Extrapolation theory with applications, Nummers 438-440

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Wadler continued to write and lecture throughout his life. Rosch-Pinnah, Eliyahu. Astrologie im hebraeischen Alphabet. Eliyahu Rosch-Pinnah is Ernst Ettisch. Moran, Hugh. The Alphabet and Ancient Calendar Signs. Our alphabet uses a visual mark letter to represent a sound of spoken language, rather than a complete idea i. Hugh Moran claimed to have established a definite link between the shapes of the Semitic letters of our phonetic alphabet and those of the Chinese lunar zodiac. According to Moran the underlying unifying force was religion. It is sometimes claimed that Moran was a Sinologist.

He was not. Victor Mair, Professor of Asian and Middle Eastern Studies, University of Pennsylvania, has also investigated the possible connection between the lunar calendar and the origin of the alphabet. His books were all religious in nature. He attended Wadham College, Oxford, and received a B. He was ordained in In he married Irene Hornby.

At University of Oxford Wadham College? On his permanent return to the USA, he became a member of the board of chaplains at Cornell University and remained there for over 20 years. He retired in to Los Altos Palo Alto? David Kelley ? He has a PhD in archaeology from Harvard University. He has taught at the University of Nebraska but much of his academic career has been with the University of Calgary Professor of Archaeology.

Points in his essay in the book that Kelley considered to be wrong were corrected in his paper "Indo-Greek cosmology and science in ancient Meso-America. Kelly, David. The lists of calendar names and deities day names, lunar houses or constellations, and the deities connected with them in India, China, Southeast Asia Greece and Polynesia are compared with the Aztec and Maya lists. It is shown that they have degrees of similarity that can only be explained by historical contact. It is also shown that the letters of the Hebrew-Greek alphabet may be derived from the sequences of lunar mansions, and that they also show some similarity with the list of Maya day names.

The author postulates trans-Pacific contact in the period between the seventh and fourth centuries B. Gordon, Cyrus. He became a leading Semitic languages scholar and had over publications to his credit. Gordon, an authority on Ugarit and Ugaritic a Semitic language used by Late Bronze Age Canaanites at Ugarit, Syria, circa BCE; and written with a unique cuneiform alphabet , was a proponent of numerous eccentric causes. He believed he had identified a lunar zodiac in the order of the letters of the Ugarit cuneiform alphabet. Gordon's identification of a lunar zodiac in the letters of the Ugarit cuneiform alphabet has not met with general acceptance.

Life dates Bausani, Alessandro. The order followed by our own letters a, b, c, d, e Why are the signs in this order and not another? All this in an astronomical situation where the full moon of the autumn equinox occurred near the Pleiades, that is, around or B. The latter date corresponds exactly to what we have said about the beginnings of the alphabet.

In its traditional order, the alphabet would then amount to a sort of primitive calendar, however approximate, which was worked out in a region like the Near East where summer, with its attendant drought, was felt to be especially inimical. The hypothesis of the calendar-like nature of the alphabet as a whole, and in particular the order of the signs which, as Bausani says, 'probably symbolized the days of a complete lunation felt to be more benevolent than others', has been significantly confirmed by Syro-Palestinian epigraphic documentation dating from the 1st millennium B.

Numerous vases and seals, as well as various stands, present more or less complete alphabetical series, which are sometimes preceded by the preposition l, 'for'. Since these objects are always connected with religious ritual votive or funerary , the religious character taken on by the alphabet in this context is evident. The presence of the above-mentioned preposition makes it probable that the expression 'for 'bgdh When in the Apocalypse Goo introduces Himself with the words: 'I am the Alpha and Omega', using the first and last letters of the Greek alphabet to signify His eternity, we again meet the ancient Phoenician concept of the alphabet as symbolic of cyclic time, and we understand perhaps why the inventor of the alphabet wished to keep that relationship between sound and graphic sign which to us seems entirely superfluous.

Ettisch, Ernst. He received doctorate in political science from the University of Berlin in , and a doctorate of laws from the University of Freiburg in He emigrated to Palestine in For a time he was employed by the Canadian Assurance Company. He returned to Berlin in , and was employed as a legal counselor.

In he delivered a lecture on the significance of astronomy for the languages and scripts of ancient civilisations at the Universityof Frankfort. He died of heart failure in Ernst Ettisch has been categorized as a Revisionist Zionist. Anonymously reviewed in Anthropos, Volume 57, , Page The English-language book review in The World, Volume 1, Issue 11, , states it is an interesting and highly researched book. Mair, Victor. Sermonti, Giuseppe.

Teames, Sally. Paper abstract: " The Proto-Semitic alphabet is the ancestor of Greek, Etruscan, Roman, and all existing true alphabets in use today. Each of the twenty-two letters in the Proto-Semitic alphabet matches a constellation or asterism in or along the ecliptic. Not only do they match in shape and pattern, they also fall in the same general order, with only two constellations Pisces and Aries being out of sequence in the alphabetical order. The implications of the findings of this research are threefold.

First, the Proto-Semitic alphabet did not derive from the primitive Proto-Sinaitic alphabet at the turquoise mines at Serabit Al-Khadem and did not develop piecemeal, but was instead created as an organized unit of symbols designed after star patterns along the ecliptic. Second, the Proto-Semitic alphabet and the Ugaritic cuneiform alphabet may have originally been calendrical numbering systems perhaps based on lunar stations.

Third, similarities existing between the Proto-Semitic and the Ugaritic letter shapes, both being patterned after the same constellations and following the same general sequence, imply that the origin of the two may have been geographically close. Serafini, Stefano.

Alfabeto scende dalle stelle. Sull'origine della scrittura. Pellar, Brian. On the Origins of the Alphabet. Sino-Platonic Papers, Number , December. Part 1 of a 2-part paper. Sino-Platonic Papers is a n occasional series whose chief focus is on the intercultural relations of China with other peoples. Thesis Page 3 : "In , completely unaware of Moran and the others' work, I discovered that if you rotate the Phoenician alphabet ninety degrees counter-clockwise, and join the twenty-two letters into sequential couplets, a pattern appears that resembles the eleven constellations of the Egyptian solar zodiac.

The alphabet doesn't follow a simple circular pattern, but instead follows a more complex pattern that incorporates letter reversals at the solstices. It also forms two loops that meet at the constellation Gemini. All of these manifestations will be discussed in the course of this study. Academic qualifications as of B. Art, University of California, Irvine, ; B. Psychology, University of California, Irvine, ; M.

English, University of California. Irvine, Literally unbelievable - and unreliable. The author is unable to find his way through the maze of claims about early astronomy. But he doesn't seem to provide any evidence that the people who created the signs over a wide expanse of space and time connected the stars with lines the same as or similar to the lines he drew.

However, Pellar is prepared to arbitrarily change their interpretations. Regarding Figure on Plate Santillana-Dechend Hamlet's Mill have the caption: "The Mesopotamian cylinder seal shows in the upper part the "God Boat"; in the lower part people are building a ziggurat, the proposition being that the boat is bringing the me from Eridu-Canopus, the measure of creation. Then moving left, Pisces, Aries, and finally Taurus with a vessel on its back indicating the vernal equinox Santillana One again we have uninformed interpretations based on mistaken beliefs in an early zodiac.

Also, Pellar is not acknowledged as an authority on cylinder seals. The subjective nature of Pellar's reasoning - and dubious nature of his claims - lies with his claim Page 35 that the so-called Kananovo disc depicts a constellation zodiac. The rudimentary forms of the 12 modern Western zodiacal constellations no less. Incredibly, Pellar accepts a single amateur source that has not been subject to a peer-review process.

My understanding - personal communications - is that Richard Flavin does not currently support these claims and will revise the article when he has time available. The Karanovo Seal www. Karanovo is the best known of these settlement mounds. The excavation, made by Bulgarian archaeologist Georgi I. Georgiev, has revealed artefacts and house plans of three millennia.

In fact tell at Karanovo has accumulated 12 meters of cultural deposits from the Neolithic to the Bronze age. This tell was formed in layers over the centuries as wattle-and-mud houses were levelled and rebuilt about once each generation. The disk measures six centimetres in diameter, is two centimetres thick and with a handle 2 cm long.

It is inscribed with the ancient European script and for this reason it was probably an object of prestige, placed in a prominent position and possibly used in religious ceremonies. The Karanovo seal was discovered in the remains of a house destroyed by fire; an incident which slightly scorched the seal, but ultimately has contributed to its fine state of conservation.

The signs inscribed on the Karanovo seal are divided into four groups by the arms of a cross. The signs are straight, abstract and it is impossible to connect them to any forms belonging to the "real" world. This inscription is 6, years old. Richard Flavin had proposed that the incised characters from Karanovo bear a remarkable resemblance to the constellations which make up the western zodiac, in a somewhat sequential order.

Needless to say, the so-called Karanovo Seal has various interpretations. From his analysis of the Karanovo Seal the mathematician Vassil Vasko Georgiev Institute of Mathematics, Bulgarim Academy of Sciences, Bulgaria found that 8 characters show similarity with 8 characters of the Cretan hieroglyphic script. This is ignored by unknown to? See the discussion "Balkan Neolithic Scripts.

Incredibly, Pellar also accepts that iconography dating variously to the 3rd-millennium and 5th-millennium BCE indicates star groupings that match the shape of modern Western star groups. Pellar believes that the classical zodiac - our modern equally divided constellation zodiac including Aries - existed in its established form at least by the 3rd-millennium BCE Page 18 , and likely by the 5th-millennium BCE Page This period lasted until around the time of Christ, and was approximately yeas in length. We are currently residing near the end of the Age of Pisces, with the Age of Aquarius not far away.

