his first year the examination for the bachelor’s degree at Tübingen, returning for two more years as a “veteran” to Maulbronn before being admitted as a resident student at Tübingen. The three years thus spent at Maulbronn were marked by recurrences of several of the diseases from which he had suffered in childhood, and also by family troubles at his home. His father went away after a quarrel with his wife Catherine, and died abroad. Catherine herself, who seems to have been of a very unamiable disposition, next quarrelled with her own relatives. It is not surprising therefore that Kepler after taking his M.A. degree in August, 1591, coming out second in the examination lists, was ready to accept the first appointment offered him, even if it should involve leaving home. This happened to be the lectureship in astronomy at Gratz, the chief town in Styria. Kepler’s knowledge of astronomy was limited to the compulsory school course, nor had he as yet any particular leaning towards the science; the post, moreover, was a meagre and unimportant one. On the other hand he had frequently expressed disgust at the way in which one after another of his companions had refused “foreign” appointments which had been arranged for them under the Duke’s scheme of education. His tutors also strongly urged him to accept the lectureship, and he had not the usual reluctance to leave home. He therefore proceeded to Gratz, protesting that he did not thereby forfeit his claim to a more promising opening, when such should appear. His astronomical tutor, Maestlin, encouraged him to devote himself to his newly adopted science, and the first result of this advice appeared before very long in Kepler’s “Mysterium Cosmographicum”. The bent of his mind was towards philosophical speculation, to which he had been attracted in his youthful studies of Scaliger’s “Exoteric Exercises”. He says he devoted much time “to the examination of the nature of heaven, of souls, of genii, of the elements, of the essence of fire, of the cause of fountains, the ebb and flow of the tides, the shape of the continents and inland seas, and things of this sort”. Following his tutor in his admiration for the Copernican theory, he wrote an essay on the primary motion, attributing it to the rotation of the earth, and this not for the mathematical reasons brought forward by Copernicus, but, as he himself says, on physical or metaphysical grounds. In 1595, having more leisure from lectures, he turned his speculative mind to the number, size, and motion of the planetary orbits. He first tried simple numerical relations, but none of them appeared to be twice, thrice, or four times as great as another, although he felt convinced that there was some relation between the motions and the distances, seeing that when a gap appeared in one series, there was a corresponding gap in the other. These gaps he attempted to fill by hypothetical planets between Mars and Jupiter, and between Mercury and Venus, but this method also failed to provide the regular proportion which he sought, besides being open to the objection that on the same principle there might be many more equally invisible planets at either end of the series. He was nevertheless unwilling to adopt the opinion of Rheticus that the number six was sacred, maintaining that the “sacredness” of the number was of much more recent date than the creation of the worlds, and could not therefore account for it. He next tried an ingenious idea, comparing the perpendiculars from different points of a quadrant of a circle on a tangent at its extremity. The greatest of these, the tangent, not being cut by the quadrant, he called the line of the sun, and associated with infinite force. The shortest, being the point at the other end of the quadrant, thus corresponded to the fixed stars or zero force; intermediate ones were to be found proportional to the “forces” of the six planets. After a great amount of unfinished trial calculations, which took nearly a whole summer, he convinced himself that success did not lie that way. In July, 1595, while lecturing on the great planetary conjunctions, he drew quasi-triangles in a circular zodiac showing the slow progression of these points of conjunction at intervals of just over 240° or eight signs. The successive chords marked out a smaller circle to which they were tangents, about half the diameter of the zodiacal circle as drawn, and Kepler at once saw a similarity to the orbits of Saturn and Jupiter, the radius of the inscribed circle of an equilateral triangle being half that of the circumscribed circle. His natural sequence of ideas impelled him to try a square, in the hope that the circumscribed and inscribed circles might give him a similar “analogy” for the orbits of Jupiter and Mars. He next tried a pentagon and so on, but he soon noted that he would never reach the sun that way, nor would he find any such limitation as six, the number of “possibles” being obviously infinite. The actual planets moreover were not even six but only five, so far as he knew, so he next pondered the question of what sort of things these could be of which only five different figures were possible and suddenly thought of the five regular solids.[2] He immediately pounced upon this idea and ultimately evolved the following scheme. “The earth is the sphere, the measure of all; round it describe a dodecahedron; the sphere including this will be Mars. Round Mars describe a tetrahedron; the sphere including this will be Jupiter. Describe a cube round Jupiter; the sphere including this will be Saturn. Now, inscribe in the earth an icosahedron, the sphere inscribed in it will be Venus: inscribe an octahedron in Venus: the circle inscribed in it will be Mercury.” With this result Kepler was inordinately pleased, and regretted not a moment of the time spent in obtaining it, though to us this “Mysterium Cosmographicum” can only appear useless, even without the more recent additions to the known planets. He admitted that a certain thickness must be assigned to the intervening spheres to cover the greatest and least distances of the several planets from the sun, but even then some of the numbers obtained are not a very close fit for the corresponding planetary orbits. Kepler’s own suggested explanation of the discordances was that they must be due to erroneous measures of the planetary distances, and this, in those days of crude and infrequent observations, could not easily be disproved. He next thought of a variety of reasons why the five regular solids should occur in precisely the order given and in no other, diverging from this into a subtle and not very intelligible process of reasoning to account for the division of the zodiac into 360°. The next subject was more important, and dealt with the relation between the distances of the planets and their times of revolution round the sun. It was obvious that the period was not simply proportional to the distance, as the outer planets were all too slow for this, and he concluded “either that the moving intelligences of the planets are weakest in those that are farthest from the sun, or that there is one moving intelligence in the sun, the common centre, forcing them all round, but those most violently which are nearest, and that it languishes in some sort and grows weaker at the most distant, because of the remoteness and the attenuation of the virtue”. This is not so near a guess at the theory of gravitation as might be supposed, for Kepler imagined that a repulsive force was necessary to account for the planets being sometimes further from the sun, and so laid aside the idea of a constant attractive force. He made several other attempts to find a law connecting the distances and periods of the planets, but without success at that time, and only desisted when by unconsciously arguing in a circle he appeared to get the same result from two totally different hypotheses. He sent copies of his book to several leading astronomers, of whom Galileo praised his ingenuity and good faith, while Tycho Brahe was evidently much struck with the work and advised him to adapt something similar to the Tychonic system instead of the Copernican. He also intimated that his Uraniborg observations would provide more accurate determinations of the planetary orbits, and thus made Kepler eager to visit him, a project which as we shall see was more than fulfilled. Another copy of the book Kepler sent to Reymers the Imperial astronomer with a most fulsome letter, which Tycho, who asserted that Reymers had simply plagiarised his work, very strongly resented, thus drawing from Kepler a long letter of apology. About the same time Kepler had married a lady already twice widowed, and become involved in difficulties with her relatives on financial grounds, and with the Styrian authorities in connection with the religious disputes then coming to a head. On account of these latter he thought it expedient, the year after his marriage, to withdraw to Hungary, from whence he sent short treatises to Tübingen, “On the magnet” (following the ideas of Gilbert of Colchester), “On the cause of the obliquity of the ecliptic” and “On the Divine wisdom as shown in the Creation”. His next important step makes it desirable to devote a chapter to a short notice of Tycho Brahe.

Footnote 2: Since the sum of the plane angles at a corner of a regular solid must be less than four right angles, it is easily seen that few regular solids are possible. Hexagonal faces are clearly impossible, or any polygonal faces with more than five sides. The possible forms are the dodecahedron with twelve pentagonal faces, three meeting at each corner; the cube, six square faces, three meeting at each corner; and three figures with triangular faces, the tetrahedron of four faces, three meeting at each corner; the octahedron of eight faces, four meeting at each corner; and the icosahedron of twenty faces, five meeting at each corner.

Chapter III.

Tycho Brahe.

The age following that of Copernicus produced three outstanding figures associated with the science of astronomy, then reaching the close of what Professor Forbes so aptly styles the geometrical period. These three Sir David Brewster has termed “Martyrs of Science”; Galileo, the great Italian philosopher, has his own place among the “Pioneers of Science”; and invaluable though Tycho Brahe’s work was, the latter can hardly be claimed as a pioneer in the same sense as the other two. Nevertheless, Kepler, the third member of the trio, could not have made his most valuable discoveries without Tycho’s observations.

