what went wrong? units

Translation of the allegory designed by the Ikhwān al-Ṣafā’ to explain the calculation of the Great Year.

Know further, my brother, that the wise men of India have formulated an allegory of the revolutions of these stars around the Earth, so that the comprehension of this may be brought closer to those who learn and that its representation may be made easier to those who meditate. According to their report, a king among kings built a city with a circumference of 60 parasangs and sent seven individuals to revolve around it at different paces: one of them 1 parasang every day, another one 2 parasangs every day, the third 3 parasangs every day, the
fourth 4 parasangs every day, the fifth 5 parasangs every day, the sixth 6 parasangs every day, and the seventh, 7 parasangs every day. And the king told them: ‘Revolve around this city, starting from this door; when, by the number of your revolutions, you join together at the same door, come [to me] and announce to me how many revolutions each one you will have completed’. Whoever has understood the computation of the revolutions of these individuals around the city and succeeded to figure it by imagination is likely to understand the revolutions of these stars around the Earth, and to figure out how many revolutions it will take them to join together in the first [minute] of Aries, in the same way as they were when they started.
As for the computation [concerning] these individuals, it is as follows. After 60 days, 6 individuals join together at the door of the city. The first one has completed 1 revolution, the second 2 revolutions, the third 3 revolutions, the fourth 4 revolutions, the fifth 5 revolutions, the sixth 6 revolutions. As for the one who completes, every day, 7 [parasangs], he has completed 8 revolutions plus 4/7 of the parasangs of a revolution, so that these individuals must renew the revolution. After 120 days, they join together once again at the door
and each one has completed his first count once again, but the seventh has completed 17 revolutions plus 1/7 of the parasangs [of a revolution], so that they must renew the revolution. After 180 days, the six join together for the third time and each one has completed, for the third time, his first count, but the seventh companion has completed 25 revolutions plus 5/7, so that they must renew the revolution. After 240 days, they join together for the fourth time and each one of them has completed his first count, but the seventh companion has completed 34 revolutions plus 2/7, so that they must renew the revolution. After 300 days, they join
together for the fifth time, but the seventh companion has completed 42 revolutions plus 6/7, so that they must renew the revolution. After 360 days, they join together for the sixth time and each one of them has completed his first count for the sixth time, but the seventh companion has completed 51 revolutions plus 3/7 of the parasangs [of the revolution], so that they must renew the revolution. After 420 days, all of them join together at the door of the city: the first one has completed 7 revolutions, the second 14 revolutions, the third 21 revolutions, the fourth 28 revolutions, the fifth 35 revolutions, the sixth 42 revolutions, and the seventh has completed 60 revolutions. This is the allegory that the wise men of India have invented in order to account for the revolutions of the spheres and the stars around the Earth. Thus, the Earth is like this city that was built with a circumference of 60 parasangs. The seven planets and their revolutions around the Earth are like these seven individuals. The difference in speed and slowness between their movements are like the differences between the courses of these individuals. As for the king, he is the God, the Creator, the Founder – Blessed be the God, Lord of the Worlds.

Head and Tail of the Dragon : The Lunar Nodes

The Classical Astrologer


From the point of view of own planet, the Sun and the Moon appear to be about the same size. This is because the Sun is four hundred times larger than the Moon, but also about four hundred times the distance from Earth. From earliest times this has exerted a strong psychological relationship to the two luminaries, encouraging the belief that they are of equal but different importance. The Sun may be the source of all energies, but it is the Moon that determines the tides and has the most immediate effect on our moods and earthly activities. The Moon in astrological practise is therefore of great significance. In fact in most systems, she is given much more importance than the current frenzy for Sun sign astrology would imply! The Moon’s Nodes are not planets. They are points in space indicating the intersection of two orbits.

The elements to be considered are a) the Ecliptic…

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by Cleve Moler

Online Resources

Book Summary

Numerical Computing with MATLAB is a textbook for an introductory course in numerical methods, MATLAB, and technical computing. It emphasizes the informed use of mathematical software. Topics include matrix computation, interpolation and zero finding, differential equations, random numbers, and Fourier analysis.

Based on MATLAB, the textbook provides more than 70 M-files. Many of the more than 200 exercises involve modifying and extending these programs. The book also makes extensive use of computer graphics, including interactive graphical expositions of numerical algorithms.

Two editions of the book have been published. An electronic edition, published by The MathWorks, is available from this Web site. A traditional textbook print edition, published by the Society for Industrial and Applied Mathematics, is available from the SIAM Web site.