The Apis bull was believed to be the incarnation of the Egyptian god Ptah. It has not been demonstrated that the Apis bull or similar has anything to do with astronomy. The identification of native Egyptian constellations is mostly uncertain. The Apis bull has never been linked with a native Egyptian bull constellation at any time - let alone during a supposed "Age of Taurus". Chronologically the cult was more popular during the supposed "Age of Aries" than it was during the supposed "Age of Taurus. However, from the cult's beginning's Amun could be depicted as either a Nile goose or as a ram, or as a ram-headed man.

Chronologically the cult originated in the supposed "Age of Taurus. There is no solid evidence that the Greeks possessed a complete zodiac until the 5th-century BCE. We know the Greek zodiac was formalised by the latter half of the 5th-century BCE because the two Greek astronomers Meton and Euctemon both used it in their parapegmata i.

Its purpose lay with the establishment of the solar calendar. The evidence is clear that the Greek introduction of such was that of a scheme borrowed from the Babylonians - excepting Aries and Libra. The Ram was an important cult figure in both ancient Egyptian and Babylonian civilizations - but perhaps not a constellation. See: Belmonte, Juan. The zodiacal Ram is a Greek constellation. When the Greeks borrowed the zodiacal system from Babylonian uranography the Babylonian constellation of the "Hired Man" was replaced by the Ram.

The Greeks changed the Babylonian zodiacal constellation "Hired Man" into Aries and the Romans later reintroduced the Babylonian zodiacal constellation Libra. The constellation Libra was included in the Babylonian zodiac but was later described by Hellenistic astronomers, such as Ptolemy, as "'the claws' of the great Scorpio.

GA Akk. A third variant MUL. Seals depicting rams en passant , with heads forward or reversed, are known from throughout the Hellenistic period in Uruk. The concept of precession-based zodiacal "world ages" is largely a 19th-century Theosophical concept invented by the occultist Helena Blavatsky. Nick Campion identifies that the concept draws "partly on Hesiod's sequence of ages outlined in the Works and Days, the Hindu Yugas, some 19th century studies of comparative religion and Madame Blavatsky's own theory of racial and spiritual evolution Hastro-L, 13 April, Additionally, the constellations are all of uneven size and we have no knowledge of the boundaries of any early constellations.

There is nothing in any early astronomical texts to prove a Twins-, Bull-, and Ram-period of precession. To return to the Karanovo Seal momentarily. These are divided into 4 groups by the arms of a cross, comprising 6 in one sector, 5 in another sector, 3 in another sector, and 4 in the remaining sector. On another issue, Pellar Page 28 states: " Figure Cylinder Seal that shows the Gemini gate between Leo and Aries.

British Museum; Kramer On the right of the gate is Enki, the water god, who is Aquarius. Pellar does not acknowledge and seems unaware that the interpretation of scenes in ancient Mesopotamian art is immensely difficult. The Mesopotamian cylinder seal being referred to is the cylinder seal of Adda the scribe, one of the most famous cylinder seals known from the Ancient Near East.

The identification is made from an inscription 'Adda, scribe' is written in cuneiform above a lion, identifying the owner as a high official , and the seal is dated by Dominique Collon Assistant Keeper in the Department of Western Asiatic Antiquities in the British Museum to circa BCE. The Akkadian greenstone seal is 3. Depicted on it are the 5 major deities of the Mesopotamian pantheon. From left to right when looking at the rolled-out seal image there is Nergal?

Enki Ea is often depicted with a vase from which the water flows or the waters of the Apsu the Abyss is depicted flowing from his shoulders , reminiscent of Aquarius, the Water Bearer. Enki Ea , the god of subterranean waters and of wisdom and the Mesopotamian equivalent of Hermes, is accompanied by his attendant the 2-faced god Usimu or Isimud , who was the gatekeeper of the Underworld.

Grade 8 Math #14.2c, Predict from Trend lines - Interpolate vs Extrapolate

There are, however, a number of additional figures on the seal. A correct description of the seal iconography necessarily collapses his "Gemini gate and BCE date" argument. Also, the late Edith Porada, a major authority on ancient Near East imagery, and well aware of the water association with the syncretic god Enki-Ea, proposed that Aquarius originated from GU. LA, a naked hero depicted with streams of water flowing from his shoulders. The hero figure GU. LA first appears in the 3rd-millennium BCE- without streams.

The concept of retro calculation also falls into the category of speculation knowing no restraint. On the left peak of Mount Mashu a plant is depicted. This plant is not identified with certainty. The concept of axis mundi popularised by Mircea Eliade is also in need of critical assessment.

According to Edith Porada, the late eminent authority on ancient Near East imagery, mythological, ritual, or other scenes remain unintelligible due to our inability to identify most of the principal figures and their actions. These include the goddess Ishtar and the syncretic god Enki-Ea. All alphabets in use in Northern Europe can be traced back to the ancient Greek alphabet - which has a Phoenician origin.

For the type of evidence discussed Pellar's remarks concerning the possible origin and loss of early developments of the alphabet occurring in ancient Northern Europe are wildly speculative. Pellar is only able to give 2 very dubious examples: 1 glyphs appearing on a European stag bone from BCE Spain , and 2 glyphs appearing on a Karanovo stamp from circa BCE Bulgaria. Neither is from Northern Europe unless we introduce additional speculation and neither is convincing evidence for his conclusion. Pellar is very strong-minded passionate in defending the content of his publication and is currently preparing to publish a second part.

He also appears to believe that inaccuracies such as those discussed above have no consequences for the credibility of his central thesis. Thus the division between the Northern sky and the Southern. In proper perspective, Sino-Platonic Papers is actually Victor Mair's Professor of Asian and Middle Eastern Studies, University of Pennsylvania own private usually non-peer-reviewed publication that has a very small circulation.

The editor seeks and encourages speculative contributions hence the absence of the peer review process. As of August, - the time the main body of this review was posted - Victor Mair now appears to state the Sino-platonic Papers are all peer-reviewed. From Wikipedia : "An academic journal is a peer-reviewed periodical in which scholarship relating to a particular academic discipline is published. Academic journals serve as forums for the introduction and presentation for scrutiny of new research, and the critique of existing research.

Content typically takes the form of articles presenting original research, review articles, and book reviews. Academic or professional publications that are not sic peer-reviewed are usually called professional magazines. Scholars can be expert only in a limited area of their fields; they rely upon peer-reviewed journals to provide reliable, credible research upon which they can build subsequent, related research.

The purpose of the series is to make available to specialists and the interested public the results of research that, because of its unconventional or controversial nature, might otherwise go unpublished. The editor actively encourages younger, not yet well established, scholars and independent authors to submit manuscripts for consideration. Sino-Platonic Papers prefers lively work that, while taking reasonable risks to advance the field, capitalizes on brilliant new insights into the development of civilization.

Volume 1, Pages Lebeuf, Arnold. Proceedings of a conference held October , in Venezia, Italy. Part One. A speculative essay in which the author manages to ignore critical sources. The author does not seem familiar with many of the sources he does choose to use. The controversial issue of precessional re-alignment of Babylonian temples.

Dombart, Theodor. Zikkurrat und Pyramide. Discusses astronomical theories. Teil: Zikkurrat by Theodor Dombart Die Kultrichtung in Mesopotamien. Februar After gaining his PhD Martiny only rarely used the title "Doctor. Martiny Architectura, 1. Jahrgang, Heft 1 Number in the Bibliography but I cannot presently access this journal. In his doctoral dissertation published in as Die Kultrichtung in Mesopotamien Martiny described in detail the methods upon which he believed they based the orientation of their temples. In his thesis Martiny tabulated the orientation of Assyrian temples for which the dates of foundation could be established the oldest being circa BCE.

Martiny maintained that Assyrian temples had been reconstructed at least the foundations altered during the period of their use; the orientations being varied according to the angle of the precession of the equinoxes. According to Martiny the particular star would have heliacally risen on the Assyrian New Year. Martiny charted the changes of orientation of temples at Assur from BCE and believed he had identified a "steady" eastward movement in their orientations. In attempting to interpret Martiny's archaeological data Neugebauer and Schott initially proposed that Neo-Babylonian i.

As a result of his review of Dr. Neugebauer later discarded this explanation when he could not find evidence that the star alpha Virginis Spica had importance in Mesopotamian astronomy. However, Neugebauer's changed explanation held that a clearly recognisable line is formed by the stars alpha Cassiopeiae, eta Cephei, beta Ursae Minoris, alpha Draconis, eta Ursae Majoris, and eta Virginis, and this line formed the meridian used in the centuries before and after circa BCE, as the basis for the orientation of Neo-Babylonian Assyrian temples.

The thesis of Martiny is based on figures from Warka that were later seriously questioned. One of the methodological issues with the figures produced by Martiny is that he charted the orientation of the Gimilsin Temple and the Palace Chapel according to True Magnetic North rather than True North and he made adjustments accordingly.

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Also, he gave no explanatory information regarding the data on the orientation of the other temples included in his orientation chart. There is no persuasive evidence that Mesopotamian temples, of any period, were systematically aligned to any directions. Circa Asger Aaboe advised that Martiny's claims were purely hypothetical and that still today nothing is known about if and how the ancient Mesopotamians used astronomy to orient their temples. The topic of temple orientation has never gained popularity in Assyriology.

Only recently, November, , after years of systematic searching, have I located a source that provides core biographical details. I have seen offered as his year of birth. In one publication he is referred to as Architekt [i. Gunter Martiny's early work involved the investigation of the construction and alignment of Mesopotamian temples. He was, it appears, also connected with the Staatliche Museen zu Berlin.

