To the chief editor of the magazine
"Newsletter of the Russian Academy of Sciences"
academician N.А.Plate
Dear Nikolay Alfredovich!
You answered with refusal to my request to you to publish on the pages of "Newsletter of the Russian Academy of Sciences" my open letter to the President of the Russian Academy of Sciences academician Yu.S.Osipov with answer to critics. Instead of this you proposed me to write an article with an answer to publication of Yu.N.Efremov, Yu.А.Zavenyagin with a foreword of V.L.Ginzburg, which appeared in the Newsletter of the RAS, 1999, issue 12, pp.1081-1092. I am sending you an answer to this publication and ask you to publish it in your magazine.
Yours sincerely, academician А.Т.Fomenko
March 21, 2000
ANSWER to the publication of
Yu.N.Efremov, Yu.А.Zavenyagin with a foreword
of academician V.L.Ginzburg,
which appeared in the Newsletter of the RAS, 1999, issue 12
А.Т.Fomenko, G.V.Nosovskiy.
In the article of Yu.N.Efremov, Yu.А.Zavenyagin the following arguments against our dating of the star catalogue of Almagest are stated [м1], [м2].
1. The authors don't agree with our notice, that the starting point of longitudes in the catalogue of Almagest is not so single-valued. Half of the section "Almagest and its dating" of the article of Yu.N.Efremov, Yu.А.Zavenyagin is devoted to discussion of this question. This is the content of the second point of accusations, stated on p.1088 of the article [м13].
ANSWER. In our method of dating of the star catalogue of Almagest the position of the starting point of longitudes is not used anywhere. A notice regarding his point, provided in our book [м1], [м2] which induced such wordy comment of Yu.N.Efremov and Yu.А.Zavenyagin, is absolutely inconsiderable for our method of dating.
Dating of the catalogue of Almagest by longitudes, by means of proper motions, was done by us in [м2], pp.176-178. Another question is that its accuracy turned to be considerably lower, than by latitudes. The reason is that longitudes in Almagest are less accurate than latitudes, what Yu.N.Efremov and Yu.А.Zavenyagin should know. So, there is no reason for them to convince the readers that we refused the dating by longitudes [м13], p.1083. It is not so.
As for the dating of the catalogue based on precession, see below point 2.
This is, actually, the only direct "objection" in the article [м13] against our dating of the catalogue of Almagest. All the other objections are of an indirect character and come down to the following: your dating can't be correct, as other calculations, not resting on the catalogue of Almagest, according to Yu.N.Efremov and Yu.А.Zavenyagin, contradict with it. See point 2 on this issue.
2. Yu.N.Efremov and Yu.А.Zavenyagin, referring to the works of different researchers, who tried to date Almagest and other old evidences of astronomical content, point at the contradictions between these works and our dating. The following examples are given.
2а. Dating of the catalogue of Almagest by longitudes based on precession gives the I century A.D.
2б. Dating by declinations of the stars gives the epoch of around the beginning of the Common Era (See accusation number 5 on p.1088 in [м13]).
2в. Babylonian astronomical documents "uniquely prove the antiquity of the ancient history" [м13], p.1088. (See accusation number 1 on p.1088 in [м13]).
ANSWER. We in principle looked for such methods of dating of Almagest, which are based on astronomical characteristics and principles, unknown right until the XVIII century. The reasoning for such methodology is a separate question, which we have no possibility to discuss here. In any case, this principle was clearly expressed by us in our book [м1],[м2] and consistently realized. That's why we didn't use for dating inclinations of stars, positions of the Sun, and, moreover, precessions of longitudes. All such characteristics and based on them calculated dates easily could be calculated to the past by the astronomers of the XVII century, and by means of longitude precession – much earlier. We know that close to the Scaligerian datings appear based on such data. The discovered by us fact is that the use of the data of other type, which couldn't be calculated in the XVII century, leads to absolutely different datings. That's why "objections" of Yu.N.Efremov and Yu.А.Zavenyagin are connected just with misunderstanding or unwillingness to understand, the general principles of our approach.
As for the "Babylon astronomical notes", they are not mentioned in our book about the dating of Almagest. This is a separate topic, which requires a deep analysis, not just mentioning in passing [м13], p.1088. We would note that researchers, who date and interpret such old documents, as a rule, don't doubt in the traditional chronology and very often, consciously or subconsciously, use ones or others its consequences. We came across a big number of such examples. Babylon tables are not an exception. But, once again, this topic has no relation to our book on the dating of the catalogue of Almagest.
3. Discussing our dating of covers of the stars with planets, Yu.N.Efremov and Yu.А.Zavenyagin are outraged with the fact, that we didn't use calendar indications about months and days, provided in Almagest (accusation number 6 on p.1088 in [м13]).
ANSWER. This is connected with the same circumstance, which we explained above, in point 2. Month and day – this is actually a position of the Sun, which is a characteristic, calculated with the means of the XVII century. Besides, a traditional interpretation of provided in Almagest names of months and their conversion into modern calendar – is not an obvious thing, which requires a separate conversation.
4. Yu.N.Efremov and Yu.А.Zavenyagin, apparently, didn't understand our research on definition of the participants of the homogeneity of systematical mistake in the catalogue of Almagest. They write: "Contradictory to all known data... supposition that lists of the catalogue were received by different observers, is one of the main arguments of Fomenko at the choice of the sky fields, which allegedly were better observed" [м13], p.1086. Seems that exactly this misunderstanding served a reason for rather misty accusation in point 3 on p.1088 of the work [м13].
ANSWER. Yu.N.Efremov and Yu.А.Zavenyagin lie, ascribing us such suppositions. We have no "suppositions" of such type. As "our suppositions" Yu.N.Efremov and Yu.А.Zavenyagin give our explanations of possible (but not obligatory) reasons of found by us statistical fact of inhomogeneity of systematical mistake in the catalogue of Almagest. The reasons could be different. The presence of different observations is just one of them. This could be or not. Our method and our results absolutely don't depend on this. Such "objection" of Yu.N.Efremov and Yu.А.Zavenyagin simply looks strange. Appears a doubt, that they understand the core of the issue.