Of noble family, born a twin on 14th December, 1546, at Knudstrup in Scania (the southernmost part of Sweden, then forming part of the kingdom of Denmark), Tycho was kidnapped a year later by a childless uncle. This uncle brought him up as his own son, provided him at the age of seven with a tutor, and sent him in 1559 to the University of Copenhagen, to study for a political career by taking c........................................6 1.3 Limitations............................................6 2. The Need for Exclusive XML Canonicalization.............7 2.1 A Simple Example.......................................7 2.2 General Problems with re-Enveloping....................8 3. Specification of Exclusive XML Canonicalization........10 3.1 Constrained Implementation (non-normative)............11 4. Use in XML Security....................................12 5. Security Considerations................................13 5.1 Target Context........................................13 5.2 'Esoteric' Node-sets..................................14 6. References.............................................14 7. Acknowledgements (Informative).........................15 Authors Addresses.........................................16 Full Copyright Statement..................................17 Expiration and File Name..................................17 J. Boyer, D. Eastlake 3rd, J. Reagle [Page 3] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 1. Introduction The XML Recommendation [XML] specifies the syntax of a class of objects called XML documents. The Namespaces in XML Recommendation [XML-NS] specifies additional syntax and semantics for XML documents. It is normal for XML documents and subdocuments which are equivalent for the purposes of many applications to differ in their physical representation. For example, they may differ in their entity structure, attribute ordering, and character encoding. The goal of this specification is to establish a method for serializing the XPath node-set representation of an XML document or subset such that: 1. The node-set is minimally affected by any XML context which has been omitted. 2. The canonicalization of a node-set representing well-balanced XML [XML-Fragment] will be unaltered by further applications of exclusive canonicalization. 3. It can be determined whether two node-sets are identical except for transformations considered insignificant by this specification under [XML, XML-NS]. An understanding of the Canonical XML Recommendation [XML-C14N] is required. 1.1 Terminology The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in RFC 2119 [Keywords]. The XPath 1.0 Recommendation [XPath] defines the term node-set and specifies a data model for representing an input XML document as a set of nodes of various types (element, attribute, namespace, text, comment, processing instruction, and root). The nodes are included in or excluded from a node-set based on the evaluation of an expression. Within this specification and [XML-C14N], a node-set is used to directly indicate whether or not each node should be rendered in the canonical form (in this sense, it is used as a formal mathematical set). A node that is excluded from the set is not rendered in the canonical form being generated, even if its parent node is included in the node-set. However, an omitted node may still impact the rendering of its descendants (e.g. by affecting the namespace context of the descendants). A document subset is a portion of an XML document indicated by an XPath node-set that may not include all of the nodes in the document. As defined in [XPath] every node (e.g., element, attribute, and namespace), has exactly one parent, which is either an element node J. Boyer, D. Eastlake 3rd, J. Reagle [Page 4] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 or the root node. An apex node is an element node in a document subset having no element node ancestor in the document subset. An orphan node is an element node whose parent element node is not in the document subset. The output parent of an orphan node that is not an apex node is the nearest ancestor element of the orphan node that is in the document subset; an apex node has no output parent. The output parent of a non-orphan node is the parent of the node. An output ancestor is any ancestor element node in the document subset. For example given a document tree with three generations under the root node A and where capitalization denotes the node is in the document subset (A,E,G). Pictorial Representation: [diagram of nodes, http://www.w3.org/TR/xml-exc-c14n/exc-c14n.png] Textual Representation: A-+-b `-c-+-d `-E-+-f `-G The following characteristics apply: * A is an apex node, output parent of E, and output ancestor of (E,G); * E is an orphan node and the output parent of G. An element E in a document subset visibly utilizes a namespace declaration, i.e. a namespace prefix P and bound value V, if E or an attribute node in the document subset with parent E has a qualified name in which P is the namespace prefix. A similar definition applies for an element E in a document subset that visibly utilizes the default namespace declaration, which occurs if E has no namespace prefix. The namespace axis of an element contains nodes for all non-default namespace declarations made within the element as well as non-default namespace declarations inherited from ancestors of the element. The namespace axis also contains a node representing the default namespace if it is not the empty string, whether the default namespace was declared within the element or by an ancestor of the element. Any subset of the nodes in a namespace axis can be included in a document subset. The method of canonicalization described in this specification receives an InclusiveNamespaces PrefixList parameter, which lists namespace prefixes that are handled in the manner described by the J. Boyer, D. Eastlake 3rd, J. Reagle [Page 5] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 Canonical XML Recommendation [XML-C14N]. The exclusive canonical form of a document subset is a physical representation of the XPath node-set, as an octet sequence, produced by the method described in this specification. It is as defined in the Canonical XML Recommendation [XML-C14N] except for the changes summarized as follows: * attributes in the XML namespace, such as xml:lang and xml:space are not imported into orphan nodes of the document subset, and * namespace nodes that are not on the InclusiveNamespaces PrefixList are expressed only in start tags where they are visible and if they are not in effect from an output ancestor of that tag. The term exclusive canonical XML refers to XML that is in exclusive canonical form. The exclusive XML canonicalization method is the algorithm defined by this specification that generates the exclusive canonical form of a given XML document subset. The term exclusive XML canonicalization refers to the process of applying the exclusive XML canonicalization method to an XML document subset. 1.2 Applications The applications of Exclusive XML Canonicalization are very similar to those for Canonical XML [XML-C14N]. However, exclusive canonicalization, or equivalent means of excluding most XML context, is necessary for signature applications where the XML context of signed XML will change. This sort of change is typical of many protocol applications. Note that in the case of the SignedInfo element of [XML-DSig], the specification of an appropriate canonicalization method is the only technique available to protect the signature from insignificant changes in physical form and changes in XML context. 1.3 Limitations Exclusive XML Canonicalization has the limitations of Canonical XML [XML-C14N] plus two additional limitations as follows: 1. The XML being canonicalized may depend on the effect of XML namespace attributes, such as xml:lang, xml:space, and xml:base appearing in ancestor nodes. To avoid problems due to the non- importation of such attributes into an enveloped document subset, either they must be explicitly given in the apex nodes of the XML document subset being canonicalized or they must J. Boyer, D. Eastlake 3rd, J. Reagle [Page 6] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 always be declared with an equivalent value in every context in which the XML document subset will be interpreted. 2. Applications that use the XML being canonicalized may depend on the effect of XML namespace declarations where the namespace prefix being bound is not visibly utilized. An example would be an attribute whose value is an XPath expression and whose evaluation therefore depends upon namespace prefixes referenced in the expression. Or, an attribute value might be considered a QName [XML-NS] by some applications, but it is only a string- value to XPath: 10.09. To avoid problems with such namespace declarations, o the XML must be modified so that use of the namespace prefix involved is visible, or o the namespace declarations must appear and be bound to the same values in every context in which the XML will be interpreted, or o the prefixes for such namespaces must appear in the InclusiveNamespaces PrefixList. 2. The Need for Exclusive XML Canonicalization In some cases, particularly for signed XML in protocol applications, there is a need to canonicalize a subdocument in such a way that it is substantially independent of its XML context. This is because, in protocol applications, it is common to envelope XML in various layers of message or transport elements, to strip off such enveloping, and to construct new protocol messages, parts of which were extracted from different messages previously received. If the pieces of XML in question are signed, they need to be canonicalized in a way such that these operations do not break the signature but the signature still provides as much security as can be practically obtained. 2.1 A Simple Example As a simple example of the type of problem that changes in XML context can cause for signatures, consider the following document: content this is then enveloped in another document: J. Boyer, D. Eastlake 3rd, J. Reagle [Page 7] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 content The first document above is in canonical form. But assume that document is enveloped as in the second case. The subdocument with elem1 as its apex node can be extracted from this second case with an XPath expression such as: (//. | //@* | //namespace::*)[ancestor-or-self::n1:elem1] The result of applying Canonical XML to the resulting XPath node-set is the following (except for line wrapping to fit this document): content Note that the n0 namespace has been included by Canonical XML because it includes namespace context. This change which would break a signature over elem1 based on the first version. 2.2 General Problems with re-Enveloping As a more complete example of the changes in canonical form that can occur when the enveloping context of a document subset is changed, consider the following document: And the following which has been produced by changing the enveloping of elem2: J. Boyer, D. Eastlake 3rd, J. Reagle [Page 8] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 Assume an XPath node-set produced from each case by applying the following XPath expression: (//. | //@* | //namespace::*)[ancestor-or-self::n1:elem2] Applying Canonical XML to the node-set produced from the first document yields the following serialization (except for line wrapping to fit in this document): However, although elem2 is represented by the same octet sequence in both pieces of external XML above, the Canonical XML version of elem2 from the second case would be (except for line wrapping so it will fit into this document) as follows: Note that the change in context has resulted in lots of changes in the subdocument as serialized by the inclusive Canonical XML [XML- C14N]. In the first example, n0 had been included from the context and the presence of an identical n3 namespace declaration in the context had elevated that declaration to the apex of the canonicalized form. In the second example, n0 has gone away but n2 has appeared, n3 is no longer elevated, and an xml:space declaration has appeared, due to changes in context. But not all context changes have effect. In the second example, the presence at ancestor nodes of an xml:lang and n1 prefix namespace declaration have no effect because of existing declarations at the elem2 node. On the other hand, using Exclusive XML Canonicalization as specified herein, the physical form of elem2 as extracted by the XPath expression above is (except for line wrapping so it will fit into this document) as follows: in both cases. 3. Specification of Exclusive XML Canonicalization The data model, processing, input parameters, and output data for Exclusive XML Canonicalization are the same as for Canonical XML [XML-C14N] with the following exceptions: 1. Canonical XML applied to a document subset requires the search of the ancestor nodes of each orphan element node for attributes in the XML namespace, such as xml:lang and xml:space. These are copied into the element node except if a declaration of the same attribute is already in the attribute axis of the element (whether or not it is included in the document subset). This search and copying are omitted from the Exclusive XML Canonicalization method. 2. The Exclusive XML Canonicalization method may receive an additional, possibly null, parameter InclusiveNamespaces PrefixList containing a list of namespace prefixes and/or a token indicating the presence of the default namespace. All namespace nodes appearing on this list are handled as provided in Canonical XML [XML-C14N]. 3. A namespace node N with a prefix that does not appear in the InclusiveNamespaces PrefixList is rendered if all of the conditions are met: 1. Its parent element is in the node-set, and 2. it is visibly utilized by its parent element, and 3. the prefix has not yet been rendered by any output ancestor, or the nearest output ancestor of its parent element that visibly utilizes the namespace prefix does not have a namespace node in the node-set with the same namespace prefix and value as N. 4. If the token representing the default namespace is not present in InclusiveNamespaces PrefixList, then the rules for rendering xmlns="" are changed as follows. When canonicalizing the namespace axis of an element E that is in the node-set, output xmlns="" if and only if all of the conditions are met: 1. E visibly utilizes the default namespace (i.e., it has no namespace prefix), and 2. it has no default namespace node in the node-set, and 3. the nearest output ancestor of E that visibly utilizes the default namespace has a default namespace node in the node- set. (This step for for xmlns="" is necessary because it is not J. Boyer, D. Eastlake 3rd, J. Reagle [Page 10] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 represented in the XPath data model as a namespace node, but as the absence of a namespace node; see Section 4.7 Propagation of Default Namespace Declaration in Document Subsets [XML-C14N].) 3.1 Constrained Implementation (non-normative) The following is a (non-normative) method for implementing the Exclusive XML Canonicalization method for many straightforward cases -- it assumes a well-formed subset and that if an element is in the node-set, so is all of its namespace axis; if the element is not in the subset, neither is its namespace axis. 1. Recursively process the entire tree (from which the XPath node- set was selected) in document order starting with the root. (The operation of copying ancestor xml: namespace attributes into output apex element nodes is not done.) 2. If the node is not in the XPath subset, continue to process its children element nodes recursively. 3. If the element node is in the XPath subset then output the node in accordance with Canonical XML except for namespace nodes which are rendered as follows: 1. ns_rendered is a copy of a dictionary, off the top of the state stack, of prefixes and their values which have already been rendered by an output ancestor of the namespace node's parent element. 2. Render each namespace node if and only if all of the conditions are met: 1. it is visibly utilized by the immediate parent element or one of its attributes, or is present in InclusiveNamespaces PrefixList, and 2. its prefix and value do not appear in ns_rendered. 3. Render xmlns="" if and only if all of the conditions are met: 4. 1. The default namespace is visibly utilized by the immediate parent element node, or the default prefix token is present in InclusiveNamespaces PrefixList, and 2. the element does not have a namespace node in the node-set declaring a value for the default namespace, and 3. the default namespace prefix is present in the dictionary ns_rendered. 5. Insert all the rendered namespace nodes (including xmlns="") into the ns_rendered dictionary, replacing any existing entries. Push ns_rendered onto the state stack and recurse. 6. After the recursion returns, pop the state stack. J. Boyer, D. Eastlake 3rd, J. Reagle [Page 11] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 4. Use in XML Security Exclusive Canonicalization may be used as a Transform or CanonicalizationMethod algorithm in XML Digital Signature [XML-DSig] and XML Encryption [XML-Enc]. Identifier: http://www.w3.org/2001/10/xml-exc-c14n# http://www.w3.org/2001/10/xml-exc-c14n#WithComments Just as with [XML-C14N] one may use the "#WithComments" parameter to include the serialization of XML comments. This algorithm also takes an optional explicit parameter of an empty InclusiveNamespaces element with a PrefixList attribute. The value of this attribute, which may be null, is a white space delimited list of namespace prefixes, and where #default indicates the default namespace, to be handled as per [XML-C14N]. The list is in NMTOKENS format (a white space separated list). For example: indicates the exclusive canonicalization transform, but that namespaces with prefix "dsig" or "soap" and default namespaces should be processed according to [XML-C14N]. Schema Definition: ]> DTD: 5. Security Considerations This specification is used to serialize an XPath node-set under certain assumptions given in [XML-C14N] and this specification. Three such examples include: 1. implementations of [XML-C14N] and this specification do not render an XML declaration; 2. implementations of this specification only render attributes from the "XML" namespace (e.g., xml:lang, xml:space, and xml:base) when they are in the subset being serialized; 3. implementations of this specification do not consider the appearance of a namespace prefix within an attribute value to be visibly utilized. While such choices are consistent with other XML specifications and satisfy the Working Group's application requirements it is important that an XML application carefully construct its transforms such that the result is meaningful and unambiguous in its application context. In addition to this section, the Limitations of this specification, the Resolutions of [XML-C14N], and the Security Considerations of [XML-DSig] should be carefully attended to. 5.1 Target Context The requirement of this specification is to satisfy applications that "require a method which, to the extent practical, excludes ancestor context from a canonicalized subdocument." Given a fragment being removed from its source instance, this specification satisfies this requirement by excluding from the fragment any context from its ancestors that is not utilized. Consequently, a signature [XML-DSig] over that fragment will remain valid in its source context, removed from the source context, and even in a new target context. However, this specification does not insulate the fragment against confused interpretation in a target context. J. Boyer, D. Eastlake 3rd, J. Reagle [Page 13] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 For example, if the element is signed in its source instance of and then removed and placed in the target instance , the signature should still be valid, but won't be if is interprated as belonging to the http://example.org/bar namespace: this is dependent on how nodes are processed. This specification does not define mechanisms of removing, inserting, and "fixing up" a node-set. (For an example of this sort of specification, see the processing required of Creating the Result Infoset (section 4.5) when an [XInclude] is performed.) Instead, applications must carefully specify the XML (i.e., source, fragment, and target) or define the node-set processing (i.e., removal, replacement, and insertion) with respect to default namespace declarations (e.g., xmlns="") and XML attributes (e.g., xml:lang, xml:space, and xml:base). 5.2 'Esoteric' Node-sets Consider an application that might use this specification or [XML- C14N] to serialize a single attribute node. An implementation of either specification will not emit a namespace declaration for that single attribute node. Consequently, a "carefully constructed" transform should create a node-set containing the attribute and the relevant namespace declaration for serialization. This example is provided to caution that as one moves beyond well- formed [XML] and then well-balanced XML [XML-Fragment], it becomes increasingly difficult to create a result that "is meaningful and unambiguous in its application context." 6. References [Keywords] - RFC 2119. Key words for use in RFCs to Indicate Requirement Levels. S. Bradner. Best Current Practice, March 1997. Available at http://www.ietf.org/rfc/rfc2119.txt [URI] - RFC 2396 . Uniform Resource Identifiers (URI): Generic Syntax. T. Berners-Lee, R. Fielding, and L. Masinter. Standards Track, August 1998. Available at http://www.ietf.org/rfc/rfc2396.txt [XML] - Extensible Markup Language (XML) 1.0 (Second Edition). T. Bray, E. Maler, J. Paoli, and C. M. Sperberg-McQueen. W3C Recommendation, October 2000. Available at http://www.w3.org/TR/2000/REC-xml-20001006 . J. Boyer, D. Eastlake 3rd, J. Reagle [Page 14] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 [XML-C14N] - RFC 3076. "Canonical XML", J. Boyer, March 2001. Also a W3C Recommendation available at http://www.w3.org/TR/2001/REC-xml- c14n-20010315 [XML-DSig] - XML-Signature Syntax and Processing. D. Eastlake, J. Reagle, and D. Solo. IETF Draft Standard/W3C Recommendation, August 2001. Available at http://www.w3.org/TR/2002/REC-xmldsig- core-20020212/ [XML-Fragment] - XML Fragment Interchange. P. Grosso, and D. Veillard. W3C Candidate Recommendation, February 2001. Available at http://www.w3.org/TR/2001/CR-xml-fragment-20010212 [XInclude] - XML Inclusions (XInclude) Version 1.0. J. Marsh, and D. Orchad. W3C Candidate Recommendation, February 2002. Available at http://www.w3.org/TR/2002/CR-xinclude-20020221/ [XML-NS] - Namespaces in XML. T. Bray, D. Hollander, and A. Layman. W3C Recommendation, January 1999. Available at http://www.w3.org/TR/1999/REC-xml-names-19990114/ [XML-Enc] - XML Encryption Syntax and Processing. D. Eastlake, and J. Reagle. W3C Candidate Recommendation, March 2002. Available at http://www.w3.org/TR/2002/CR-xmlenc-core-20020304/ [XML-schema] - XML Schema Part 1: Structures D. Beech, M. Maloney, N. Mendelsohn, and H. Thompson. W3C Recommendation, May 2001. Available at http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/ [XPath] - XML Path Language (XPath) Version 1.0. J. Clark and S. DeRose. W3C Recommendation, November 1999. Available at http://www.w3.org/TR/1999/REC-xpath-19991116. 7. Acknowledgements (Informative) The following people provided valuable feedback that improved the quality of this specification: * Merlin Hughes, Baltimore * Thomas Maslen, DSTC * Paul Denning, MITRE * Christian Geuer-Pollmann, University Siegen * Bob Atkinson, Microsoft J. Boyer, D. Eastlake 3rd, J. Reagle [Page 15] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 Authors Addresses John Boyer PureEdge Solutions Inc. 4396 West Saanich Rd. Victoria, BC, Canada V8Z 3E9 Phone: 1-888-517-2675 EMail: jboyer@PureEdge.com Donald E. Eastlake 3rd Motorola 155 Beaver Street Milford, MA 01757 USA Telephone: +1-508-634-2066 (h) +1-508-851-8280 (w) EMail: Donald.Eastlake@motorola.com Joseph M. Reagle Jr., W3C Massachusetts Institute of Technology Laboratory for Computer Science NE43-350, 545 Technology Square Cambridge, MA 02139 Phone: +1.617.258.7621 EMail: reagle@w3.org J. Boyer, D. Eastlake 3rd, J. Reagle [Page 16] INTERNET-DRAFT Exclusive XML Canonicalization August 2002 Full Copyright Statement Copyright (C) 2002 The Internet Society & W3C (MIT, INRIA, Keio), All Rights Reserved. This document and translations of it may be copied and furnished to others, and derivative works that comment on or otherwise explain it or assist in its implementation may be prepared, copied, published and distributed, in whole or in part, without restriction of any kind, provided that the above copyright notice and this paragraph are included on all such copies and derivative works. However, this document itself may not be modified in any way, such as by removing the copyright notice or references to the Internet Society or other Internet organizations, except as needed for the purpose of developing Internet standards in which case the procedures for copyrights defined in the Internet Standards process must be followed, or as required to translate it into languages other than English. The limited permissions granted above are perpetual and will not be revoked by the Internet Society or its successors or assigns. This document and the information contained herein is provided on an "AS IS" basis and THE INTERNET SOCIETY AND THE INTERNET ENGINEERING TASK FORCE DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Expiration and File Name This draft expires February 2003. Its file name is draft-ietf-xmldsig-xc14n-00.txt. J. Boyer, D. Eastlake 3rd, J. Reagle [Page 17] he earth. They are attracted, but in a less degree, and so are driven outwards by the heavy bodies; which being done, they stop, and are kept by the earth in their own place. But although the attractive virtue of the earth extends upwards, as has been said, so very far, yet if any stone should be at a distance great enough to become sensible compared with the earth’s diameter, it is true that on the motion of the earth such a stone would not follow altogether; its own force of resistance would be combined with the attractive force of the earth, and thus it would extricate itself in some degree from the motion of the earth.” The above passage from the Introduction to Kepler’s “Commentaries on the Motion of Mars,” always regarded as his most valuable work, must have been known to Newton, so that no such incident as the fall of an apple was required to provide a necessary and sufficient explanation of the genesis of his Theory of Universal Gravitation. Kepler’s glimpse at such a theory could have been no more than a glimpse, for he went no further with it. This seems a pity, as it is far less fanciful than many of his ideas, though not free from the “virtues” and “animal faculties,” that correspond to Gilbert’s “spirits and humours”. We must, however, proceed to the subject of Mars, which was, as before noted, the first important investigation entrusted to Kepler on his arrival at Prague.