There is reason to believe that in he was practicing as an architect whose designs were highly regarded in Dresden, Germany. He was residing in Berlin at least by circa He may have also briefly resided in Munich. I have also seen him referred to as an engineer but this is incorrect. In several publications briefly mentioning him the terms Diplomingenieurs, Architekten Dipl. At that period of time the German Diplom-Ingenieur Dipl.

It is an engineering degree in the sense that technical studies were undertaken. His Dr. His Dipl, Ing. Amongst other things he was an architectural historian. He was interested in the sacred architecture of ancient Mesopotamia at least during his younger days. At least those temples for which foundation dates could be established. This particular German expedition was conducted from until The German Research Foundation Deutschen Forschungsgemeinschaft getragen at one time also supported excavations at Warka. However, I am presently unsure of the years. Since the DAI allocated an annual grant for the promotion of young academics in archaeology and its neighboring sciences.

In general, the focus was classical antiquity and the Mediterranean region, but also included the Middle East. The duration of the scholarship is usually one year but can be extended in justified exceptions for a further one year. During the time he was a member of this expedition he also spent time at Assur with the Deutschen Orient-Gesellschaft.

This journal first began publication in Leo Adler, a prominent German voice in architectural discussion between and , could be described as an architectural theorist. Note: For a while Martiny was a proponent of Assyrian temple alignment. However, his methods lacked rigour. At one time his ideas were supported by the astronomer P. The asserted discovery by P. But archaeology has failed to provide consistent results. Neugebauer later changed his mind. The subject of the orientation of of Assyrian temples has been suspect. His excavations on Ithaca were largely financed by a wealthy Dutch gentleman, Adrian Goekoop.

Martiny was, at least prior to World War II, a significant shareholder in the specialist glassworks manufacturer, S[amuel]. In Istanbul, Martiny shared a house with the German assyriologist Rudolf Fritz Kraus , and also 3 Scottish students who were to assist in the excavations a Czech couple with a young daughter, and others. It is indicated that Martiny also lived in the same accommodation as the German Consul General.

It is considered to be an exhaustive study of the construction and decoration of this 6-domed mosque. In it he gives a complete description of the plan, construction, and decoration of this Arab-Islamic mosque. He also published another small book, titled Istanbul , in the same year. In his article "Die Piyale Pasha Moschee. Martiny asserted that the architect must be someone of the school of the famous Ottoman architect Sinan, and not actually the master.

During the s Martiny focused on Byzantine architecture. For a time in the s he lived in Istanbul. However, he worked in Turkey earlier than this date. He was in Istanbul in One source states his research was carried out in and His later work involved the investigation of medieval Ottoman mosque architecture. His involvement in this latter work would place him within the job category of an excavation architect.

Circa he worked as an archaeologist and architect on the excavation of Justinian's palace built circa mid-5th-century CE in Istanbul. In or earlier? In the late 's he appears to have resided for a time in England. From through 4 seasons to the summer of he was part of a British or rather international, as a Turkish archaeologist was also included archaeological 'dig,' led by Professor J. One source indicates that at some time, possibly in the s, Martiny was also involved at a 'dig' at Tell Asmar, the mound being the site of the ancient Mesopotamian city of Eshnunnu located in the Diyala Plain of Iraq.

Also, another source located him in Istanbul in , which would be the St Andrews University 'dig. The first part of the report of these 4 seasons of archaeological excavations was published in as The Great Palace of the Byzantine Emperors pages. This is likely the Istanbul expedition. It is also mentioned that at one excavation of a building in ? David Russell was a Fife paper manufacturer, entrepreneur, philanthropist, and benefactor. He had a deep interest in history, archaeology, etc. David Russell partly? Up to this period very little field-work had been carried out in Istanbul.

A source has indicated that in the early? Another source indicates he was concealing certain information from them marriage details? This is indicated as part reason for his emigration. In Martiny emigrated from Germany in the sense that he permanently resided elsewhere for nearly half a decade. His residence for approximately 5 years was likely divided between Turkey and the United Kingdom. For 4 years he was involved in archaeological excavation work in Turkey, and then for years he lived in London.

From to circa he was in London. In late he returned to Germany. From to circa ? In he lectured at the Theological Seminary in Rimini, Italy. Volume 92, , indicates that Martiny may have returned to Istanbul in at least. However, in A. Haller and W. Also see his article: " Astronomisches zum Babylonischen Turm, 11 bis. His conclusions were that Mesopotamian temples were oriented with respect the north celestial pole. The circular path traced by this point in the sky due to the precession of the equinoxes determined the orientation of Mesopotamian temples and the orientation thus testified to the date.

Martiny's proof was that the angular values of the corrections periodically introduced into the axial alignments of many Near Eastern sanctuaries equalled and thus offset those of the precessional shift for the same period. See the Italian-language review in Rivista degli studi orientali Dombart, Theodore. Lanfranchi, Giovanni. Archeologia e astronomia: esperienze e prospettive future. Accademia nazionale dei Lincei. Argued that written evidence supports the importance of temple orientation to southeast and its role in the politics of the Neo-Assyrian Period.

Nadali, Davide. The controversial issue of precessional re-alignment of Greek temples. After the investigative work of Francis Penrose, mostly in the last decade of the 19th-century, there was very little work done on Greek temple alignments until the beginning of the 21st-century. Something that can be considered a lapse in classical scholarship. However, using 'alignments' as evidence instead of finding evidence for the cultural practice of 'alignments' poses a problem which archaeologists were quick to point out.

It is rather easy to find supposed 'alignments' i. I have not identified which papers - if any - deal with the possibility of significant solar orientations such as sunrise on significant dates i. There is evidence that later Greek temples were oriented at a very slight angle to one built say half a century earlier. The later temple of Nemesis is oriented at a very slight angle to the temple of Themis built half a century earlier, in a manner that caused considerable architectural difficulties.

Francis Penrose born at Bracebridge — died at Wimbledon M. Penrose was educated at Bedford Grammar and Winchester College and afterwards worked briefly for an architectural firm. He studied architecture under Edward Blore from to He attended Magdalene College, Cambridge University, as an undergraduate, studying astronomy among other subjects and completed his degree in He then studied abroad under the Cambridge designation of "travelling bachelor" from to He became surveyor of St.

Paul's Cathedral in , and it was there that he did his main architectural work. In at least he also carried out excavations there. Francis Penrose became a Fellow of Magdalene in From to and from to he was Director of the present British School in Athens which he designed without charging a fee. He was president of the RIBA from to He was appointed architect and antiquary to the Royal Academy in He was the author of a number of books and papers, including: Principles of Athenian Architecture the first complete publication on the subject an enlarged edition appeared in , and a work predicting eclipses.

Francis Penrose published extensively in the Philosophical Transactions of the Royal Society on the astronomical alignment of Greek temples in the Mediterranean. Penrose was also an amateur astronomer and published a work on Saturn in Under the influence of Norman Lockyer he combined his interests in architecture, archaeology, and astronomy to study how astronomical phenomenon determined the design of ancient buildings, including Stonehenge.

Penrose was - and still is by some - considered a careful observer. He believed he had demonstrated that Greek temples had particular foundation alignments, either solar or stellar, and due to the effects of precession the temple foundations were periodically realigned in order to keep the original orientations.

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However, Penrose's work on the dating of Greek temples using the coincidence of the heliacal rising of a star with the rising sun on a particular date has been discredited by by modern archaeology. Penrose attempted to do for Greek temples what Lockyer had done for Stonehenge and some Egyptian temples. Lockyer's book, The Dawn of Astronomy , depends on Penrose for his statements about Mediterranean alignments.

Citations of published papers on astronomical alignments by Francis Penrose - by various sources - indicate some possible confusion with exact titles, dates, publications, and page numbers, and so do not always seem to be accurate. For the beginnings of a more modern discussion of the issues see: The Architecture of Ancient Greece by William Dinsmoor Revised and expanded edition, Republished Penrose, Francis.

Series A, Volume , Pages Lockyer, Norman. An early review is:. A study of Greek temples in Sicily. From the Introduction: "It has long been proposed that classical temples may have been aligned with respect to sunrise on certain dates. The idea was first proposed by Nissen [1] in This idea was developed further by other authors such as Penrose [2,3,4] and Dinsmoor, [5] who argued that a temple could be dated from its astronomical alignment. This explanation was rebutted in the s by Herbert [6] on the grounds that plenty of Greek temples did not face east.

Following a survey of Sicilian and southern Italian temples Aveni and Romano [7] reasserted that there is an astronomical pattern to the alignment of Greek temples, but the two most recently published statements on the subject [8,9] both state that there was no evidence of astronomical intent. At best, there is no consensus about the answer, though a more accurate summary would be that opinion is shifting away from the notion of astronomical alignments being embedded within Greek temples.

His book in 3 parts 3 monographs , Orientation, Studien zur Geschichte der Religion 3 Volumes, , basically first appeared as articles. His book, Das Templum. The ideas contained in these 4 articles were printed later in revised form in a series of monographs Herbert, Sharon. Boutsikas, Efrosyn. Retallack, Gregory.

Astronomy and Ancient Greek Cult : an application of archaeoastronomy to Greek religious architecture, cosmologies and landscapes by Efrosyn Boutsikas , doctoral thesis. An extract of her review of Francis Penrose; Page "He [Penrose] argues that archaeologically there is evidence of another temple earlier than the surviving one on the same site, but with a different orientation. Archaeologically, in fact, there is no evidence for the presence of a temple around BC, the date Penrose deduces for the construction of the earliest temple.