5. Especially strange is accusation number 4 on p.1088 in [м13].
Yu.N.Efremov and Yu.А.Zavenyagin write the following. We can't but quote fully this fragment. <<Why all, reached us, ancient, including Arabian, catalogues, in which the coordinates of the stars were just recalculated on the corresponding epochs with the coordinates of "Almagest", came from the same ancient epoch of the catalogue of "Aalmagest"?>> [м13], p.1088. It is asked, how Yu.N.Efremov and Yu.А.Zavenyagin learned the "content of the drafts" and intermediate calculations of the medieval authors? It is absolutely clear, that their confirmation fully rests on an absolute belief in the Scaligerian chronology, based on which such conclusions are made.
6. In conclusion we will settle on the afterword of Yu.N.Efremov, in which he propose his own, together with A.K.Dambis, dating of the catalogue of Almagest. As a result of his studies Yu.N.Efremov on p.1090 provides two graphs. The first is dependence of the received by him by means of proper motions of the epoch of the catalogue of Almagest from the number of used stars in decreasing order of the value of their proper motion. The second is the same, but dependence is from the number of excluded from the analysis fast stars in the order of decrease of the value of their proper motion. Around "precise datings" there are drawn intervals, which Yu.N.Efremov calls "intervals of mean square errors". And which, according to Yu.N.Efremov, estimate the mistakes of his method. This comes out from the text on p.1090. Even at a fleet glance at the graphs it immediately strikes the eye that the accuracy of "the method of Efremov" in a strange way doesn't change in case of the first graph, or nearly doesn't change in case of the second graph, when putting of the most fast stars. How Yu.N.Efremov and A.V.Dambis manage to date the catalogue of Almagest with the accuracy plus-minus 400 years, having put off twenty most fast stars, that is all noticeably moving stars of Almagest, - remains a mystery. This is the same as to date the catalogue of Almagest by means of proper motions, using motionless stars, that is stars, which nearly have no proper motion. In case when Yu.N.Efremov and A.V.Dambis consider all the stars of Almagest, including the fastest of them, the accuracy of their dating is absolutely fantastic – allegedly plus-minus around 100 years. Elementary estimations, received by means of dividing a characteristic mistake of Almagest on the speed of movement of the fastest, reliably identified in Almagest stars, show that we can't reach the accuracy of dating according to proper motions better, than plus-minus 300-350 years. Moreover, there are few such stars, just several. Most stars are nearly motionless. That's why, having put off 20 fastest stars, Yu.N.Efremov and A.V.Dambis can try to date the catalogue with the accuracy not better, than plus-minus several thousand years. Yu.N.Efremov once already made a serious mistake in the estimation of the accuracy of his "method" in work [м12]. The mistake of Yu.N.Efremov was in detail discussed by us in the book [м1],[м2] and in the article [м5]. Nevertheless, Yu.N.Efremov again repeats the mistake in the same place. We once again have to provide for Yu.N.Efremov and A.K.Dambis a simple arithmetical calculation, showing the absurdness of his claims for the accuracy of dating of the catalogue of Almagest by means of proper motions.
It is clear that the accuracy of any method of dating by means of proper motion of a fast star is estimated from below with an individual mistake of the position of the considered fast star in Almagest, divided into the speed of its proper motion. If there were many such stars - N items, then we could increase the accuracy, dividing for around a root of N. But, as we already told, there are few fast stars in the catalogue of Almagest and the speed of proper motions in their row quickly falls. That's why an invariable estimation from the above for the accuracy of the method will be a calculation by the fastest from the reliably identified stars - Arcturus. In general, for dating by means of proper motions really it is possible to use not more than 20 stars of Almagest, as the others are nearly motionless. Yu.N.Efremov actually acknowledges this, saying: "All the 1022 stars were used, slow stars set a system of coordinates" [м13], p.1089. So, slow stars are useful only for setting a system of coordinates, but not for dating.
All the stars of Almagest are measured with some mistakes. Of course, this is also related to slow stars, setting for Yu.N.Efremov and A.K.Dambis a system of coordinates. But, let's suppose for a second, that slow stars are ideally precisely measured in Almagest. Even in an ideal case a mistake in the position of Arcturus in Almagest can't be considered less than 10 minutes at any of the coordinates. Such is the coordinates scale gradation of the star catalogue of Almagest. Really the border should be increased due to inaccuracy of the coordinates of the surrounding stars.
With this a mistake in the arch distance is around 14 minutes of the arch. If for each coordinate a possible mistake is 10 minutes of the arch, than for the hypotenuse – according to the theorem of Pythagor – it is equal to 14 minutes of the arch. The speed of the proper motion of Arcturus is around 2 seconds of the arch per year. So, Arcturus passes the distance of 14 minutes of the arch for around 420 years. So, plus-minus 400 years – this is just a rough estimation of the accuracy of dating by Arcturus, with the use of the arch distances, which are latitudes together with longitudes. The use of latitudes separately lets to increase a bit the accuracy of the method and to get a date with the accuracy until plus-minus 300 years. Further increase of the accuracy of dating by means of proper motions of any stars in case of the catalogue of Almagest is in principle impossible. The use for such purpose of fast, but unreliably identified in Almagest stars, leads to a vicious circle. As, for example, in case with Omicron -2 Eridana.
After all this a strange impression produce the words of academician V.L.Ginzburg from his foreword, that in the work of Yu.N.Efremov and Yu.A.Zavenyagin he finally met a "clear and accurate analysis of the mistakes of А.Т.Fomenko" [м13], p.1081. It is reasonable to ask – what exactly in a purely demagogical work of Yu.N.Efremov and Yu.A.Zavenyagin seemed to academician V.L.Ginzburg clear and accurate? Did he enter into the core of the problem?
Literature:
[м1] Fomenko A.T., Kalashnikov V.V., Nosovsky G.V. Geometrical and Statistical Methods of Analysis of Star Configurations. Dating Ptolemy's Almagest. - CRC Press. 1993, USA.