The time taken from one opposition of Mars to the next is decidedly unequal at different parts of his orbit, so that many oppositions must be used to determine the mean motion. The ancients had noticed that what was called the “second inequality,” due as we now know to the orbital motion of the earth, only vanished when earth, sun, and planet were in line, i.e. at the planet’s opposition; therefore they used oppositions to determine the mean motion, but deemed it necessary to apply a correction to the true opposition to reduce to mean opposition, thus sacrificing part of the advantage of using oppositions. Tycho and Longomontanus had followed this method in their calculations from Tycho’s twenty years’ observations. Their aim was to find a position of the “equant,” such that these observations would show a constant angular motion about it; and that the computed positions would agree in latitude and longitude with the actual observed positions. When Kepler arrived he was told that their longitudes agreed within a couple of minutes of arc, but that something was wrong with the latitudes. He found, however, that even in longitude their positions showed discordances ten times as great as they admitted, and so, to clear the ground of assumptions as far as possible, he determined to use true oppositions. To this Tycho objected, and Kepler had great difficulty in convincing him that the new move would be any improvement, but undertook to prove to him by actual examples that a false position of the orbit could by adjusting the equant be made to fit the longitudes within five minutes of arc, while giving quite erroneous values of the latitudes and second inequalities. To avoid the possibility of further objection he carried out this demonstration separately for each of the systems of Ptolemy, Copernicus, and Tycho. For the new method he noticed that great accuracy was required in the reduction of the observed places of Mars to the ecliptic, and for this purpose the value obtained for the parallax by Tycho’s assistants fell far short of the requisite accuracy. Kepler therefore was obliged to recompute the parallax from the original observations, as also the position of the line of nodes and the inclination of the orbit. The last he found to be constant, thus corroborating his theory that the plane of the orbit passed through the sun. He repeated his calculations no fewer than seventy times (and that before the invention of logarithms), and at length adopted values for the mean longitude and longitude of aphelion. He found no discordance greater than two minutes of arc in Tycho’s observed longitudes in opposition, but the latitudes, and also longitudes in other parts of the orbit were much more discordant, and he found to his chagrin that four years’ work was practically wasted. Before making a fresh start he looked for some simplification of the labour; and determined to adopt Ptolemy’s assumption known as the principle of the bisection of the excentricity. Hitherto, since Ptolemy had given no reason for this assumption, Kepler had preferred not to make it, only taking for granted that the centre was at some point on the line called the excentricity (see Figs. 1, 2).