He further attempts to define the orientation of the older temple. In fact, there is no archaeological evidence that an earlier temple shared the same orientation as the masonry wall. However, the lack of historical evidence and the faintness of the star result in him revisiting this claim. Regarding William Dinsmoor:. Dinsmoor, William. The Architecture of Ancient Greece. Papathanassiou, Maria. Abstract page Abstract: "We present the results of orientation studies of the majority 44 of the temples and the sanctuaries on Delos which date from different periods of the history of the island.

Here are some examples of our study. The three Apollo temples built side by side to one another and dating from the end of the 6th century B. Two days 11 March 13 October in a year the SunApollo has the corresponding declination and sets exactly at the point of the orientation of these temples. Therefore it is very likely that the orientation of Apollo temples is related to the setting Sun in March rather than in October.

A possible explanation based on a stellar orientation towards delta Orionis or generally Orion's belt is not convincing as it is not supported by some historical archaeological evidence. The Artemision 7th century B. The Oikos Naxion is a unique building with two doors, the oldest 2nd half of the 7th century B. These orientations are also related to the rising and setting Sun. But there are other cases, e. In our study we try to examine and elucidate every case according to the measured orientation in relation to the topography of the island Kynthos mountain, port, etc and the archaeological evidence.

A list of members of Scientific and Local Organizing Commitees, a list of Sponsors as well as 2 Sections with content has been given. Murdin as well as the abstracts of the Panel written by L. Woltjer, J. Trumper, R. Fosbury, J. Swings, F. Sanchez, P.

Lequeux, R. Giacconi, P. Murdin see: separate entries. The second section of the highlits sic includes Reports from the Convenors of the Sections: J. Hadjidemetriou, L. Vlahos, X. Moussas, P. Laskarides, N. Voglis, P. Niarchos, J. Ventura, N. Spyrou, S. A list of Participants has been included at the end of the booklet. Mickelson, Michael. Major exceptions, such as the temple of Apollo at Bassae which faces north-south, are always noted in the handbooks, but many other temples are scattered across the Greek landscape in a variety of orientations.

Although no surviving ancient author ever discusses the criteria for placing or orienting temples, we may assume from scattered remarks that Greeks had reasons for choosing the sites and orientations. In the last century, archaeologists and architects such as Nissen , Penrose and Dinsmoor , have measured the alignments of Greek temples on the Greek mainland, the west coast of Turkey, and the Aegean islands.

Their data have varying degrees of precision and accuracy, as a recent paper by Papathanassiou makes clear. Parallel work done in Italy on Etruscan temples by Aveni and Romano provides further stimulus to re-investigate Greek temples. We have undertaken two field seasons in Greece to make preliminary measurements for a number of temples associated with Athena, Apollo, and Zeus. These temples were chosen for a number of reasons. The structures have to be well enough preserved to allow determination of the orientation of foundations, location of doorways and other openings, placement of cult statues etc.

By focusing on these three gods, we may be able to discover patterns in the orientation and placement for specific divinities. For some of these questions, we are dependent on literary and inscriptional evidence, such as the work of the Greek travel writer, Pausanias. This paper describes the preliminary measurements made over our two field seasons in Greece. Field methods and analysis of the data will be presented along with proposed applications. Research supported by the Denison University Research Foundation.

Liritzis, Ioannis. Ioannis, Liritzis. Kravaritou, Sophia. The Hellenic Astronomical Society Hel. Nissen Dinsmoor , that still preoccupies many recent studies on Archaeoastronomy Papathanasiou and Hoskin A main factor to the determination of the orientation of those temples is the date of the festivals celebrated in situ in honor of their divine owners.

In reality, apart from the festivals with a fixed date, epigraphic evidence implies existence of local homonymous festivals with various and maybe movable dates, including seasonal rites associated to the agricultural year and celebrated in honor of Demeter, the chief deity of agriculture Kravaritou , Our problematic is oriented towards the eventual impact of this fact to the orientation of Demeters' temples.

Here we provide new evidence based on measurements of 15 monuments attributed to Demeter at Eleusis, Athens, Thebes, Delphes, Eretria, Corinth, Delos, Tenos, Thassos, Mytilene, Kos, Rhodes, Knossos, Andros, and Sicily, that date to various historical periods and provide literary evidence attesting local celebrations. Azimuth Values, angular altitude of skyline, and geographical coordinates were measured by magnetic compass, a clinometer, a theodolite and a GPS, and comparison was made with measurements obtained by other scholars.

In the interpretation, use was made of the SkyMap Pro The question of the orientation of Greek temples is actually revisited, on the basis of investigating the impact and the eventual interference of both local festivities and panhellenic customs cf. Bibliography A. Kravaritou, "Greek 'Calendars' and symbolic representation of the cosmic order: seasonal rites for Demeter", Mediterranean Archaeology and Archaeometry, Special Issue, Vol.

Niessen, Orientation. Studien zur Geschichte der Religion, Berlin Penrose, "A preliminary statement of an investigation of the dates of some of the Greek temples as derived from their orientation", Nature 45, , Salt, Alun. Hannah, Robert. Under the umbrella of the new multi-discipline of archaeoastronomy, the analyses of orientations have become more sophisticated, with a wider range of celestial phenomena now included, along with information derived from cult myths and rituals, so that celestial phenomena are identified through being relevant on a local, cultural and cultic level.

Ironically, this revival of interest in temple orientations largely matches in its methodology that of Dinsmoor [], yet his work has lost favour because of his insistence on the priority of solar phenomena. Ranieri, Marcello. Akragas Valley of the Temples as a case study. In fact, although a general tendency to orientation within the arc of the rising sun is undeniable, specific patterns and true meaning remain obscure.

With the aim of shedding light on this problem we present here a new complete, high precision survey of the temples of Akragas, the so called Valley of the Temples UNESCO site. Our results include a important temple which was essentially yet unpublished, and most of all show that very different reasons influenced the orientation choices, some symbolical, but others by far more practical, besides the general rule of orienting to the rising sun.

In particular, the existence of temples orientated in accordance with the towns grid, as well as to the cardinal points irrespectively from the sun's declination associated to true east at the uneven horizon, is evidenced. Finally, for two temples having anomalous orientations a stellar and a lunar proposal are made respectively.

Boutsikas, Efrosyni. Handbook of Archaeoastronomy and Ethnoastronomy. Chapter , Pages Note: It is indicated that in the mi d s two German astronomers published an article in one of the early volumes of the German periodical Sterne und Weltraum, on a Greek temple in Athens? The controversial "void zone" theory of constellation origins. The reviewers do not mention the "void zone" argument.

The "void zone" arguments have now been critically demolished by two recent articles by the astronomer and historian Bradley Schaefer. See: 1 Schaefer, Bradley. The former paper establishes that the southern Greek constellations originated in the first millennium BCE. The latter paper establishes that the astronomical lore of Eudoxus was of Babylonian origin circa BCE. Proctor, Richard. Myths and Marvels of Astronomy.

Peck, William. The constellations and how to find them. See the English-language obituary by H. A Popular Handbook and Atlas of Astronomy. He uses several of the arguments and conclusions employed by the "void zone" proponents i. In this publication the author believed that the very earliest constellations were established by the Egyptians circa 15, BCE and were further developed by the "Chaldeans" circa BCE.

Maunder, Edward. Astronomy Without a Telescope. Maunder, Annie. Evershed, Mary. In her short article, "The Origin of the Constellations. Rejects a 3rd-millennium BCE date for the Greek constellations. Dates the Greek constellations to the 1st-millennium BCE. Orr, Mary. Dante and the Early Astronomers. Follows the ideas of Edward Maunder. See also; "M. Orr Mrs John Evershed , astronomer and Dantist.

Crommelin, Andrew. See the English-language obituaries by P. Also see the English-language biographical entry in Who Was Who, , published , Page Her method is only partially explained and the void zone is not mentioned. The author used the translation of Aratos' Phaenomena by G. Mair whom she incorrectly identifies as G.

Mair in the Loeb's Classical Library Edition, She reinforced her view in a short note answering Query 9, under Astronomical Queries. Davis, George. See the authors remarks at the end of the article references. Ovenden, Michael. Mason and Others, Astronomy. Inventory Identifier Box Number 12 Series 2. Peter Mason was a physicist and science broadcaster. Michael Ovenden was an astronomer. The authors ideas appeared earlier, for example in a talk on "The Origin of the Constellations" given in at an ordinary general meeting of the British Astronomical Association, and appeared in the Journal of the British Astronomical Association, Volume 71, , Pages [Some sources give pages The article is unreliable and contains significant errors and the same applies to the talk.

The University of Glasgow was the same university where Ovenden was teaching at the time. Wasserstein was Assistant in Greek at Glasgow University from to and then Lecturer in Greek at Glasgow University from to Wasserstein then then, circa , moved to Leicester University as Professor of Classics. The core of Wasserstein's advice was the information which the works of Aratus and Hipparchus could give about the origin of the constellation figures.

In addition to astronomy Abraham Wasserstein also had a deep interest in Greek mathematics and was a Fellow of the Royal Astronomical Society. Life dates for Michael Ovenden: See: "Obituary: Professor Abraham Wasserstein. He was a man of extremely wide scholarship; there can be few professors of Classics who are also Fellows of the Royal Astronomical Society, and his inaugural lecture at Leicester, delivered a short while after he had taken up his appointment, moved broadly but confidently over the whole field of Hellenic endeavour. Greek mathematics, astronomy, musical theory, philosophy, and history - all were brought together to illustrate what he termed "Economy and Elegance".