Mitochondrial Eve

Allan Charles Wilson (18 October 1934 – 21 July 1991) was a Professor of Biochemistry at the University of California, Berkeley, a pioneer in the use of molecular approaches to understand evolutionary change and reconstruct phylogenies, and a revolutionary contributor to the study of human evolution. He was one of the most controversial figures in post-war biology; his work attracted a great deal of attention both from within and outside the academic world. He is the only New Zealander to have won the MacArthur Fellowship.[1]

He is best known for experimental demonstration of the concept of the molecular clock (with his doctoral student Vincent Sarich), which was theoretically postulated by Linus Pauling and Emile Zuckerkandl, revolutionary insights into the nature of the molecular anthropology of higher primates and human evolution, called Mitochondrial Eve hypothesis (with his doctoral students Rebecca L. Cann and Mark Stoneking).[2][3]

Wilson joined the UC Berkeley faculty of biochemistry in 1964, and was promoted to full professor in 1972.[8] His first major scientific contribution was published as Immunological Time-Scale For Hominid Evolution in the journal Science in December 1967.[16] With his student Vincent Sarich,[17][18] he showed that evolutionary relationships of the humanspecies with other primates, in particular the Great Apes (chimpanzees, gorillas, and orangutans), could be inferred from molecular evidence obtained from living species, rather than solely from fossils of extinct creatures. Their microcomplement fixation method (see complement system) measured the strength of the immune reaction between an antigen(serum albumin) from one species and an antibody raised against the same antigen in another species. The strength of the antibody-antigen reaction was known to be stronger between more closely related species: their innovation was to measure it quantitatively among many species pairs as an “immunological distance“. When these distances were plotted against the divergence times of species pair with well-established evolutionary histories, the data showed that the molecular difference increased linearly with time, in what was termed a “molecular clock“. Given this calibration curve, the time of divergence between species pairs with unknown or uncertain fossil histories could be inferred. Most controversially, their data suggested that divergence times between humans, chimpanzees, and gorillas were on the order of 3~5 million years, far less than the estimates of 9~30 million years accepted by conventional paleoanthropologists from fossil hominids such as Ramapithecus. This ‘recent origin’ theory of human/ape divergence remained controversial until the discovery of the “Lucy” fossils in 1974.[15]

Wilson and another PhD student Mary-Claire King subsequently compared several lines of genetic evidence (immunology, amino acid differences, and protein electrophoresis) on the divergence of humans and chimpanzees, and showed that all methods agreed that the two species were >99% similar.[4][19] Given the large organismal differences between the two species in the absence of large genetic differences, King and Wilson argued that it was not structural gene differences that were responsible for species differences, butgene regulation of those differences, that is, the timing and manner in which near-identical gene products are assembled during embryology and development. In combination with the “molecular clock” hypothesis, this contrasted sharply with the accepted view that larger or smaller organismal differences were due to large or smaller rates of genetic divergence.

In the early 1980s, Wilson further refined traditional anthropological thinking with his work with PhD students Rebecca Cann and Mark Stoneking on the so-called “Mitochondrial Eve” hypothesis.[20] In his efforts to identify informative genetic markers for tracking human evolutionary history, he focused on mitochondrial DNA (mtDNA) — genes that are found in mitochondria in the cytoplasm of the cell outside the nucleus. Because of its location in the cytoplasm, mtDNA is passed exclusively from mother to child, the father making no contribution, and in the absence of genetic recombination defines female lineages over evolutionary timescales. Because it also mutates rapidly, it is possible to measure the small genetic differences between individual within species by restriction endonuclease gene mapping. Wilson, Cann, and Stoneking measured differences among many individuals from different human continental groups, and found that humans from Africa showed the greatest inter-individual differences, consistent with an African origin of the human species (the so-called “Out of Africa” hypothesis). The data further indicated that all living humans shared a common maternal ancestor, who lived in Africa only a few hundreds of thousands of years ago. This common ancestor became widely known in the media and popular culture as the Mitochondrial Eve. This had the unfortunate and erroneous implication that only a single female lived at that time, when in fact the occurrence of a coalescent ancestor is a necessary consequence of population genetic theory, and the Mitochondrial Eve would have been only one of many humans (male and female) alive at that time.[2][3] This finding was, like his earlier results, not readily accepted by anthropologists. Conventional hypothesis was that various human continental groups had evolved from diverse ancestors, over several million of years since divergence from chimpanzees. The mtDNA data, however, strongly suggested that all humans descended from a common, quite recent, African mother.[4][15]

Proc Natl Acad Sci U S A. 1969 Aug;63(4):1088-93.

A molecular time scale for human evolution.


We discuss published molecular evidence concerning the relationship of man to African apes and Old World monkeys. Quantitative comparisons of their serum albumins, transferrins, hemoglobins, and DNA show that man is genetically much more similar to the African apes than to the Old World monkeys. The amino acid sequences of hemoglobins from humans, chimpanzees, gorillas, and rhesus monkeys are consistent with the hypothesis that the probability of an amino acid substitution occurring in a given interval of time is the same for every hemoglobin lineage. This allows the use of these data as a hemoglobin evolutionary clock, just as we have previously done with the albumins. It is shown that concordance exists between the hemoglobin and albumin results and that both support the suggestion that the human lineage diverged from that leading to the African apes far more recently than is generally supposed. Considering both the albumin and hemoglobin data, we would set the most probable date at 4 to 5 million years.

[PubMed – indexed for MEDLINE]

Free PMC Article