[м2] Kalashnikov V.V., Nosovskiy G.V., Fomenko A.T. Dating of the star catalogue of "Almagest". Statistical and geometric analysis. – Moscow, publishing house "Factorial", 1995.
[м3] Kalashnikov V.V., Nosovskiy G.V., Fomenko A.T. Geometry of flexible configurations of stars and dating of Almagest. – Problems of steadiness of stochastic models. Works of the seminar. M.,
VNIISI, 1988, pp.59-78.
[м4] Kalashnikov V.V., Nosovskiy G.V., Fomenko A.T. Statistical analysis and dating of observations, forming the basis of the star catalogue from "Almagest". – Theses from the reports of the 5th International conference on the theory of probabilities and mathematical statistics. Vilnius, Institute of mathematics and cybernetics of the AS of Lithuanian SSR, 1989, v.3, pp.271-272.
[м5] Kalashnikov V.V., Nosovskiy G.V., Fomenko A.T. Dating the Almagest by variable star configurations. – Reports of the AS USSR, 1989, v.307, No.4, pp.829-832.
English translation: Fomenko A. T., Kalashnikov V. V., Nosovsky G. V. Dating the Almagest by variable star configurations. - Soviet Phys. Dokl. vol.34, 1989, No.8, pp.666-668.
[м6] Kalashnikov V.V., Nosovskiy G.V., Fomenko A.T. The star catalogue of Ptolemy is dated by mathematics. - "Suppositions, forecasts. Future of science". International annual. 1990. Issue 23. Moscow, publishing house "Znaniye", pp.78-92.
[м7] Kalashnikov V.V., Nosovskiy G.V., Fomenko A.T. Statistical analysis of the star catalogue of "Almagest". – Reports of the AS USSR. 1990, v.313, No.6, pp.1315-1320.
[м8] Fomenko A.T., Kalashnikov V.V., Nosovsky G.V. When was Ptolemy's star catalogue in "Almagest" compiled in reality?
Preprint, No. 1989-04, ISSN 0347-2809. Dept. of Math. Chalmers Univ. of Technology, The University of Goteborg. Sweden.
[м9] Fomenko A.T., Kalashnikov V.V., Nosovsky G.V. When was Ptolemy's star catalogue in "Almagest" compiled in reality?
Statistical Analysis. - Acta Applicandae Mathematical. 1989. Vol.17, pp.203-229.
[м10] Fomenko A.T., Kalashnikov V.V., Nosovsky G.V. The dating of Ptolemy's Almagest based on the coverings of the stars and on lunar eclipses. - Acta Applicandae Mathematicae. 1992. vol.29, pp.281-298.
[м11] Fomenko A.T., Kalashnikov V.V., Nosovsky G.V. Statistical analysis and dating of the observations on which Ptolemy's "Almagest" star catalogue is based. - In: Probability theory and mathematical statistics. Proc.of the Fifth Vilnius Conference. 1990, Moklas, Vilnius, Lithuania; VSP, Utrecht, The Netherlands, vol.1, pp.360-374.
[м12] Efremov Yu.N., Pavlovskaya E.D. Dating the "Almagest" by means of proper motions of stars. – DAS USSR, 1987, v.294, No.2, pp.310-313.
[м13] Yu.N.Efremov, Yu.А.Zavenyagin (with a foreword of V.L.Ginzburg). About so called "new chronology" of А.Т.Fomenko. – Newsletter of the RAS, 1999, v.69, issue 12, pp.1081-1092.
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Our answer, provided above, was published in the "Newsletter of RAS", issue 9, 2000.
REVIEW OF THE ARTICLE OF YU.N.EFREMOV "NEW, BUT FALSE CHRONOLOGY" [р19], pp.142-146
Around half of the text of the article of Yu.N.Efremov consists of emotionally colored statements, expressing an unlimited confidence of Yu.N.Efremov to the chronology of Scaliger-Petavius and school course of history. So, for example, according to Yu.N.Efremov a "generally accepted chronology doesn't need new checking and confirmation" [р19], p.142. Next, Yu.N.Efremov is sure that against new chronology "irresistible arguments are still politely published by historians... but politeness doesn't help" [р19], p.142. For this reason Yu.N.Efremov desided to refuse the rules of politeness, accepted in scientific discussions and, as he writes, "to call things with their names" [р19], p.142. Well, an extremely rough mood of polemic is characteristic for most articles from [р19], [р20], not only for the article of Yu.N.Efremov. In this sense the article of Yu.N.Efremov is more a rule, than an exception in [р19], [р20].
As a fun we would note, that according to Yu.N.Efremov, for example, the following "strong argument" successfully works against new chronology. Quote: "The spirit of the epoch has different taste. Vergil is not similar to Dante, Julius Caesar - to Charlemagne, and gothic cathedrals - to the Parthenon. WITHOUT DISCUSSIONS IT IS CLEAR that many centuries of evolution of the mankind separate them" [р19], p.142. Yu.N.Efremov has strange logics. Let's say, St. Basil's Cathedral on the Red Square and Annunciation Cathedral of the Moscow Kremlin absolutely don't resemble each other. And nevertheless they are built at the same epoch. Where did Yu.N.Efremov get his immovable sureness ("without discussions it is clear"), that not resembling each other buildings should be definitely separated by "many centuries of evolution"? Many examples show that this is not so.
Now let's turn to discussion of the own results of Yu.N.Efremov in the field of chronology. Please note, that Yu.N.Efremov personally was engaged in dating of the star catalogue of Almagest by means of proper motions of stars. He got a result, which, as it seemed him, confirmed the Scaligerian chronology [р21], [р22]. Unfortunately, the works of Yu.N.Efremov on the dating of the catalogue of Almagest have a mistake for around one thousand years in the estimation of the accuracy of the received by him dates. This fully devaluates dating of the catalogue of Almagest, received by Yu.N.Efremov. The mistakes in the works of Yu.N.Efremov [р21], [р22] on dating of the star catalogue of Almagest were in detail reviewed by us in the books [р6], [р7], [р8]. Here we will not repeat the review once again.