That year he was appointed as Assistant to the Department of Greek in Glasgow, where he stayed for nine years, subsequently as Lecturer in Greek; in later years he emphasised the debt he owed to the patterns of Glasgow's teaching and regarded it as a model to be followed. In Wasserstein applied for the Chair of Classics in Leicester, with so little confidence in himself that he did not even wait to hear the results of the interview. But his stay in Leicester from to was to prove memorable for him. He made many friends in the academic community and for him and his wife these were very happy years.

These were stirring years too in the university, culminating in the first manifestations of student unrest in Britain. Much later colleagues still recalled how he stood single-handedly at the entrance to the library to prevent students extending their "sit-in" into its precincts. But when some students asked to be allowed to enter and retrieve their possessions, promising not to abuse this permission, he accepted their word and his trust was not betrayed. Brown, Peter. What star is that? Life dates for Peter Lancaster Brown: ? Pomerance, Leon. The least squares normal equations are Z w!

To conserve storage the four distinct diagonals of this matrix are stored as columns of a '' k! The turns of both sides of 6 are performed sequentially so that all of the ranges and weights are not needed In core simultaneously. He have found that the use of the unweleht- ed least squares start will usually result In convergence of the IRWLS cubic spline to a good fit with outliers correctly Identified, but that many fewer Iterations are required If a robust choice of Initial weights Is used.

The Initial weights for the robust cubic spline fit are chosen on a localized th basis. Several methods for robustly fitting the linear curve have been tried. Most methods performed about equally well on the data sequences tested.. The results of some of thesu tests are given In Appendix A. Because of Its simplicity, the modified method of Thell was selected for routine application.

This method is described in the following paragraph. Let R be the median of the observations In the K— knot interval. Slnee there Is an average of maybe five receivers on each data pass, the robust preprocessing method described has been used on more than one thousand measurement sequences. The method has performed successfully on all of these sequences. Mott of these sequences are rather uneventful, having only a few Isolated outliers. There have been some sequences which have some rather dense bursts of outliers.

These sequences best Illustrate the ability of the method described to detect outliers. Fig 1 presents a range measurement sequence and Fig 2 the robust cubic spline fit to this sequence. Note that the outliers In Fig 1, which have been darkened, occur In many sizes. The outliers at the top of the graph were added by hand since they all occurred far off scale at the top. All outliers have been successfully detected and removed by the robust spline fit.

The measurements In Fig 1 have two dense burst of outliers, one In the Interval The measurement sequence In Fig 3 has outlier bursts In the Intervals The sequence In Fig 5 has bursts of outliers during the Intervals Any points away from the main curve should be considered outliers In Figs 1, 3, and 5. Note also In Figs 1, 3, and 5 that there are time breaks In the measurement sequences, another Important consideration In preprocessing.

Note that some of the time breaks have been filled with fitted data points. The filling of the time breaks Is controlled by the length of the time breaks In relation to the sample rate and the proportion of outliers found In a knot Interval. The robust cubic spline preprocessor has deleted all outliers from the measurement sequence, generated measurements during the time breaks as desired, and synchronized different measurement sequences If desired.

In addition the measurement variances are available for further processing. This fairly rapid convergence Is dependent on a robust method for choosing good Initial weights. Surprisingly, the IRWLS cubic spline Iteration for these examples also converges using an unweighted least squares start, but at the expense of more Iterations. For the measurement sequences displayed here the IRWLS cubic spline fit converged In 7 - 8 Iterations using an unweighted least squares start. Thus, at least In these examples, a good choice of the Initial weights results only In a significant Improvement In computing efficiency and not In an Improvement of fit.

NPTO must be large enough so that Is likely that only a fraction, say less than one fourth of the data points In any Interval will be outliers. On the other hand, If NPTO Is too large, the robust linear curve fit may not he a good enough representation of the variation of the data In the Interval. Nested Medians Nested or repeated medians Is a robust regression method recently described by Siegel [l]. Siegel shows that tMs method has the highest breakdown method of any known method.

This method Is particularly easy to apply for a linear fit. It Is similar to the Thell method and modified Thell Method already described. The following data sets were taken from the knot intervals of the data sequences used previously to Illustrate the application of the robust range measurement preprocessing. The measurements In this set are from the time Interval In both of those date sets there are eight outliers In the sample of twenty. Each of the robust linear methods seem to have no difficulty In Identifying the outliers In these data sets. Andrew F. The ultimate camouflage being that of a target with no reatrictiona on time or manpower in its erection or retraction, uaing standard materials and methods.

This ultimate condition ia not normally measured in field studies where prototype camouflage is evaluated against a base line target. This paper concerns a reel field study. The camoufleur then ascertains true values of the prototype camouflage, or How Good is Cood. In all cases, the developers goal is to produce the best camouflage possible within the restraints of time and manpower.

One method of camouflage evaluation is through the conduct of field teats. The resulting data is analyzed, enabling the developer to determine the effectiveness of his product. The problem with this method of evaluation is that while it provides a good estimate of the prototype camouflage performance, as compared against the base line camouflage condition, no information is available for a comparison against the ultimate camouflage condition. This range had an average elevation of 68 feet above mean sea level, and is flat in topography with mixed Oak, Pine, and Palmetto tree hammocks dispersed in sawgrasa.

The land is primarily swampy in nature and displays a light green color throughout. The equipment was tactically emplaced along a tree line one was emplaced outside the tree line and one was partially concealed by the treeline. This color blend consisted of 45 percent forest green, 45 percent dark green, 5 percent light green and 5 percent black. These camouflage kits were constructed of special supports, and modified lightweight net screening.

Oak leaf mulch was dispersed around the edge of the camouflage screens to break ufp the straight line edge- effect. Palmetto fronds and aawgraas clumps were placed over the batten spreaders of the support system to reduce shine and were woven into the camouflage screens to simulate the appearance of natural foliage. The target location was identical for each camouflage condition.

The end product was three strips of imagery at each of the scales of , and , The , scale of imagery was 17 frames long while the , scale imagery was 5 frames in length. They were instructed on the purpose and tasks to be performed. Each team had three-quarters of an hour to detect targets on one of the strips of imagery scaled , They were allowed 15 minutes to determine an identification. No team of II's viewed more than one camouflage condition.

The item not as deeply embedded in the trees was identified by more II's and yielded significant differences between camouflage conditions as follows: o The pattern painted item was detected significantly a- 0. Ultimate Condition The percentages of identification for each of the two teat items were determined for each of the three conditions. Ultimate Conditions 7. However, the trade off of the amount of time and manpower required to decrease the number of detections must be considered.

There was no significant, diffarance between the number of identifications for the prototype and ultimate camouflage conditions. This study was conducted in Avon Perk, Florida. The result! Vol 1, TM, December 4. Natrella, Mary G. National Bureau of Standards, Handbook 91, U. Department of Commerce, Washington, D. Although universally valid, the formulae are particularly apropos when straitened test circumstances are dictated.

Programming suggestions are Included. This paper Is a sequel to the one entitled Estimating Reliability from Small Samples and presented before the twenty-second conference on the Design of Experiments In October [4]. The Poisson distribution Is treated In a manner parallel to that afforded the binomial distribution In the earlier paper. Probability statistics require the Identification of a unit commonly called event or trial. Often this Identification Is self-evident.

It Is to be noted that the failure ratio is dimensionless; l. Identification Is not always so clear-cut. For example, suppose an operator of heavy trucks notices that In the preceding six months, he has experienced 13 major mechanical breakdowns--one every two weeks, on the average. The definition of failure Is obvious, but what Is a euooeeel To what do we add k to get n, the sample size?

But this Is also a useless answer; no realistic test design could require an infinite sample size. To avoid facing this dilemma, let us arbitrarily define event In some convenient unit different from that In which k Is expressed. As a consequence, we no longer have a failure ratio. In Its place we substitute a failure rate--of k per event. Thus the failure rate depends upon an observed k, but upon a defined event. It Is sometimes reqarded as preferable to express the reciprocal of the failure rate, calling It mean life.

The term event can be defined In any of a variety of units—area, volume, weight, time--almost anything that can be measured. Unfortunately, however, It does not suffice. In most test designs, It will be possible to define event arbitrarily and to observe the value of k exactly, but nothing will be known about x. Usually, In fact, x will be the principal value sought. A probability function In x Is required. Pi Hill fill' i 1 m 11! LV tr. The slope there Is unity. If the Itself. When a function Is defined by or can be described as a definite Inteqral, very frequently It will be found that repeated Integration by parts will produce an expansion suitable for computing.

In fact, as In the Instance at hand, It may be possible to expand In either ascending or descending factorials or powers, as the case may be , thereby producing two different expansions, both of vrtilch are valid. Usually, one will appear In the familiar form of a power series which converges more rapidly for smaller values of the argument. The other will be the associated asymptotic expansion. Can the Indefinite Integral fixed, positive Integer? It may be more correct to say that the sum reduces to the constant of Integration. Is It true that.

See Table 1. It enables us to select a series for computlnq whose terms are known to decrease monotonic ally. This results in worthwhile economy for larqer values of k. There are two cases to consider. Under the stated conditions, this is always less than unity. For large values of k, the interior series can be summed as though it were an infinite series, thus achieving a laudable saving in the number of terms required.

The value of z may be derived from any source, or It may be arbitrarily specified. But when k Is very large, two difficulties arise. First: The number of terms becomes excessive. If the series Is summed as though it were an infinite ser1es--i. Second: Large factorials will overflow the computer. To circumvent this, the first term of the series is computed by logarithms. Instead, successive approximations z 0 , zi, Z2,. Newton's method serves very well. See the discussion In [4] pp. Noting that IILlL. Li f'U k - 2, the formula can be simplified to az f'Ui U3 The process Is stable when started from the right-hand point of Inflection; l.