A marked improvement in residuals was the result of this step, proving, so far, the correctness of Ptolemy’s principle, but there still remained discordances amounting to eight minutes of arc. Copernicus, who had no idea of the accuracy obtainable in observations, would probably have regarded such an agreement as remarkably good; but Kepler refused to admit the possibility of an error of eight minutes in any of Tycho’s observations. He thereupon vowed to construct from these eight minutes a new planetary theory that should account for them all. His repeated failures had by this time convinced him that no uniformly described circle could possibly represent the motion of Mars. Either the orbit could not be circular, or else the angular velocity could not be constant about any point whatever. He determined to attack the “second inequality,” i.e. the optical illusion caused by the earth’s annual motion, but first revived an old idea of his own that for the sake of uniformity the sun, or as he preferred to regard it, the earth, should have an equant as well as the planets. From the irregularities of the solar motion he soon found that this was the case, and that the motion was uniform about a point on the line from the sun to the centre of the earth’s orbit, such that the centre bisected the distance from the sun to the “Equant”; this fully supported Ptolemy’s principle. Clearly then the earth’s linear velocity could not be constant, and Kepler was encouraged to revive another of his speculations as to a force which was weaker at greater distances. He found the velocity greater at the nearer apse, so that the time over an equal arc at either apse was proportional to the distance. He conjectured that this might prove to be true for arcs at all parts of the orbit, and to test this he divided the orbit into 360 equal parts, and calculated the distances to the points of division. Archimedes had obtained an approximation to the area of a circle by dividing it radially into a very large number of triangles, and Kepler had this device in mind. He found that the sums of successive distances from his 360 points were approximately proportional to the times from point to point, and was thus enabled to represent much more accurately the annual motion of the earth which produced the second inequality of Mars, to whose motion he now returned. Three points are sufficient to define a circle, so he took three observed positions of Mars and found a circle; he then took three other positions, but obtained a different circle, and a third set gave yet another. It thus began to appear that the orbit could not be a circle. He next tried to divide into 360 equal parts, as he had in the case of the earth, but the sums of distances failed to fit the times, and he realised that the sums of distances were not a good measure of the area of successive triangles. He noted, however, that the errors at the apses were now smaller than with a central circular orbit, and of the opposite sign, so he determined to try whether an oval orbit would fit better, following a suggestion made by Purbach in the case of Mercury, whose orbit is even more eccentric than that of Mars, though observations were too scanty to form the foundation of any theory. Kepler gave his fancy play in the choice of an oval, greater at one end than the other, endeavouring to satisfy some ideas about epicyclic motion, but could not find a satisfactory curve. He then had the fortunate idea of trying an ellipse with the same axis as his tentative oval. Mars now appeared too slow at the apses instead of too quick, so obviously some intermediate ellipse must be sought between the trial ellipse and the circle on the same axis. At this point the “long arm of coincidence” came into play. Half-way between the apses lay the mean distance, and at this position the error was half the distance between the ellipse and the circle, amounting to .00429 of a radius. With these figures in his mind, Kepler looked up the greatest optical inequality of Mars, the angle between the straight lines from Mars to the Sun and to the centre of the circle.[3] The secant of this angle was 1.00429, so that he noted that an ellipse reduced from the circle in the ratio of 1.00429 to 1 would fit the motion of Mars at the mean distance as well as the apses.