But in his article, published in [р19] and reviewed by us here, Yu.N.Efremov states, that his new work together with A.K.Dambis [р23] already without mistakes (as he thinks) confirms the Scaligerian dating of the star catalogue of Almagest, and therefore – the Scaligerian chronology. Moreover, Yu.N.Efremov straightly writes that his previous method of dating of Almagest, already several times reviewed in detail by us in print, already "lost its value in the light of the results" of the article [р23]. See. [р19], p.145. Otherwise, according to Yu.N.Efremov, all his previous mistakes in the dating of Almagest now were improved, and the same result was received – confirming the Scaligerian chronology. No details regarding his new method of dating of Almagest Yu.N.Efremov doesn't give in [р19] and just addresses a reader to English language publication [р23] in the magazine Journal for History of Astronomy.
That's why we address to the indicated article of Yu.N.Efremov and A.K.Dambis [р23]. According to the authors, even two new methods of dating of the star catalogue of Ptolemy are proposed in it. Of course, both methods, according to the authors [p23], "fully confirm" the Scaligerian chronology. But analysis of the work [р23] shows that, unfortunately, Yu.N.Efremov together with his co-author A.K.Dambis again and again repeats the same his old mistake. He incorrectly evaluates the accuracy of the received by him approximate dates.
The first of the two new methods of dating of the catalogue of Almagest, proposed by Yu.N.Efremov and A.K.Dambis, is described in [р23] in the section "Results of Mutual Distances Method". The method is just taken from our book [р8], about what Yu.N.Efremov and A.K.Dambis directly say [р23], p.121. According to their opinion, we, having proposed this method, allegedly didn't notice ourselves, what a "good" result it gave [р23], p.121. But in our book, devoted to the dating of the star catalogue of Almagest [р6], [р7], [р8], we rather clearly explained, why the indicated method, as well as a number of other simple approaches to the dating of the catalogue of Almagest, DOESN'T GIVE ANY NONTRIVIAL RESULT. The reason is that the ACCURACY of the given by these methods datings is too low and, as a consequence, the datings themselves have too big dispersion. As a result datings of the catalogue of Almagest with such rather simple modes turn to be uninformative, trivial. As for the method, taken from our book by Yu.N.Efremov and A.K.Dambis, we address the reader to paragraph 3, chapter 3 of the edition [р6] or to section 3.3 of the edition [р7] of our book on the dating of Almagest. See also section 7.4 "Dating of the catalogue of Almagest by widened informative core" in the last edition [р7] of our book on the dating of Almagest.
Here we once again come across the fact that Yu.N.Efremov in a strange way doesn't place importance on the evaluation of the accuracy in the problem of dating of the star catalogue of Almagest. Evaluations of the accuracy of approximate datings of the catalogue composing are not given by Yu.N.Efremov at all – as in the reviewed case, - or are wrong. The provided example of borrowing by Yu.N.Efremov and A.K.Dambis of a method of dating from our book – besides the method, thrown away by us due to its low accuracy, - brightly characterizes the relation of Yu.N.Efremov to the issue of evaluation of accuracy in general. By the way, evaluation of accuracy is the key question in this problem. See details in [р6], [р7].jm?.
Let's turn to the next section of the article [р23]. It is named "The Case of o^2 Eri". Here the authors directly write: "The fastest of the Almagest stars, o^2 Eri is important for catalogue dating by means of proper motions". This is really so. But in order to use the star o^2 Eri in the dating of Almagest, at least it is necessary to be sure, that this star is really included in the catalogue of Almagest. In order to prove this Yu.N.Efremov and A.K.Dambis refer to the works of a number of astronomers, who searched for identification of the star number 779 from Almagest (in numeration of Baily), which was called by Ptolemy in Almagest just "medium star". This not remarkable star of Almagest was really identified by the most researchers with also not remarkable star o^2 Eri of the modern sky. But we underline, only based on the fact, that at the epoch of the II century A.D., where the Scaligerian chronology places Ptolemy, the star o^2 Eri better than her neighbors corresponded to the coordinates, prescribed in Almagest to the star number 779. We underline, that no other arguments for the mentioned identification, except correspondence of the coordinates, were used in case with the star number 779. This star is marked neither by brightness nor by own name or more or less detailed description in Almagest.
But let's recall, that the star o^2 Eri has rather big proper motion. It considerably changes its position on the sky through centuries.
And if in the beginning of the Common era o^2 Eri really could be the best candidate for identification with the star number 779 from Almagest, then in the other epochs it is not so. The fact, that astronomers stopped at the identification of the star number 779 from Almagest with the star o^2 Eri, is a trivial consequence of the circumstance, that astronomers had already used information about proper motions of the stars. Moreover, they, of course, used the Scaligerian dating of Almagest. Otherwise, the mentioned identification – extremely important for Yu.N.Efremov – is just a consequence of the Scaligerian dating of Almagest. To use it for dating of Almagest, as Yu.N.Efremov does, - it is the same as to solve a reverse problem and to restore according to the results of the work of astronomers of the XVIII-XX centuries that the Scaligerian dating of Almagest, which they used in their work on identification of the stars of Ptolemy. But such dating is well known – it is, we will repeat, the Scaligerian dating. Of course, with such approach Yu.N.Efremov couldn't receive any other date for Almagest, but the Scaligerian one. Here we come across a vicious circle in the reasonings of Yu.N.Efremov, who as a consequence insistently gives forwarding.
Many times we explained to Yu.N.Efremov, that using of o^2 Eri for dating of the catalogue of Almagest is senseless, as it leads to a vicious circle. In our book [р6], [р7], [р8] it is explained in detail, corresponding drawings of the positions of Ptolemian and real stars in the constellation of Eridan are provided. Nevertheless, Yu.N.Efremov insistently continues to date Almagest according to o^2 Eri, each time repeating the same vicious circle. Finally, it becomes wearisome to explain him this.
The next section [р23] under the name "The Bulk Method" finishes the substantial part [р23]. The remaining sections of the article [р23] are devoted to conclusions and gratitudes.