There are, of course, an unlimited number of solution oalrs a, s which satisfy this equation. This form properly is used to test for compliance with an Imposed standard, Sometimes, however, that standard is absent, unrealistic, or even erroneous. Put It Is still required to make a meaningful statement about the failure rate. Essentially, that solution pair a, s Is chosen which minimizes the difference s - a.

Some properties of the Best Estimate of the Failure Rate are: a. The limits of integration lie on opposite sides of the maximum; i. The ordinates at a and 5 are equal; i. The solution is unique. There are several steps in the solution. Step One. Step Two. Then Aa 15 The process Is repeated until f a and hence a Is found to the desired accuracy.

Step Three. The Incremental area always will appear in two separate parts. The ratio of these areas can be estimated quite closely by the slopes. It Is convenient to express the ratio In terms of the ordinates. It should be noted that the methods developed In this paper are virtually Independent of the definition of Event. Event often will be synonymous with Duration of Test. Suitable values of a and s or z, as the case may be having been found, It Is apparent that they should be expressed In the units failures per event.

If at this point the definition of event is Imposed, the results can be expressed In failures per mile or failures per hour or whatever. Now the simple reciprocal converts to mean life, it should be remembered that taking the reciprocal reverses the sense of inequality signs.

When x Is an Integer, the series terminates. Let us replace x with k In the formulae In order that x can be employed as a variable of Integration. Thus the formulae restated appear as follows: P k, 2. This case of k being a positive Integer was studied at length In Its application to sampling distributions by Helmert and K. Pearson Thus arose the statistics of the x 2 distribution. The exponent 2 In x 2 has little significance beyond ensuring that the parameter Is non-negative. As a general rule, the only variable of observation will oe k, tne numberof failures.

The variable of Integration will be x, with z one of Its extreme values limits of integration. It is necessary to define event In some suitable unit time, distance, mass, volume, etc. Event often Is synonymous with Duration of Teet. Many formulae of interest are greatly simplified If expressed as functions of f x or of f z. For large values of z, compute f z by logarithms.

Stirling's formula for IngU Is useful here. If k does not change, It need be computed but once. Equations 4 , 5 , 9 , 10 and Enter data subroutine for zero failures Compute In k. T2] Boole, G. Publishing Co. Chelsea [3] Olver, F. New York, In Army R. In Proc. The survival probabilities are.

Both parametric and non-parametrio statistical models are evaluated with respeot to their desirability in obtaining the allowables. The analysis also provides a process for recognising either poor testing procedures or inferior material processing. The Informative Quantile Punotion is applied to the test data in order to seleot the function that best represents the data. The required conservatism in this region is satisfied while alao satisfying criteria for acceptability of the data representation.

A simple computer oode has been made available so that ML estimate of Meibull parameter can be determined thereby resulting in direct computation of the allowables. The current statistical procedures used in the MXL-HDBK-5 is not considered applicable in the determination of the allowables for composite materials. The sslaotion of an adequate statistical model parametric or non- parametrio for representing material strength data can result in either a conjectural approach or s costly teat program.

Often times the normal distrl" bution will be seleoted since the allowables can be readily determined from tables. Most conventional tests for determining model aooeptabllity will rarely rejeat the normal funotion. Zn most instances the 2 parameter Maibull funotion is recommended.

This extrema value funotion will usually provide acceptable estimates of the allowables. The IQ results provided an excellent description of test data in terms of a specified distribution function. Zn applying the Weibull distribution it is important to recognise data with outliers in vicinity of higher ordered values in addition to multi-modal behavior. Data contaminated in this manner will usually reduce accuracy in the Meibull allowable computation.

The Quantile Box Plot 13 was used in determining outliers and multi-modality. This method proved to be more reliable then the conventional robust procedures currently being suggested for determining outliers. Zt is poaaibla that in tastirtq cartain com- poaita material!

In exploring aoma recent taat results from Kevlar, Graphite and Glass oocpositea in addition to ceramic materials resulted in occasional bimodality behavior. In most cases errors in testing or materials conditioning and processing have accounted for this situation. The following robust method for applying exploratory procedural! The disadvantage results from arbitrariness in selection of scale and the tuning constant. In some instances where a large amount of skewnese or a small date sets exists, then the Quantile Box Plot will be dispersion.

Milted The outliers are determined in a formal manner by applying a robust method involving application of the MX. The oomputed weights describe the relative importance of the data points. Por example, a aero weight should indicate exclusion of a point.

The removal of outliers bad data will essentially define robust data. Initially the Huber technique is applied in order to determine a robust location parameter weighted mean. It should be noted that Andrew's function was selected for its ability to describe outliers as data with essentially sero weights. Prom tha ordered statiatio X. Tha-box bcundarias ara dafinad aa Q. Plat slots in Q u indioata modas. Tha obvious biaodality shown in figure 2 ia raprasantad by tha Quantila Box Plot displayed in figure 3. The results shown in figures 2, 3, 4, 5 ara not raprasantativa of typical data seta.

In many instance multi-modality and outliers ara not obvious from routine examination of tha data. Thia mathod providaa an aoourata and slapla approach to tha problam of identifying which function chould rapraaant tha data. With tha idantification oomplatad than mora alaborata procedures ara racommandad inordar to provide varification of tha asaumad nodal. If ona of tha conventional distribution cannot be established as an aocaptabla modal than tha Maximum Penalised Likelihood MPL approach is suggested. Tha details of MPL will ba discussed later in tha text. The Informative Quantile Function is simply defined as 10 where Q u was previously defined in aquation 8.

An example of IQ vs U plot for normal distribution is shown in figure 6. Tha straight line joined at IQ of -. This is introduced in order to provide for an easier identification of the unknown distribution. F x s can be simply and accurately evaluated using the following polynomial representation. Equation IS must be solved in an iterative manner by using the oomputer oode listed in Appendix A. Parametric determination of three parameter and aenaored data requires a more elaborate analysis.

These procedures will not be outlined in this text. Non-oaraaetrie method. In the parametric case, the reliance is on an assumed distribution function which provides extrapolated results for the probability of survival values. The penalty for applying the non-parametrio method is the need for relatively large amounts of data e. In the oase where data values are available, the sixth lowest ranked data value determines the "B a allowable. Table 3 shows which ranked data value should be used for a particular sample sise.

The importance of using sample sixes greater than the required 29 for the "B" allowable is shown in figure 9. By removing the outlier, the allowable t s 4. Iquivalently, wa may maximise the Ln L f ] which separates into two terms - a log likelihood plus a log penalty tern. Values of o that are "too small" result in bumpy estimates while a "too large" overanootha. Zn practice, use an a as small as possible without introducing excessive bumps.

Numerical integration determines the cumulative density values probability of survival for the prescribe percentile, SO or 9S depending on the desired allowable. The teats recommended for determining significant differences are the conventional t or Mann-Whitney Teat and two non-parametric tests for the K - sample oase. The Kruakal-Wallls multi-sample test for identical populations is applied such that X is corrected for ties.

The null hypothesis identical populations is rejected at the 2t level. Where tiea are removed by applying a randomisation process. The null hypothesis of randomness is rejected at the 21 level, that is, aooeptance of dlfferenee in populations at this level.

Statistical Evaluation of Data Graphical Display. In Figure 4, a plot of probability failure Pf versus failure stress see narked circles of empirical failure data were shown. The Pf values were determined from the Rj ranking. The four candidate functions are listed on the graph with their corresponding line fora representations.

In addition to obtaining a visual inspection of best fit, the RMS error provides a quantitative evaluation for the three rankings and the density functions. The sample evaluation for the three rankings and the density functions. The Weibull shape and normalising Char, value parameters are tabulated with confidence intervals. The origin represents one peroent probability of failure with lower confidence limit for that number. Thia representation is the A allowable. If one wants to lnorease or decrease origin percentage, it can be done by applying methods described in 2. The word "origin" will equal sero if two parameter Weibull funotion was considered, otherwise, three parameter Weibull function was used.


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Zn the radical parameter tabulation, A, B and C are ooeffioients obtained from a least squared fit routine. Bxp 8 N and Bxp C R are the corresponding exponents determined in the fitting of the date. That is, the smallest and largest projected values determined by the function. Zn applying the robust procedures, it is important to have a rationale for ignoring the determined outlier, otherwise, erroneous estimates of survival probability computations could result.

The robust scheme can be applied to data from relatively small slse specimens, where errors in machining, twsting, etc. The code oan be altered to include other ellowablee by using the simple relationships outlined in Zn the last three boxes the allowable estimates are tabulated for the weibull, normal and lognormal functions. Results and Discussion Zn figure 4, the results of tension tests on Kevlar composite material Hexcel Co. This is an ideal situation, since selecting ellovables from any of the functions will provide acceptable results.

The RMs error indicates the weibull funtion would be an acceptable representation of data. The results shown in figure 10, describe the existence of an outlier at highest ordered value. See figure 11 display of Box Plot for verification of outlier value It is obvious from figure 10 that Weibull funotion does not represent lower tail region particularly well. The resultant design "B" allowable of The "A" allowable result, Weibull vs. Zn figure 12 tha outlier haa bean removed from tha data diaplayad in figure Note tha aubatantial inoraaaa in tha Waibull ahapa paranatar from 5.