Footnote 3: This is clearly a maximum at AMC in Fig. 2, when its tangent AC / CM = the eccentricity.

It is often said that a coincidence like this only happens to somebody who “deserves his luck,” but this simply means that recognition is essential to the coincidence. In the same way the appearance of one of a large number of people mentioned is hailed as a case of the old adage “Talk of the devil, etc.,” ignoring all the people who failed to appear. No one, however, will consider Kepler unduly favoured. His genius, in his case certainly “an infinite capacity for taking pains,” enabled him out of his medley of hypotheses, mainly unsound, by dint of enormous labour and patience, to arrive thus at the first two of the laws which established his title of “Legislator of the Heavens”.

Figures Explanatory of Kepler’s Theory of the Motion of Mars.

[Illustration: Fig. 1.]
Fig. 1.
[Illustration: Fig. 2.]
Fig. 2.

Fig. 1.—In Ptolemy’s excentric theory, A may be taken to represent the earth, C the centre of a planet’s orbit, and E the equant, P (perigee) and Q (apogee) being the apses of the orbit. Ptolemy’s idea was that uniform motion in a circle must be provided, and since the motion was not uniform about the earth, A could not coincide with C; and since the motion still failed to be uniform about A or C, some point E must be found about which the motion should be uniform.

Fig. 2.—This is not drawn to scale, but is intended to illustrate Kepler’s modification of Ptolemy’s excentric. Kepler found velocities at P and Q proportional not to AP and AQ but to AQ and AP, or to EP and EQ if EC = CA (bisection of the excentricity). The velocity at M was wrong, and AM appeared too great. Kepler’s first ellipse had M moved too near C. The distance AC is much exaggerated in the figure, as also is MN. AN = CP, the radius of the circle. MN should be .00429 of the radius, and MC / NC should be 1.00429. The velocity at N appeared to be proportional to EN ( = AN). Kepler concluded that Mars moved round PNQ, so that the area described about A (the sun) was equal in equal times, A being the focus of the ellipse PNQ. The angular velocity is not quite constant about E, the equant or empty focus, but the difference could hardly have been detected in Kepler’s time.

Kepler’s improved determination of the earth’s orbit was obtained by plotting the different positions of the earth corresponding to successive rotations of Mars, i.e. intervals of 687 days. At each of these the date of the year would give the angle MSE (Mars-Sun-Earth), and Tycho’s observation the angle MES. So the triangle could be solved except for scale, and the ratio of SE to SM would give the distance of Mars from the sun in terms of that of the earth. Measuring from a fixed position of Mars (e.g. perihelion), this gave the variation of SE, showing the earth’s inequality. Measuring from a fixed position of the earth, it would give similarly a series of positions of Mars, which, though lying not far from the circle whose diameter was the axis of Mars’ orbit, joining perihelion and aphelion, always fell inside the circle except at those two points. It was a long time before it dawned upon Kepler that the simplest figure falling within the circle except at the two extremities of the diameter, was an ellipse, and it is not clear why his first attempt with an ellipse should have been just as much too narrow as the circle was too wide. The fact remains that he recognised suddenly that halving this error was tantamount to reducing the circle to the ellipse whose eccentricity was that of the old theory, i.e. that in which the sun would be in one focus and the equant in the other.

Having now fitted the ends of both major and minor axes of the ellipse, he leaped to the conclusion that the orbit would fit everywhere.

The practical effect of his clearing of the “second inequality” was to refer the orbit of Mars directly to the sun, and he found that the area between successive distances of Mars from the sun (instead of the sum of the distances) was strictly proportional to the time taken, in short, equal areas were described in equal times (2nd Law) when referred to the sun in the focus of the ellipse (1st Law).

He announced that (1) The planet describes an ellipse, the sun being in one focus; and (2) The straight line joining the planet to the sun sweeps out equal areas in any two equal intervals of time. These are Kepler’s first and second Laws though not discovered in that order, and it was at once clear that Ptolemy’s “bisection of the excentricity” simply amounted to the fact that the centre of an ellipse bisects the distance between the foci, the sun being in one focus and the angular velocity being uniform about the empty focus. For so many centuries had the fetish of circular motion postponed discovery. It was natural that Kepler should assume that his laws would apply equally to all the planets, but the proof of this, as well as the reason underlying the laws, was only given by Newton, who approached the subject from a totally different standpoint.