According to the authors [р23], in this section they propose a method of dating of the catalogue of Almagest by means of proper motions, considerably differing from the previous method of Yu.N.Efremov [р21], [р22]. An important difference of such mode from the previous method of Yu.N.Efremov, as Yu.N.Efremov and A.K.Dambis write in [р23], is that this time for dating of the catalogue of Ptolemy all the fast stars from Almagest were used together. While in the previous method of Yu.N.Efremov each of the fast stars was used for dating separately [р23], p.125.
But it is surprising, that a transition to new modified method of dating didn't improve, but instead, worsened a bit the accuracy of the received by Yu.N.Efremov dating. So, in his previous work [р21] Yu.N.Efremov dated Almagest with 13th year A.D. allegedly with plus-minus 100 years. And in the work [р23], having developed and improved his method of dating, Yu.N.Efremov could date Almagest "only" with the accuracy plus-minus 122 years. The result of new dating of Almagest of Yu.N.Efremov is such: 90 year B.C. plus-minus 122 years [р23], p.128. So, the method was improved, and the accuracy, which it gives, - worsened. How this could be understood?
The answer is that both in work [р21] and in work [р23] Yu.N.Efremov in a wrong way evaluates the accuracy of the received by him datings.
Whimsicality of the stated by Yu.N.Efremov order of accuracy of the received by him datings of the catalogue of Ptolemy was explained by us in detail else within the review of the previous works of Yu.N.Efremov. See also our book [р6], [р7], where this question is discussed in detail [р6], pp.99-102; [р7], pp.200-212. A simple calculation shows, that a real order of the accuracy of the method of Yu.N.Efremov is not 100-120 years, as he for some reason thinks, but around THOUSAND years [р6], pp.99-102; [р7], pp.200-212.
By the way, in his first work on dating of Almagest [р21] Yu.N.Efremov told in detail, how did he receive his estimation of accuracy. This gave a possibility to find a mistake in his reasonings, which was shown to him [р6], pp.99-102; [р7], pp.200-212. But in the last work of Yu.N.Efremov [р23] on the dating of Almagest by means of proper motions not less fantastic estimations of the accuracy are given without any ground. No formulas, no algorithms, based on which the estimations were received, are given in [р23]. Other, more detailed works on this topic Yu.N.Efremov, apparently, also doesn't have. At least neither in [р19], nor in [р23] there are no links on such works. That's why it is difficult to indicate the specific mistakes in the estimation of accuracy, made by Yu.N.Efremov in [р23]. But there is no need in it. The presence of mistake in the estimations of accuracy of the datings in [р23] follows from the analysis of accuracy characteristics of the catalogue of Almagest, provided by us in [р6], [р7]. The characteristics are such, that the accuracy of dating of the catalogue of Almagest by means of proper motions of the stars with the mehod of Yu.N.Efremov can't be better, than plus-minus 400-500 years with the use of arch closing errors or, at least, plus-minus 300 years with the use of latitude closing errors [р7], p.206, [р7].
Moreover, it is not excluded, that like in his previous works [р21], [р22], in the work [р23] Yu.N.Efremov made a task oriented preliminary selection of the environments of fast stars, what provided the "required" answer. At least, from the text of the article [р23] it is not absolutely clear, according to which rule environments of one or another fast star were chosen when getting the final dating. As the method of Yu.N.Efremov is unstable to the choice of the stars environment, then with the help of the choice of suitable environment it is possible to receive the desired in advance date from the catalogue of Almagest. See details in our review of the method of Yu.N.Efremov [р6], pp.99-102; [р7], pp.200-212.
In general, new method of dating of Almagest by means of proper motions, proposed in [р23] doesn't differ much from the initial variant of the method from [р21], [р22]. The difference is only in the fact, that earlier Yu.N.Efremov received datings for each fast star separately (with some choice of its environment). We will clarify, that in the method of Yu.N.Efremov a position of a fast star is determined relatively to its environment, which consists of close to it stars. We found out that change in the composition of the environment is able to change considerably the received by such method dating [р6], pp.99-102; [р7], pp.200-212. Now, in the work [р23], Yu.N.Efremov proposes to get a single date for all fast stars.
With this some not very clear from the text [р23] rule of the choice of environments is used. Yu.N.Efremov and A.K.Dambis determine the searched single date as follows [р23], p.125.
Ecliptic coordinates on the celestial sphere for the epoch of the beginning of the Common era are considered. One of the coordinates, latitude or longitude, is fixed. Next one of the datings, according to the initial method of Yu.N.Efremov, is presented as a point on a plane. On horizontal axis the component of the speed of proper motion of the given fast star for this coordinate is given (with some amendment for the speed of the stars of the environment, what doesn't change the essence of the issue). A closing error for this coordinate lies over the vertical axis for an average distance from this fast star to its environment. A closing error is taken between an average distance, calculated by Almagest, and the same distance, calculated on the estimated sky for the beginning of the Common era. A point on a plane appears. After this, dating according to the method of Yu.N.Efremov for this fast star and for this environment is expressed with a decline of a line, drawn from the beginning of the coordinates to the received point.
The indicated procedure is done for each of the ecliptic coordinates – latitude and longitude – and for all the fast stars and their different environments.
A field of points on a plane appears. It is clear that if the catalogue of Almagest contained ideally precise coordinates of the stars, then all such stars would lie on one straight line. A decline, which would represent the date of the catalogue. But as coordinates of the stars in Almagest contain mistakes, there are no points on this line. Yu.N.Efremov and A.K.Dambis had an idea to use the method of linear regression and to determine the date of the catalogue from the decline of a regression line, drawn through the received field of points.
The idea itself is rather reasonable. But that field of points for Almagest, which Yu.N.Efremov and A.K.Dambis got, [р23], p.125, pic.5, doesn't let to estimate the decline of a regression line with the stated by them accuracy. What is, of course, not surprising due to essential inaccuracy of their method.