These raaulta indicate need for exploring data prior to applying functional repreaentation. If tha outlier actually exiata, than allowable obtained from noraal distribution should be considered. Figure 13 describes the statistical results from compression tests on Kevlar material obtained from the three manufacturers. An approximate 61 reduction in allowable estimate with addition of the forth data set. The Kruakal-Wallis test for determining K - sample difference indicated the forth monufaoturer data differed slgnifloantly from tha other samples at. Bven though homogeneous data is not available for determining allowables the committee considers it more important to represent the difference among manufacturers particularily so for composite materials.

In figure 15 the higher ordered values appeared to be from a different mode of failure. The Box Plot example shown in figure 1 is a display of this data which essentially oonfirms the existence of blmodallty. It should be noted in figure 15 that none of the candidate functions adequately represent the ranked data. The allowables are too conservative particularly for the Weibull and normal representative.

A suggested alternative in determining allowables for this data is the Maximum Penalised Likelihood Methodic described in the text. The results are shown in this figure are an excellent repreaentation of the ranked data including the sizable bump. The allowables of 5. The later method may have merit if selected percentiles of the distribution are matched with the corresponding ranked values in a manner that guaranteed a good fit.

Zn this case representation of Weibull shape parameter value of 6. The results from MPL method are also shown in the figure including tabulation of the allowables. Manufacturer of this material has recently indicated that lower mode data was incorrectly added to iqppar node data dua to different autoolavaa uaad In processing of material. Figure 17 shows tha raaulta fron Tanaion taat on Ravlar material Cycoo Co.

Figure The existence of bimodality displayed in the sample was a common occurence among several of the graphite test samples. Exploratory Data Procedure should ba applied prior to acceptance of statistical model used in the allowable computation. Quantile Box Plot provides an excellent summary of test data results in addition to location outliers and multi-modality in the sample.

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The authors rocotmand using the Welbull distribution function for obtaining the allowables, if outliers higher ordered values or multi-modality, do not exist in data set. An extreme value distribution Welbull provides a degree of security if pooled samples do not represent general data population, a common occurrence.

Nortal distribution is recommended for allowable computation if outliers exist in data set at upper tail region. In multi-modality case, the Maximum Penalized Likelihood Method le suggested for the allowable determination. At present the authors recommend pooling all samples made available even though significant difference teat indicated otherwise.

If a sample contains modality then this data set would be excluded. The development uf tables 1 and 2 required a considerable amount of effort most complete tabulation for tha Welbull function.

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Non-parametric procedures are always the most desirable in obtaining the allowables for the given sample if properly applied. A sufficiently large number of sources in obtaining test data is more important in determining allowables than size of individual sample. Royal Boo. II, , D. Van Moatrand Company, Inc. The specification for a widely developed theory of "alphabetic optimality" for response surface applications is analysed and found to be unduly limiting.

Ways in which designs might be chosen to satisfy a set of criteria of greater scientific relevance are suggested. Problems are discussed of checking for lack of fit, sequential assembly, orthogonal blocking, estimation of error, estimation of transformations, robustness to bad values, using minimum numbers of points, and employing simple data patterns. Box University of Wisconsin - Madison 1.

Pisher was engaged by Russell 16] on a temporary basis at Rothamstad Experimental Station in "to examine our data and elicit further information that we had missed. In his analysis , [24] , Pisher attempted to relate yield to fertiliser combination, to weather, and in particular, to rainfall.

The method he used was multiple regression with distributed lag models, involving an ingenious employment of orthogonal polynomials which led to important advances in the theory of regression analysis, and in particular its distribution theory. With only the crudest of computations!

A comprehension of the logical problems in drawing conclusions from such analyses led naturally to speculation on how some of the difficulties might be overcome by appropriate ilirii'in. Thereafter, as Fisher gradually acquired more influenoe in the setting up of field trials, the principles of replication, randomisation and their application to randomised blocks, latin squares and factorial designs quickly evolved out of the actual planning, running, and analysis of a series of experimental designs of increasing complexity and baauty. Tha praotloal context of scientific experimentation continued to produce important theoretical advances whtn Yatss cams to Rothamstad in , leading in particular to important developments in tha design and analysis of aomplex factorial designs and thsir associated systems of confounding 44 , 46 and to the introduction of incomplete block designs.

My own experience with experimental design began during tha Saoond world War. I worked at the Chemical Defense Experimental Station in England with a group of medical research workers who were attempting, using animals and volunteers, to find ways to combat the effects of poison gss end other toxic agents. At this time it was believed that these agents might be used not only against the military, but also against the civilian population.

It was important therefore that our work should progress as rapidly as possible. I found myself a part of evolving investigations which employod sequences of experimsnts which 1 designed and whose nature needed to adapt to changing needs at different stages of the study. This lad to the development of whet has come to be oalled response surface methodology. Let us call a design suitable for estimating a general polynomial of degree d a dth order design in k variables. Thus a design. For reasons I will explain, I have reservations about the usefulness of this approach so far as response surface designs are concerned.

For completeness, a brief summary of some of the main ideas are set out below 32 , 33], 34], 35], 36 , 42], A motivation for simplification is provided by considering the confidence region 1 for 0 Obviously there are also parallel fiducial and Bayesian rationalisations. In particular we might choose, for m m two of these. It is traditionally assumed that the p p-1 elements ooncerned with orientation of the ellipsoid ars of no intersst, and attsntlon has besn conosntratsd on partioular oritsria which measure in some way or another the sisee of the eigenvalues, measuring some combination of sise and sphericity of the confidence ellipsoid.

But in practical situations, each of these criteria will take smaller and hence more desirable values as the ranges for the experimental variables x are taken larger and larger. To oope with this problem it is usually assumed that the experimental variables x may vary only within some exaotly known region in the spaoe of X, but not outside it.

I will call this permissible region RO. A further important result of Kiefer and Wolfowits linked the problem of estimating B with that of estimating the raeponse n via the property of m "O-optimality. The design is set out in Figure 4 b. Zn the hope of stimulating paw initiative, I have set out below what I believe is the scientific context for response surface studies and indicated soma possible lines of development.

Problems of nonlinear experimental design then arise which are of considerable interest although they have received comparatively ILttle attention hou Cor example 13 , 18 , For example when e first degree polynomial approximation could be employed it might, via the method of stewpeat ascent, be ueed to find a new region of interact where, aay, the yield was higher. Alao a maximum in many variablea le often represented by 4 some rather complicated ridge system and e second degree polynomial approximation when suitably analysed might be used to elucidate, describe, and exploit such a eystem.

Thus we are typically involved in using e sequence of designs, each making use of information gleaned from earlier experiments — a characteristic typioal of a much wider field of scientific investigation. This provides the 4 Empirical evidence suggests this. Also, integration of sets of differential equations which describe the kinetics of chemical systems almost invariably leads to ridge systems 4 , IS , 2b], Thus in any realistio view of the process of investigation the dimensions, identity, location and metrics of measurement of reqions of interest in the experimental space ere all iteratively evolving.

The problem of choosing suitable experimental designs in such e context ia a difficult one. Soma properties 5 , [8] of a response surface design, any, all or some of which might in different circumstances be of importance in the above context are given in Table 1. TABLE 1. In the response surface context a number of questions arise concerning the appropriateness of the specification set out in Section 2 of this paper for alphabetic optimality. In particular, it is obvious that this must be so for any investigation in which we allow the possibility that results of one design may allow progress to a different unexplored region.

Consequently X believe that formulation in terms of an RO region which assumes that R and 0 are mm mm identical is artificial and limiting. Also einoe typically R is only vaguely known, we will want to consider the information function over a wider region, as is done for example in Figure 5 for Pedorov's second order D-optimal design.

The information function for this design may now be compared over this wider region with that for the 3 2 factorial in Figure 3. For response surface studies however, it is far from clear how desirable is ths property of G-optimality itself. In some subregions one design is slightly better, and in others the other design is slightly better. Zf the idea of tho desirability ot designs possessing a particular kind of information profile is basic, then it seams unsatisfactory that the nature of that profile should depend so vtry much on ths order of the design.

Indeed, the relevanoe of the minimax criterion which produces U- optimality ie arguable. But as we have seen i D- optlmality is only one of many single-valued criteria that might ba used In attempts to describe some important characteristic of ths X'X matrix. But I would argue that sines the Information function Itself is ths most direct measure of desirability so far as the single issue of variance properties Is concerned, our best course ie to choose our design directly by picking s suitable information function, and not indirectly by finding soma extremum for A, E, D, or other rtutvary criterion.

Sensitivity of crlts rla to size and shape In the process of scientific Investigation, the investigator and the statistician must do a great deal of guesswork. In matohing the region of interest R and the degree of complexity of the approximating function, they j must try to take into account, for example, that a more flexible second degree l approximating polynomial can be expected to be adequate over a larger region K than a first degree approximation.

Obviously different experimenters would have different ideas of appropriate looatione and ranges for experimental variables, in particular, ranges could easily differ from one experimenter to R another by a factor of two or more. For example, suppose each dimension of a dth order experimental design is increased 7 by s fsotor o.

This convention la not entirely satisfactory, but will of course give vary different results from those whiah assume design points to be all included in the same region RO. It is important to be aware that the apparent superiority of ono design over another will often disappear if tha method of scaling tha design ia changed. In particular this applies to comparieone suoh as those made by Nallmov at al [40] and Lucas [36].

Equivalently here, to illustrate soale seneitivity, we concentrate attention on the factor c by which each scale would naad to be inflated to achieve the same value of the D criterion. Obviously no experimenter can guess to anything approaching this accuracy what are suitable ranges ovar which to vary these factors. Obviously choice of region and choict of information function arc closely interlinked. For example, net of N - k-t-1 points in k-spaco which have no coplanarities is obviously a D-optimal first order design for some ellipsoidal region. Thus the prob 1 m of design ie not so much a question of choosing the design to increabe cotal Information ns spreading the total information around in the man.