This commentary on Mars was published in 1609, the year of the invention of the telescope, and Kepler petitioned the Emperor for further funds to enable him to complete the study of the other planets, but once more there was delay; in 1612 Rudolph died, and his brother Matthias who succeeded him, cared very little for astronomy or even astrology, though Kepler was reappointed to his post of Imperial Mathematician. He left Prague to take up a permanent professorship at the University of Linz. His own account of the circumstances is gloomy enough. He says, “In the first place I could get no money from the Court, and my wife, who had for a long time been suffering from low spirits and despondency, was taken violently ill towards the end of 1610, with the Hungarian fever, epilepsy and phrenitis. She was scarcely convalescent when all my three children were at once attacked with smallpox. Leopold with his army occupied the town beyond the river just as I lost the dearest of my sons, him whose nativity you will find in my book on the new star. The town on this side of the river where I lived was harassed by the Bohemian troops, whose new levies were insubordinate and insolent; to complete the whole, the Austrian army brought the plague with them into the city. I went into Austria and endeavoured to procure the situation which I now hold. Returning in June, I found my wife in a decline from her grief at the death of her son, and on the eve of an infectious fever, and I lost her also within eleven days of my return. Then came fresh annoyance, of course, and her fortune was to be divided with my step-sisters. The Emperor Rudolph would not agree to my departure; vain hopes were given me of being paid from Saxony; my time and money were wasted together, till on the death of the Emperor in 1612, I was named again by his successor, and suffered to depart to Linz.”

Being thus left a widower with a ten-year-old daughter Susanna, and a boy Louis of half her age, he looked for a second wife to take charge of them. He has given an account of eleven ladies whose suitability he considered. The first, an intimate friend of his first wife, ultimately declined; one was too old, another an invalid, another too proud of her birth and quarterings, another could do nothing useful, and so on. Number eight kept him guessing for three months, until he tired of her constant indecision, and confided his disappointment to number nine, who was not impressed. Number ten, introduced by a friend, Kepler found exceedingly ugly and enormously fat, and number eleven apparently too young. Kepler then reconsidered one of the earlier ones, disregarding the advice of his friends who objected to her lowly station. She was the orphan daughter of a cabinetmaker, educated for twelve years by favour of the Lady of Stahrenburg, and Kepler writes of her: “Her person and manners are suitable to mine; no pride, no extravagance; she can bear to work; she has a tolerable knowledge of how to manage a family; middle-aged and of a disposition and capability to acquire what she still wants”.

Wine from the Austrian vineyards was plentiful and cheap at the time of the marriage, and Kepler bought a few casks for his household. When the seller came to ascertain the quantity, Kepler noticed that no proper allowance was made for the bulging parts, and the upshot of his objections was that he wrote a book on a new method of gauging—one of the earliest specimens of modern analysis, extending the properties of plane figures to segments of cones and cylinders as being “incorporated circles”. He was summoned before the Diet at Ratisbon to give his opinion on the Gregorian Reform of the Calendar, and soon afterwards was excommunicated, having fallen foul of the Roman Catholic party at Linz just as he had previously at Gratz, the reason apparently being that he desired to think for himself. Meanwhile his salary was not paid any more regularly than before, and he was forced to supplement it by publishing what he called a “vile prophesying almanac which is scarcely more respectable than begging unless it be because it saves the Emperor’s credit, who abandons me entirely, and with all his frequent and recent orders in council, would suffer me to perish with hunger”.

In 1617 he was invited to Italy to succeed Magini as Professor of Mathematics at Bologna. Galileo urged him to accept the post, but he excused himself on the ground that he was a German and brought up among Germans with such liberty of speech as he thought might get him into trouble in Italy. In 1619 Matthias died and was succeeded by Ferdinand III, who again retained Kepler in his post. In the same year Kepler reprinted his “Mysterium Cosmographicum,” and also published his “Harmonics” in five books dedicated to James I of England. “The first geometrical, on the origin and demonstration of the laws of the figures which produce harmonious proportions; the second, architectonical, on figurate geometry and the congruence of plane and solid regular figures; the third, properly Harmonic, on the derivation of musical proportions from figures, and on the nature and distinction of things relating to song, in opposition to the old theories; the fourth, metaphysical, psychological, and astrological, on the mental essence of Harmonics, and of their kinds in the world, especially on the harmony of rays emanating on the earth from the heavenly bodies, and on their effect in nature and on the sublunary and human soul; the fifth, astronomical and metaphysical, on the very exquisite Harmonics of the celestial motions and the origin of the excentricities in harmonious proportions.” The extravagance of his fancies does not appear until the fourth book, in which he reiterates the statement that he was forced to adopt his astrological opinions from direct and positive observation. He despises “The common herd of prophesiers who describe the operations of the stars as if they were a sort of deities, the lords of heaven and earth, and producing everything at their pleasure. They never trouble themselves to consider what means the stars have of working any effects among us on the earth whilst they remain in the sky and send down nothing to us which is obvious to the senses, except rays of light.” His own notion is “Like one who listens to a sweet melodious song, and by the gladness of his countenance, by his voice, and by the beating of his hand or foot attuned to the music, gives token that he perceives and approves the harmony: just so does sublunary nature, with the notable and evident emotion of the bowels of the earth, bear like witness to the same feelings, especially at those times when the rays of the planets form harmonious configurations on the earth,” and again “The earth is not an animal like a dog, ready at every nod; but more like a bull or an elephant, slow to become angry, and so much the more furious when incensed.” He seems to have believed the earth to be actually a living animal, as witness the following: “If anyone who has climbed the peaks of the highest mountains, throw a stone down their very deep clefts, a sound is heard from them; or if he throw it into one of the mountain lakes, which beyond doubt are bottomless, a storm will immediately arise, just as when you thrust a straw into the ear or nose of a ticklish animal, it shakes its head, or runs shudderingly away. What so like breathing, especially of those fish who draw water into their mouths and spout it out again through their gills, as that wonderful tide! For although it is so regulated according to the course of the moon, that, in the preface to my ‘Commentaries on Mars,’ I have mentioned it as probable that the waters are attracted by the moon, as iron by the loadstone, yet if anyone uphold that the earth regulates its breathing according to the motion of the sun and moon, as animals have daily and nightly alternations of sleep and waking, I shall not think his philosophy unworthy of being listened to; especially if any flexible parts should be discovered in the depths of the earth, to supply the functions of lungs or gills.”

In the same book Kepler enlarges again on his views in reference to the basis of astrology as concerned with nativities and the importance of planetary conjunctions. He gives particulars of his own nativity. “Jupiter nearest the nonagesimal had passed by four degrees the trine of Saturn; the Sun and Venus in conjunction were moving from the latter towards the former, nearly in sextiles with both: they were also removing from quadratures with Mars, to which Mercury was closely approaching: the moon drew near to the trine of the same planet, close to the Bull’s Eye even in latitude. The 25th degree of Gemini was rising, and the 22nd of Aquarius culminating. That there was this triple configuration on that day—namely the sextile of Saturn and the Sun, the sextile of Mars and Jupiter, and the quadrature of Mercury and Mars, is proved by the change of weather; for after a frost of some days, that very day became warmer, there was a thaw and a fall of rain.” This alleged “proof” is interesting as it relies on the same principle which was held to justify the correction of an uncertain birth-time, by reference to illnesses, etc., met with later. Kepler however goes on to say, “If I am to speak of the results of my studies, what, I pray, can I find in the sky, even remotely alluding to it? The learned confess that several not despicable branches of philosophy have been newly extricated or amended or brought to perfection by me: but here my constellations were, not Mercury from the East in the angle of the seventh, and in quadratures with Mars, but Copernicus, but Tycho Brahe, without whose books of observations everything now set by me in the clearest light must have remained buried in darkness; not Saturn predominating Mercury, but my lords the Emperors Rudolph and Matthias, not Capricorn the house of Saturn but Upper Austria, the house of the Emperor, and the ready and unexampled bounty of his nobles to my petition. Here is that corner, not the western one of the horoscope, but on the earth whither, by permission of my Imperial master, I have betaken myself from a too uneasy Court; and whence, during these years of my life, which now tends towards its setting, emanate these Harmonics and the other matters on which I am engaged.”