The field of points, provided on pic.5 of the work [р23], more or less randomly fills an area, resembling ellipsis with the center in the beginning of the coordinates. See pic.p5.3, which reproduces ill.5 from the work of Yefremov and Dambis. We just added to the picture a corresponding vertical axis, going through zero. An ellipsis, made by the field of points on pic. p5.3, is pulled a bit horizontally (correlation of semi-axis around 2:1). Yu.N.Efremov and A.K.Dambis state that the angle of decline of a regression line, determined with such «ellipsoidal» field of points, is close to zero. Moreover, they actually state that the angle allegedly can be determined with a fantastic accuracy just in several degrees [р23], p.125, ill.5. This is more than doubtful. It is obvious that Yu.N.Efremov once again made a mistake in estimation of the accuracy of the received by him date.
So, let's conclude. A new work of Yu.N.Efremov on the dating of Almagest, to which he refers in [р19], in fact, is just a variant of his old mode of dating of Almagest. In it the same mistake of Yu.N.Efremov is repeated – an incorrect estimation of the accuracy of the received by him dating. Moreover, in this work Yu.N.Efremov again uses the star o^2 Eri for the dating, the presence of which in the catalogue of Almagest could be reasoned only within a suggestion that the catalogue is composed close to the beginning of the Common era – that is in the Scaligerian epoch. It is clear that using such star for dating of the catalogue simply leads to a vicious circle.
Pic.p4.3 Illustration from the work of Yu.N.Efremov and A.A.Dambis, displaying a field of points, showing datings of Almagest by separate configurations. Yu.N.Efremov and A.A.Dambis draw through a field of points a regression line, decline of which gives them dating of Almagest by their method. Two such lines are displayed on the picture. One of them corresponds to the Scaligerian epoch of Ptolemy, the second – to the Scaligerian epoch of Hipparch. According to Yu.N.Efremov and A.A.Dambis, this field of points determines a regression line with such high accuracy, that a variant for Ptolemy falls out, and a variant for Hipparch remains. Nevertheless rather big closeness of both variants from the point of provided by them picture. Such opinion of Yu.N.Efremov and A.A.Dambis is more than doubtful.
A field of points, like provided on their picture, obviously doesn't let to determine a decline of repression line even with an accuracy, able, let's say, to separate the epoch of the XVI century from the beginning of the Common Era.
REVIEW OF THE ARTICLE OF А.А.VENKSSTERN AND А.I.ZAKHAROV <<DATING OF "ALMAGEST" OF PTOLEMY BY PLANETS' CONFIGURATIONS>> [р19], pp.111-123, AND THE ARTICLE OF YU.D.KRASILNIKOV <<ABOUT COVERAGE OF STARS WITH PLANETS IN "ALMAGEST" OF PTOLEMY>> [р19], pp.160-165.
The first part of the article of А.А.Venkstern and А.I.Zakharov is devoted to an attempt to date Almagest by 23 observations of stars, which Ptolemy prescribes to himself [р19], p.111. (А.I.Zakharov - astronomer, an employee of GAISH, А.А.Venkstern - mathematician, his scientific supervisor on mechanical-mathematical faculty of the MSU was А.Т.Fomenko). In the article a number of calculations in this direction are provided; we didn't check them, but we have no reasons to doubt in correctness of them. Let's quote the received by authors result. It doesn't anyhow contradict with our studies of Almagest.
А.А.Venkstern and А.I.Zakharov write: "Conclusion: One of the two takes place:
a) observations of planets, on which Ptolemy builds his theory, really took place in the II century A.D.; b) these observations were calculated according to some theory for the indicated date" [р19], p.111.
As for the possibility b), that is falsification of the data of Almagest, А.А.Venkstern and А.I.Zakharov tell the following: <<In order to check the possibility of fake of the data by medieval falsificators (before creation by Kepler of his theory), we decided to learn: how quickly grows a mistake in the theory of Ptolemy? Or in a different way: how far in time the falsificator could live ("Ptolemy") from the traditional time, in order to have a possibility to falsify observations, using a theory, which he explained in "Almagest"? ... Conclusion: the discussed observations couldn't be falsified based on the theory like the theory of Ptolemy, "TIME OF THE LIFE" OF SUCH THEORY IS JUST 200-300 years>> [р19], p.114.
All this perfectly agrees with our calculations and with our reconstructions. See details in the book "Astronomical analysis of the chronology" [р7]. We think [р7] that Almagest, in his famous today view, is an edition of the XVII century – that is EDITION OF THE EPOCH OF KEPLER, - some famous old astronomical work. Activity on the editing of Almagest in the XVII century was FALCIFICATION, the purpose of which was to show Almagest as a creation allegedly of the II century A.D. This epoch was taken from the Scaligerian chronological tables. Falsificators-scaligerians brought into the correspondence with the epoch of the II century A.D. those astronomical data of Almagest, which, using the theory of Kepler, they couldn't calculate for the II century A.D. For example, a planet theory of Ptolemy. What now А.А.Venkstern and А.I.Zakharov discover in their work, published in [р19]. We should give them proper respect; they clearly speak about what exactly is proved by them.
Those astronomical data, which in the XVII century nobody yet could reliably calculate, for example, sun eclipses were just excluded from Almagest. As a result, Almagest in its modern form in a strange way doesn't mention any sun eclipse (?!).
We were successful that the falsificators of the XVII century didn't exclude from Almagest an old star catalogue of Ptolemy. Most likely, they simply didn't suspect that it was possible to take the dating of Almagest from the catalogue based on such thin effect like proper motions of stars [р7]. More rough effects, as, for example, precession of longitudes, of course, they didn't take into account.
As for the precession of longitudes, it was not a difficult task to calculate it for the I century A.D. not only in the XVII, but also in the XV-XVI centuries. And today – within brackets – another critic, astronomer Yu.N.Efremov, in numerous newspaper publications tells how to date Almagest by precession of longitudes. That is how to restore the date, "sewn" by the Scaligerian editors of the XVII century and with this "successfully confirm" the Scaligerian chronology. These funny considerations of Yu.N.Efremov you can also find in [р19], p.143.
So, returning to the work of А.А.Venkstern and А.I.Zakharov, we can conclude that the received by them result doesn't contradict with the new chronology and our reconstruction of the history. But it contradicts with the Scaligerian version of chronology and history. And very strongly, although А.А.Venkstern and А.I.Zakharov themselves don't note this.