For second and higher order designs, a requirement of rotatability fixes many moment properties of the design, but V and hence I are still to some extent at our choice, and P P can be changed by changing certain moment ratios [11]. The truth seems to be that at any particular phase of an investigation the scientific decision that must contributes to the outcome of that phase it; the choice of the current region of interest involving choice of variables, locations, range? After this decision is made, and given the assumption that the model fits perfectly so that only the variance properties Figure 7 a Information function for a second order rotatable design consisting of 8 points on a circle plus 4 center points.

This aphorism Is particularly true for empirical functions such as polynomials that make no claim to do more then locally graduate the true funotion. For chemical oxamplas some idea of the adequaay of euch approximations can be gained by Q studying surfaces produced by chemical kinetio models.

An example taken from 10] is shown in Figure 8. See also One conclusion 1 reached from many such studies wee that approximations would not need to be very good for response surface methods to work. Thus within region A of Figure 8 the locally monotonic function could be crudely approximated by a plane which could indicate a useful path of ascent.

Also valuable information might be obtained about a ridge suoh as that in regLon B, even though the underlying surface was not exactly quadratic. The optimality oriteria discussed earlier which assume the response function to be exact usually produce a substantial proportion of sxpsrimsntal points on the boundary of MO.

To see this, consider again the formulation given earlier in terms of n region of interest R and a larger region 0 of operability.

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Draper and J. Figure 9 shows a situation as it might exist for a single variable when a straight line approximating function is to be used. We see how V becomes very large if the spread of the design is made very small, while if the design is made very large, V slowly approaches its minimum value of unity. The average squared bias B, on the other hand, has a minimum value when Xq shout 0,7, and increases for larger or smaller designs.

Progress may ba made however by supposing that n X is t. This would suggest that the experimenter would tend to choose the sise of his region R, and the degree of his approximating function in such a way that the integrated random error and the integrated systematic error were about equal. Thus we might suppose that a situation of particular interest is that where B is roughly equal to V.

Consequently we suggested that, if a simplification were bo be made in the design problem, it might almost be better to ignore the effeots of sampling variation rather than those of bias. As a factor in our final choice, this should oertainly receive as much attention as the indications supplied by, say, D-optimality. The former is the familiar 2 3 factorial scaled so that the points are 0,71 units from the center. The latter is a rotatable composite design with "cube" points at a distance 0. Figura 11 b A aacond ordar compoaita rotatabla daaign which niniaisaa aquarad biaa front third ordar tarma whan tha waight function ia uniform ovar a apharical ragion of intaraat of unit radiua.

Obviously in practice because of the inevitable inexactness of choosing scales exact dimensions of the designs should not be taken too seriously, but these examples illustrate the fact that as 30on as we take account of bias, design points are not chosen on the boundary of R. Choice of designs which minimize piaa Before considering the problem of choosing minimum bias designs it is desirable to generalize slightly the previous formulation.

Although it avoids limiting the location of the design points in an artificial way the idea of a region of interest R within a larger operability region 0 ie still not entirely satisfactory because it implies that we have equal interest at all points within R. Suppose as before the fitted function is a polynomial X. Lack of Fit ill. Sequential Assembly vl , Blocking v , Estimation of Error vli , Transformation Estimation iv while the adequacy of a particular approximating function to explore a region of current interest is always to some extent a matter of guesswork, simple approximations requiring fewer runs for their elucidation will usually be preferred to more complicated ones.

This leads to a strategy of building up from simpler models, rather than down from more complicated ones. A ! For illustration, Figure 13 shows the sequential assembly of a design arranged in three orthogonal blocks, each of six runs, labeled I, II, and III. Block I is a first ordsr design but also provides a check for overall curvature obtained by contrasting the average response of the center points with the average response on the cube. A single contrast of the center response is available as a gross chsck on previous information about ex lmental error.

If after analysing the results from Block I there are doubts about the adequacy uf a first degree polynomial model, Block II may be performed. The complete design also provides orthogonal checking contrasts for lack of quadraticity in each of the three directions 9 , These contrasts can also be regarded as checking the need for transformation in each of the X's. Finally if it were decided that more information about experimental error was desirable, the replication of the star in a further Block IV could furnish this, and also provide some increase in the robustness of the design to wild observations.

Robustness Approaches to the robust design of experiments have been recently reviewed by Herzberg [28]; see also [29]. In particular. Thus G-optimal designs are optimally robust in this sense. Size of the experimental design A good experimental design is one which focuses experimental effort on what is judged important in the particular current experimental context. However the importance of checking fit, blocking, and obtaining an independent estimate of error will differ in different circumstances, and the minimum value of n will thus correspondingly differ.

When 0 is larger designs larger than the i minimum design will be needed to obtain estimates of sufficient precision. The etatiatician's task as a member of a scientific team is a dual one, involving inductive criticism and deductiva estimation. The latter involves deducing in the light of the data the consequences of given assumptions estimating the fitted function , and title can certainly be dona with haphazard designs.

But tits former involves the question a of what function should be fitted in the first place, and b of how to examine residuals from the fitted function in an attempt to understand deviations from the initial model, in particular in relation to the independent variables, and so to be led to appropriate model modification. Designs such as factorials and composite response surface designs employ patterns of experimental points that allow many such comparisons to be made, both for the original observations and for the residuals from any fitted function, l-'or example, consider a 3 factorial design used to elucidate the effects of temperature and concentration on some response such as yield.

Intelligent inductive criticism is greatly enhanced by the possibility of being able to plot the original data and residuals against temperature for each individual level of concentration, and against concentration for each individual level of temperature. Optimum levels of these criteria can be useful as bench marks in judging the efficiencies of a particular design with respect to these various criteria. An optimal design is represented by a point in the multi dimensional space of the coordinates of the design and a series of different f 28 criteria will give a series of such extremal points which can be very differently located.

Obviously knowledge of the location of such extrema may tell us almost nothing about the location of good compromises. For this we would need to study the joint behaviour of the criterion functions at levels close to their extremal values. One limited but useful step would be to further investigate which criteria are in accord, such as G-optimality and robustness to wild observations and which in conflict such as variance and bias. Clearly we must learn to live with scientific arbitrariness, or slae we are in a world of make believe. But we can make the problems worse, not better, by introducing arbitrariness for purely mathematical reasons.

Planning experiments to detect Inadequate regression models. Biometrika , 59, Atkinson, A. Biometrika , 60, Atwood, C. Optimal and efficient designs of experiments. Box, G. The exploration and exploitation of response surfacesi Some general considerations and examples. Biometrics 10, Response surfaces. Sills, New York! MacMillan and Free Press. A basis for the selection of a response surface design. Statist, Assoc. The ohoioe of a second order rotatable design.

Biometrika , 50, Robust designs. Biometrika, 62, Measures of lack of fit for rseponee surface designs and predictor variable transformations. To appear in Technometrice. Empirical Model Building with Response Surfaces. Book in preparation. Multi-factor experimental designs for exploring response surfaces. Statistics tor Experimenters. Design of Experiments in nonlinear situations. Blometrlks , 46, On the experimental attainment of optimum conditions, J. B, 13, , [15] Box, G. An example of the link between the fitted surface and the basic mechanism of the system.

Biometrics , 11, On the nonrandomized optimality and the randomized nonoptimality of symmetrical designs. Optimum experimental designs.


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H , 21, Optimal design; variation in structure and performance under change of criterion. Optimal designs for large degree polynomial regression. Ann, Statist. Optimum designs in regression problems. The equivalence of two extremum problems. Canadian J. A review of response surface methodology from a biometric viewpoint. Biometrics , 31, On practical use of the conaept of D-optimality.

Tschnometries , 12, Chenu Bngr, Science. The design and analysis of factorial experiments. Imperial Bureau of Soil Scienae. A fresh look at the basic principles of the design and analysis of experiments, in Proc, 5th Derkelev Symp. Fl3her; The Life of a Scientist. J, The occurrence of replications in the optimal dasigns of experiments to estimate parameters in non-linear models. B , 30, Experiments for nonlinear functions.

Assoc, , 68, Block effoots in the determination of optimum conditions. Biometrics , 12, Sequential designs for spherical weight functions. Technometrics , 9, , [21] Fedorov, V,V. Theory of Optimal Experiments. New York and Londont Academic Press. Studies in crop variation. An examination of the yield of dressed grain from Broadbalk. The manurial response of different potato varieties. The influence of rainfall on the yield of wheat at Rothamsted. The Design of Experiments. Oliver and Boyd. L,, Pinchbeck, P. A statistical approach to catalyst development. Part 1. The effect of process variables on the vapour phase oxidation of naphthalene.

Inst, Chem, Engra. Are theoretical designs applicable? The robust design of experiments! Some considerations in the optimal design of experiments in non-optimal situations. B , 38, Recent work on the design of experiments! A bibliography and a review. J, R, Statist. A , , A review of response surface methodology? Technoinetrles, 8, Presented here is a specific case in which the design was readily apparent but where difficulties subsequently arose. A charge within the rocket is ignited, causing the skin of the rocket to peel away.

This allows the undetonated bomblets to be sprayed over the target area; as the bomblets fall to the ground, a portion of them will impact the target. Dispersing is the process by which the bomblets are delivered from the rocket to the target Tn particular, the customer was concerned with the ignition of the charge within the rocket which causes the skin of the rocket to peel away.



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