The fifth book contains a great deal of nonsense about the harmony of the spheres; the notes contributed by the several planets are gravely set down, that of Mercury having the greatest resemblance to a melody, though perhaps more reminiscent of a bugle-call. Yet the book is not all worthless for it includes Kepler’s Third Law, which he had diligently sought for years. In his own words, “The proportion existing between the periodic times of any two planets is exactly the sesquiplicate proportion of the mean distances of the orbits,” or as generally given, “the squares of the periodic times are proportional to the cubes of the mean distances.” Kepler was evidently transported with delight and wrote, “What I prophesied two and twenty years ago, as soon as I discovered the five solids among the heavenly orbits,—what I firmly believed long before I had seen Ptolemy’s ‘Harmonics’—what I had promised my friends in the title of this book, which I named before I was sure of my discovery,—what sixteen years ago I urged as a thing to be sought,—that for which I joined Tycho Brahe, for which I settled in Prague, for which I have devoted the best part of my life to astronomical computations, at length I have brought to light, and have recognised its truth beyond my most sanguine expectations. Great as is the absolute nature of Harmonics, with all its details as set forth in my third book, it is all found among the celestial motions, not indeed in the manner which I imagined (that is not the least part of my delight), but in another very different, and yet most perfect and excellent. It is now eighteen months since I got the first glimpse of light, three months since the dawn, very few days since the unveiled sun, most admirable to gaze on, burst out upon me. Nothing holds me; I will indulge in my sacred fury; I will triumph over mankind by the honest confession that I have stolen the golden vases of the Egyptians to build up a tabernacle for my God far away from the confines of Egypt. If you forgive me, I rejoice, if you are angry, I can bear it; the die is cast, the book is written; to be read either now or by posterity, I care not which; it may well wait a century for a reader, as God has waited six thousand years for an observer.” He gives the date 15th May, 1618, for the completion of his discovery. In his “Epitome of the Copernican Astronomy,” he gives his own idea as to the reason for this Third Law. “Four causes concur for lengthening the periodic time. First, the length of the path; secondly, the weight or quantity of matter to be carried; thirdly, the degree of strength of the moving virtue; fourthly, the bulk or space into which is spread out the matter to be moved. The orbital paths of the planets are in the simple ratio of the distances; the weights or quantities of matter in different planets are in the subduplicate ratio of the same distances, as has been already proved; so that with every increase of distance a planet has more matter and therefore is moved more slowly, and accumulates more time in its revolution, requiring already, as it did, more time by reason of the length of the way. The third and fourth causes compensate each other in a comparison of different planets; the simple and subduplicate proportion compound the sesquiplicate proportion, which therefore is the ratio of the periodic times.” The only part of this “explanation” that is true is that the paths are in the simple ratio of the distances, the “proof” so confidently claimed being of the circular kind commonly known as “begging the question”. It was reserved for Newton to establish the Laws of Motion, to find the law of force that would constrain a planet to obey Kepler’s first and second Laws, and to prove that it must therefore also obey the third.

Chapter VI.

Closing Years.

Soon after its publication Kepler’s “Epitome” was placed along with the book of Copernicus, on the list of books prohibited by the Congregation of the Index at Rome, and he feared that this might prevent the publication or sale of his books in Austria also, but was told that though Galileo’s violence was getting him into trouble, there would be no difficulty in obtaining permission for learned men to read any prohibited books, and that he (Kepler) need fear nothing so long as he remained quiet.

In his various works on Comets, he adhered to the opinion that they travelled in straight lines with varying velocity. He suggested that comets come from the remotest parts of ether, as whales and monsters from the depth of the sea, and that perhaps they are something of the nature of silkworms, and are wasted and consumed in spinning their own tails. Napier’s invention of logarithms at once attracted Kepler’s attention. He must have regretted that the discovery was not made early enough to save him a vast amount of labour in computations, but he managed to find time to compute some logarithm tables for himself, though he does not seem to have understood quite what Napier had done, and though with his usual honesty he gave full credit to the Scottish baron for his invention.

Though Eugenists may find a difficulty in reconciling Napier’s brilliancy with the extreme youth of his parents, they may at any rate attribute Kepler’s occasional fits of bad temper to heredity. His cantankerous mother, Catherine Kepler, had for some years been carrying on an action for slander against a woman who had accused her of administering a poisonous potion. Dame Kepler employed a young advocate who for reasons of his own “nursed” the case so long that after five years had elapsed without any conclusion being reached another judge was appointed, who had himself suffered from the caustic tongue of the prosecutrix, and so was already prejudiced against her. The defendant, knowing this, turned the tables on her opponent by bringing an accusation of witchcraft against her, and Catherine Kepler was imprisoned and condemned to the torture in July, 1620. Kepler, hearing of the sentence, hurried back from Linz, and succeeded in stopping the completion of the sentence, securing his mother’s release the following year, as it was made clear that the only support for the case against her was her own intemperate language. Kepler returned to Linz, and his mother at once brought another action for costs and damages against her late opponent, but died before the case could be tried.

A few months before this Sir Henry Wotton, English Ambassador to Venice, visited Kepler, and finding him as usual, almost penniless, urged him to go to England, promising him a warm welcome there. Kepler, however, would not at that time leave Germany, giving several reasons, one of which was that he dreaded the confinement of an island. Later on he expressed his willingness to go as soon as his Rudolphine Tables were published, and lecture on them, even in England, if he could not do it in Germany, and if a good enough salary were forthcoming.

In 1624 he went to Vienna, and managed to extract from the Treasury 6000 florins on account of expenses connected with the Tables, but, instead of a further grant, was given letters to the States of Swabia, which owed money to the Imperial treasury. Some of this he succeeded in collecting, but the Tables were still further delayed by the religious disturbances then becoming violent. The Jesuits contrived to have Kepler’s library sealed up, and, but for the Imperial protection, would have imprisoned him also; moreover the peasants revolted and blockaded Linz. In 1627, however, the long promised Tables, the first to discard the conventional circular motion, were at last published at Ulm in four parts. Two of these parts consisted of subsidiary Tables, of logarithms and other computing devices, another contained Tables of the elements of the sun, moon, and planets, and the fourth gave the places of a thousand stars as determined by Tycho, with Tycho’s refraction Tables, which had the peculiarity of using different values for the refraction of the sun, moon, and stars. From a map prefixed to some copies of the Tables, we may infer that Kepler was one of the first, if not actually the first, to suggest the method of determining differences of longitude by occultations of stars at the moon’s limb. In an Appendix, he showed how his Tables could be used by astrologers for their predictions, saying “Astronomy is the daughter of Astrology, and this modern Astrology again is the daughter of Astronomy, bearing something of the lineaments of her grandmother; and, as I have already said, this foolish daughter, Astrology, supports her wise but needy mother, Astronomy, from the profits of a profession not generally considered creditable”. There is no doubt that Kepler strongly resented having to depend so much for his income on such methods which he certainly did not consider creditable.