The issue is in the following. In their article, in section "Possibility of falsification of planet observations of Almagest on the base of other theories" [р19], pp.113-114, А.А.Venkstern and А.I.Zakharov study the question: how long a planet theory, explained in Almagest, could "live". We should explain that through time the characteristics of planetary orbits slowly change. That's why some planetary theory, which satisfactorily worked at the epoch of its creation, in several hundred years could become dreadfull. And then, of course, it would be necessary to change it for another one. Or, at least, to renew it, having improved its criteria. The question is how long the theory of Ptolemy could work?
The answer is given by А.А.Venkstern and А.I.Zakharov: not more than 300 years.
Made by them calculations showed that <<mistake of the theory of Ptolemy runs very quickly, that's why with such criteria out of the neighborhood plus-minus 300 years the theory already works too badly... "The time of the life" of such theory is just 200-300 years>> [р19], p.114.
Let's suppose now that the Scaligerian historical-chronological picture is correct. And that Almagest, in its known today view, was really written by Ptolemy around the beginning of the Commmon Era. For example, in the I-II centuries B.C. or in the I-II centuries A.D. But then it appears that the planet theory, described in Almagest, stopped to work already in the VI-VII century. We add 300 years – maximal time of life of this theory, calculated by А.А.Venkstern and А.I.Zakharov, by the Scaligerian date of finishing of Almagest (around 150 year A.D. [р24], p.430) and we receive 450. Let it be even 500 or 600 year A.D. But not later. After this time the planetary theory of Ptolemy was obliged to be out of use or to be subject to modification.
What do we read in the Scligerian textbooks on history? In the Scaligerian version it is considered that Almagest was the main source of astronomical knowledge in general and planetary theory in particular until the epoch of Copernicus, which is until the XVI century A.D. [р24], pp.445-448; [р25], pp.2-3. See also our review of the history of Almagest in the Scaligerian version [р7], p.19-21.
It turns out that HAVING FIRST DURING 200-300 YEARS GOOD PLANETARY THEORY, astronomers and mathematicians THEN DURING MORE THAN ONE THOUSAND YEARS USED AN EXTREMELY BAD PLANETARY THEORY, which already by the V-VI century A.D. finally lost its accuracy and became absolutely unsatisfactory. And only in the XVI century they finally decided to refuse it. But before this, during hundreds of years they used it, translated into other languages, studied, admired etc. And it didn't come to anybody's mind at least simply to improve the criteria of planetary orbits in it. If somebody had done it, then the calculations of А.А.Venkstern and А.I.Zakharov would give not the I century A.D., but the date of the last change.
Such picture is incredible. The only, in our opinion, reasonable explanation of the results of А.А.Venkstern and А.I.Zakharov is the fact that the planetary theory of Almagest in the view, in which we see it today, was written in it in the XVII century, at the epoch of Kepler, with the purpose of falsification of its dating. Falsification was important for the launched exactly at that time Scaligerian historical-chronological version. See details in [р7]. Of course, falsificators adapted the criteria of planetary orbits for the required date – beginning of the Common Era. That was found in the work of А.А.Venkstern and А.I.Zakharov.
In the next, last section of their article in [р19], А.А.Venkstern and А.I.Zakharov address to critics of the found by us astronomical solution of four covers of stars with planets, described in Almagest. We will remind that our decision: morning of February 14, 959 A.D. for Mars, morning of October 18, 960 for Venus, dawn of July 25, 994 for Jove and evening of August 16, 1009 for Saturn – perfectly corresponds with the dating of star catalogue of Almagest by means of proper motions of stars. Allowable interval of dating of the catalogue of Almagest by means of proper motions: from 600 to 1300 year A.D. [р7], p.392. Found by us solution for coverage comes right in the center of the interval.
Moreover, we discovered that the found by us solution for coverage of the stars with planets ideally satisfies the time of day, when, according to the words of Ptolemy, one or another coverage was seen [р7], pp.454-467. So, for example, in case of Mars, Ptolemy says that the cover was observed in the morning – and really in our solution Mars was seen only after midnight, that is only in the morning. In case with Jove Ptolemy informs that the cover was seen in the morning – and really, in our solution Jove rose around one hour before the sunrise, being always in the sunrise area of the sky. And, let's say, in "traditional", that is Scaligerian, solution Jove was seen near the star all the night, and that's why the words of Ptolemy about observation of the cover only at the sunrise become exceeding and even strange. So, a traditional solution in this place (and not only here) contains a strain. Next, in case with Saturn, Ptolemy notes that approach with a star was seen in the evening. And really, in our solution Saturn went in one hour after the sunset and, therefore, was seen only in the evening, at the sunset. But in the Scaligerian solution Saturn was seen again all the night, what makes explanation of Ptolemy about evening observation exceeding and even incomprehensible. The same fine correspondence of our solution with description of Ptolemy is regarding Venus [р7], pp.454-467.
At the same time we absolutely didn't require that the found by us solution on covers was the only possible. The issue is that there are no ideal solutions for stated by us task – as in the case with Mars, for example, as "cover" should be considered an approach of Mars with the indicated star just for 15 arc minutes. Such approach, strictly speaking, is not a cover. Moreover, Mars didn't cover the required star in the historical epoch. That's why a question about the unicity of solution becomes diffuse. There is still no ideal solution, and there will be as more close to ideal ones, as more we will weaken the terms of Ptolemy. This fact was noted by us in [р7]. It is also confirmed in the article of А.А.Venkstern and А.I.Zakharov.
But absolutely groundless and even mistaken in the article of А.А.Venkstern and А.I.Zakharov is a comparison of our solution on coverage of stars with the Scaligerian solution, which they put in a short table [р19], p.117. In the table it is stated that our solution "badly satisfies the circumstances of covers", while the Scaligerian solution "more or less satisfactorily describes the circumstances of covers" [р19], p.117. This is wrong. Examples of the reverse we just provided above. In more details this question could be learned from the book "Stars".
Extremely doubtful looks also the statement of А.А.Venkstern and А.I.Zakharov that they have found five more series of datings for covers, which satisfy the descriptions of Ptolemy not worse than the solution, which was found by us. Of course, at the absence of ideal decision one can argue, if one solution is "worse" or "better" than another one. Nevertheless, we will note, that no one from the solutions, provided by А.А.Venkstern and А.I.Zakharov in the table on page 119 of the book [р19], doesn't satisfy the indicated above conditions of the time of visibility ("morning", "evening", "sunrise"), which Ptolemy provides. This is seen at least from the column "digression from the Sun" in their table [р19], p.119.
As for our solution, also included by А.А.Venkstern and А.I.Zakharov in their table, a strange typing error in the line on Jove pays attention to itself. In the second column of the line it is indicated that at the day of cover of the star by Jove the end of the night (dawn) took place at 4:36 local time, and in the fifth column of the same line it is told that the Sun rose at 4:58 local time. But the Sun rises around one hour after the dawn that is after the end of night. It is well known to А.А.Venkstern and А.I.Zakharov, and they clearly write about this on page 117 [р19]. This is also seen from all the other lines of their table [р19], p.119. Why at this day the Sun rose just in 20 minutes after the dawn?
Probably it is just a casual typing error. But for the indicated line А.А.Venkstern and А.I.Zakharov give the following note: "The time of rise of Jove until 6 degrees under the horizon is indicated. A weak star delta Cnc is not seen due to closeness to the Sun" [р19], p.118. So, А.А.Venkstern and А.I.Zakharov, as they think, indicate a disadvantage of our solution.
In which described by Ptolemy cover "was impossible to see anywhere in the world" [р19], p.118. The same they say regarding Saturn [р19], p.118. Both statements of А.А.Venkstern and А.I.Zakharov don't correspond to the reality. But mentioned above typing error in their table makes an impression that the issue was like they say. As it appears that cover of the star by Jove was seen allegedly only 20 minutes before the sunrise. Of course, when a star couldn't be noticed by observers on already lightened sky and due to this really a cover couldn't be observed. Actually, the calculations, for example, with easy and convenient for rough calculations program Turbo-Sky, show that approaches of Jove and Saturn with the corresponding stars took place in our solution one hour before the sunrise in case of Jove and in one hour after the sunset in case of Saturn. That could be easily observed on a rather blackened sky, although not for a long time. That's why Ptolemy says about observation exactly "at the dawn" and "in the evening".
By the way, a question about the possibility of real observation of covers of stars in our solution is not essential neither for new chronology in general nor for the dating of Almagest. The issue is that as the solution was not strict (no ideal covers), then remains a theoretical possibility that the covers in fact were not observed, but calculated. So, we have to do not with reports about real observations, included into Almagest, but with the results of medieval calculations. Which, of course, were not very precise.
Let's go now to the article of Yu.D.Krasilnikov <<About covers of stars with planets in "Almagest" of Ptolemy>> [р19], pp.160-165. In it Yu.D.Krasilnikov tells about the Scaligerian solution of the task of dating of covers. In particular, he had to accept that cover of the star by Venus, which Ptolemy called "precise cover", in the Scaligerian solution turned to be only an approach on 12 angle minutes [р19], p.161. Such approach hardly could be called "precise cover", that's why here in the solution, protected by Yu.D.Krasilnikov, it is an evident strain. There are several more such strains. For example, Ptolemy underlines that cover of the star by Jove was seen at the dawn, but in a solution of 241 A.D., which Yu.D.Krasilnikov defends, the approach of Jove with the star was seen nearly all the night – around five hours [р19], p.163. This is also some strain. Indication of Ptolemy of an evening time of observation of the approach of Saturn with the star fully "hangs in the air" in the solution, which Yu.D.Krasilnikov liked. In this solution Saturn is seen all the night. A confused comment of Yu.D.Krasilnikov on this issue with rather inappropriate claims on disadvantages of a computer program, which Yu.D.Krasilnikov used for calculation of covers, could be read on page 163 of the collection book[р19].
By the way, Yu.D.Krasilnikov, as well as А.А.Venkstern and А.I.Zakharov, for some reason is sure, that for new chronology and our dating of Almagest it is important that there was now other solution for covers, except the proposed by us one. This is not so. For us it is enough that there is a solution for covers, probably not the only one, which is well agreed with the received by us dating of the catalogue of Almagest. See details in [р7].
At the end of his article Yu.D.Krasilnikov compares the Scaligerian solution, which he liked, with our solution, trying to prove that our solution was "much worse". With this Yu.D.Krasilnikov mainly insists that we in our solution didn't take into account the Sun longitude, provided in Almagest within description by Ptolemy of the covers of stars with planets. Our answer is the following. First of all, the Sun longitude is not a part of the used by Ptolemy observations. A longitude is calculated in Almagest for each cover. Second, it is not difficult to understand that a Sun longitude – it is the same date, but in other designations.
As we have today only a falsified in the XVII century edition of Almagest, it is difficult to expect that such simple things like Sun longitude were not brought by the editors of Almagest to the necessary for them Scaligerian date. No doubt, they were thoroughly agreed with the required date. That is what Yu.D.Krasilnikov finds out now, studying the Sun longitude in Almagest. With this he thinks that he "restores" a real date of Almagest, but actually he restores only an opinion of the editors-falsificators of the XVII century about this date. We already know this opinion – the Scaligerian date is written in all the textbooks. It is strange that Yu.D.Krasilnikov doesn't understand this. Probably he simply didn't read our book [р6], [р7], where everything is explained in details.
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[р30] Morozov N.А. Christ. (The history of mankind in natural scientific light) vv. 1-7. - M.-L., Gosizdat, 1924-1932. v.1: 1924 (2nd edition 1927), v.2: 1926, v.3: 1927, v.4: 1928, v.5: 1929, v.6: 1930, v.7: 1932.
In 1998 a reprint re-edition of this work of N.А.Morozov in Moscow publishing house "Kraft". All the seven volumes were republished.