It was probably Galileo whose praise of the new Tables induced the Grand Duke of Tuscany to send Kepler a gold chain soon after their publication, and we may perhaps regard it as a mark of favour from the Emperor Ferdinand that he permitted Kepler to attach himself to the great Wallenstein, now Duke of Friedland, and a firm believer in Astrology. The Duke was a better paymaster than either of the three successive Emperors. He furnished Kepler with an assistant and a printing press; and obtained for him the Professorship of Astronomy at the University of Rostock in Mecklenburg. Apparently, however, the Emperor could not induce Wallenstein to take over the responsibility of the 8000 crowns, still owing from the Imperial treasury on account of the Rudolphine Tables. Kepler made a last attempt to secure payment at Ratisbon, but his journey thither brought disappointment and fatigue and left him in such a condition that he rapidly succumbed to an attack of fever, dying in November, 1630, in his fifty-ninth year. His body was buried at Ratisbon, but the tombstone was destroyed during the war then raging. His daughter, Susanna, the wife of Jacob Bartsch, a physician who had helped Kepler with his Ephemeris, lost her husband soon after her father’s death, and succeeded in obtaining part of Kepler’s arrears of salary by threatening to keep Tycho’s manuscripts, but her stepmother was left almost penniless with five young children. For their benefit Louis Kepler printed a “Dream of Lunar Astronomy,” which first his father and then his brother-in-law had been preparing for publication at the time of their respective deaths. It is a curious mixture of saga and fairy tale with a little science in the way of astronomy studied from the moon, and cast in the form of a dream to overcome the practical difficulties of the hypothesis of visiting the moon. Other writings in large numbers were left unpublished. No attempt at a complete edition of Kepler’s works was made for a long time. One was projected in 1714 by his biographer, Hantsch, but all that appeared was one volume of letters. After various learned bodies had declined to move in the matter the manuscripts were purchased for the Imperial Russian library. An edition was at length brought out at Frankfort by C. Frisch, in eight volumes, appearing at intervals from 1858-1870.

Kepler’s fame does not rest upon his voluminous works. With his peculiar method of approaching problems there was bound to be an inordinate amount of chaff mixed with the grain, and he used no winnowing machine. His simplicity and transparent honesty induced him to include everything, in fact he seemed to glory in the number of false trails he laboriously followed. He was one who might be expected to find the proverbial “needle in a haystack,” but unfortunately the needle was not always there. Delambre says, “Ardent, restless, burning to distinguish himself by his discoveries he attempted everything, and having once obtained a glimpse of one, no labour was too hard for him in following or verifying it. All his attempts had not the same success, and in fact that was impossible. Those which have failed seem to us only fanciful; those which have been more fortunate appear sublime. When in search of that which really existed, he has sometimes found it; when he devoted himself to the pursuit of a chimera, he could not but fail, but even then he unfolded the same qualities, and that obstinate perseverance that must triumph over all difficulties but those which are insurmountable.” Berry, in his “Short History of Astronomy,” says “as one reads chapter after chapter without a lucid, still less a correct idea, it is impossible to refrain from regrets that the intelligence of Kepler should have been so wasted, and it is difficult not to suspect at times that some of the valuable results which lie embedded in this great mass of tedious speculation were arrived at by a mere accident. On the other hand it must not be forgotten that such accidents have a habit of happening only to great men, and that if Kepler loved to give reins to his imagination he was equally impressed with the necessity of scrupulously comparing speculative results with observed facts, and of surrendering without demur the most beloved of his fancies if it was unable to stand this test. If Kepler had burnt three-quarters of what he printed, we should in all probability have formed a higher opinion of his intellectual grasp and sobriety of judgment, but we should have lost to a great extent the impression of extraordinary enthusiasm and industry, and of almost unequalled intellectual honesty which we now get from a study of his works.”

Professor Forbes is more enthusiastic. In his “History of Astronomy,” he refers to Kepler as “the man whose place, as is generally agreed, would have been the most difficult to fill among all those who have contributed to the advance of astronomical knowledge,” and again à propos of Kepler’s great book, “it must be obvious that he had at that time some inkling of the meaning of his laws—universal gravitation. From that moment the idea of universal gravitation was in the air, and hints and guesses were thrown out by many; and in time the law of gravitation would doubtless have been discovered, though probably not by the work of one man, even if Newton had not lived. But, if Kepler had not lived, who else could have discovered his Laws?”

Appendix I.

List of Dates.

  1. Johann Kepler, born 1571;
  2. school at Maulbronn, 1586;
  3. University of Tübingen, 1589;
  4. M.A. of Tübingen, 1591;
  5. Professor at Gratz, 1594;
  6. “Prodromus,” with “Mysterium Cosmographicum,” published 1596;
  7. first marriage, 1597;
  8. joins Tycho Brahe at Prague, 1600;
  9. death of Tycho, 1601;
  10. Kepler’s optics, 1603;
  11. Nova, 1604;
  12. on Comets, 1607;
  13. Commentary on Mars, including First and Second Laws, 1609;
  14. Professor at Linz, 1612;
  15. second marriage, 1613;
  16. Third Law discovered, 1618;
  17. Epitome of Copernican Astronomy, 1618-1621;
  18. Rudolphine Tables published, 1627;
  19. died, 1630.

Appendix II.

Bibliography.

For a full account of the various systems of Kepler and his predecessors the reader cannot do better than consult the “History of the Planetary Systems, from Thales to Kepler,” by Dr. J.L.E. Dreyer (Cambridge Univ. Press, 1906). The same author’s “Tycho Brahe” gives a wealth of detail about that “Phœnix of Astronomers,” as Kepler styles him. A great proportion of the literature relating to Kepler is German, but he has his place in the histories of astronomy, from Delambre and the more modern R. Wolfs “Geschichte” to those of A. Berry, “History of Astronomy” (University Extension Manuals, Murray, 1898), and Professor G. Forbes, “History of Astronomy” (History of Science Series, Watts, 1909).

Glossary.

Apogee:
The point in the orbit of a celestial body when it is furthest from the earth.
Apse:
An extremity of the major axis of the orbit of a body; a body is at its greatest and least distances from the body about which it revolves, when at one or other apse.
Conjunction:
When a plane containing the earth’s axis and passing through the centre of the sun also passes through that of the moon or a planet, at the same side of the earth, the moon or planet is in conjunction, or if on opposite sides of the earth, the moon or planet is in opposition. Mercury and Venus cannot be in opposition, but are in inferior or superior conjunction according as they are nearer or further than the sun.
Deferent:
In the epicyclic theory, uneven motion is represented by motion round a circle whose centre travels round another circle, the latter is called the deferent.
Ecliptic:
The plane of the earth’s orbital motion about the sun, which cuts the heavens in a great circle. It is so called because obviously eclipses can only occur when the moon is also approximately in this plane, besides being in conjunction or opposition with the sun.
Epicycle:
A point moving on the circumference of a circle whose centre describes another circle, traces an epicycle with reference to the centre of the second circle.
Equant:
In Ptolemy’s excentric theory, when a planet is describing a circle about a centre which is not the earth, in order to satisfy the convention that the motion must be uniform, a point was found about which the motion was apparently uniform,[4] and this point was called the equant.
Footnote 4: I.e. the angular motion about the equant was uniform.
Equinox:
When the sun is in the plane of the earth’s equator the lengths of day and night are equal. This happens twice a year, and the times when the sun passes the equator are called the vernal or spring equinox and the autumnal equinox respectively.
Evection:
The second inequality of the moon, which vanishes at new and full moon and is a maximum at first and last quarter.
Excentric:
As an alternative to epicycles, planets whose motion round the earth was not uniform could be represented as moving round a point some distance from the earth called the excentric.
Geocentric:
Referred to the centre of the earth; e.g. Ptolemy’s theory.
Heliocentric:
Referred to the centre of the sun; e.g. the theory commonly called Copernican.
Inequality:
The difference between the actual position of a planet and its theoretical position on the hypothesis of uniform circular motion.
Node:
The points where the orbit of the moon or a planet intersect the plane of the ecliptic. The ascending node is the one when the planet is moving northwards, and the line of intersection of the orbital plane with the ecliptic is the line of nodes.
Occultation:
Usually means when a planet or star is hidden by the moon, but it also includes “occultation” of a star by a planet or of a satellite by a planet or of one planet by another.
Opposition
v. Conjunction.
Parallax:
The error introduced by observing from some point other than that required in theory, e.g. in geocentric places because the observations are made from the surface of the earth instead of the centre, or in heliocentric places because observations are made from the earth and not from the sun.
Perigee:
The point in the orbit of a celestial body when it is nearest to the earth.
Precession:
Owing to the slow motion of the earth’s pole around the pole of the ecliptic, the equator cuts the ecliptic a little earlier every year, so that the equinox each year slightly precedes, with reference to the stars, that of the previous year. 





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