2d:null 2e:["$","title",null,{"children":"Babylonian numerals \n DRAFT | Toph Tucker | Observable"}] 33:T799,md`# Babylonian numerals ${draft()} ##### 2000 bc – 1st century ad ∙ [Wikipedia](https://en.wikipedia.org/wiki/Babylonian_cuneiform_numerals) ∙ [MacTutor](http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html) ---

“Finally we should look at the question of why the Babylonians had a number system with a base of 60. The easy answer is that they inherited the base of 60 from the Sumerians but that is no answer at all. I feel that all of these reasons are really not worth considering seriously. I just do not believe that anyone ever chose a number base for any civilisation.”

E. F. Robertson ${quote("mactutorBabylonian", 0)}

Based on Florian Cajori’s _A History of Mathematical Notations_ and [Wikipedia](https://en.wikipedia.org/wiki/Babylonian_numerals) (mostly latter so far tbh, see diagram layout inspo there). I’m gonna write more about all this but for now I’ll just say I wanted to try to animate the two systems to contrast a stylus inscribing clay and a pen inscribing paper. Also gee wouldn’t it suck to have to make nineteen marks with your stylus in clay when we can more dexterously write “59”? But also as I look at this it makes sense, given the medium, that you’d have used more configurations of a smaller set of primitive marks (as opposed to “strokes”). Maybe sorta like circuit voltages and binary! When you go past sixty you get into a positional place-value system. Babylonian numerals were mostly sexagesimal, so doing the under-60s is like doing just the under-10s in a decimal system. Except these are more systematic than our under-10 glyphs, ’cause they’re made up of these basic marks. Cf. earlier sketches of [our Hindu–Arabic numerals](https://observablehq.com/@tophtucker/svg-numeral-sketches). ---`34:T7ed,{ await visibility(); const decimalPaths = paths.map(scale(1 / 3)); const numbers = d3 .range(59) .map(d => d + 1) .map(getNumeral); const node = svg``; d3.select(node) .selectAll(".number-pair") .each(function(d, i) { const sel = d3.select(this); sel .select(".babylonian") .selectAll("path") .attr("transform", "scale(0)"); sel .select(".arabic") .selectAll("path") .each(function(d, i) { const len = this.getTotalLength(); d3.select(this) .attr("stroke-dasharray", `${len} ${len}`) .attr("stroke-dashoffset", len); }); }) .transition() .delay((d, i) => 1000 + i * 500) .on("end", function(d, i) { const sel = d3.select(this); sel .select(".babylonian") .selectAll("path") .transition() .duration(200) .delay((d, i) => i * 100) .attr("transform", "scale(1)"); sel .select(".arabic") .selectAll("path") .transition() .duration(500) .delay((d, i, arr) => (arr.length - i - 1) * 500) .ease(d3.easeLinear) .attr("stroke-dashoffset", 0); }); return node; }35:T855b,sources = [ { key: "menninger", date: 1934, authors: [{ name: "Karl Menninger", life: [1898, 1963] }], title: "Zahlwort und Ziffer", subtitle: "Aus der Kulturgeschichte unserer Zahlsprache, unserer Zahlschrift und des Rechenbretts", translatedTitle: "Number Words and Number Symbols", language: "German", urls: ["https://link.springer.com/article/10.1007/BF01733310"], links: [ { label: "Wikipedia", url: "https://en.wikipedia.org/wiki/Karl_Menninger_(mathematics)" } ], summary: "In 1969 Neugebauer called it “the best existing book on numbers and number systems”, though “the sections about the Babylonian number system are not always reliable.” (He is thanked in the revised and translated editions.) Note that the author is not the famous American psychiatrist." }, { key: "mkt", date: 1935, authors: [{ name: "Otto Neugebauer", life: [1899, 1990] }], title: "Mathematische Keilschrift-Texte", translatedTitle: "Mathematical Cuneiform Texts", urls: ["https://link.springer.com/book/10.1007/978-3-662-32794-4"], summary: "J. Gray wrote: “Neugebauer was to publish his 3-volume collection on mathematical tablets in the mid-1930s. They established the great richness of Babylonian mathematics, far exceeding anything one could have guessed from Greek or Egyptian sources.”" }, { key: "menningerEN", date: 1969, authors: [{ name: "Karl Menninger", life: [1898, 1963] }], translator: "Paul Broneer", title: "Number Words and Number Symbols", urls: ["https://archive.org/details/numberwordsnumbe00menn/"], relations: [{ type: "translates", key: "menninger" }], summary: "As with Neugebauer and others, I feel an immediate kinship and liking." }, { date: 1969, authors: [{ name: "Otto Neugebauer", life: [1899, 1990] }], title: "The Exact Sciences in Antiquity", urls: ["https://books.google.com/books?id=JVhTtVA2zr8C"], links: [ { label: "Wikipedia", url: "https://en.wikipedia.org/wiki/Otto_E._Neugebauer" }, { label: "MacTutor", url: "http://www-history.mcs.st-and.ac.uk/Biographies/Neugebauer.html" } ], relations: [ { type: "cites", key: "menninger" }, { type: "cites", key: "mkt" } ], summary: "Excellent account of Babylonian and Egyptian ideas in mathematics and astronomy, their transmission and influence. Thinks very highly of the contributions of the Babylonians, and very little of those of the Egyptians — though he says he would rather live among the latter. Heavy on gloss and commentary; Neugebauer is sassy, sharp, and somewhat aggrieved regarding the state of the study of the history of science. He is adamant that he cannot synthesize, nor untangle causes, nor see what we have lost, nor make it make much sense. In the appendix he carefully clarifies the true contributions of Copernicus, which have little in common with the folk story we tell.", quotes: [ [ "vii", "I am exceedingly sceptical of any attempt to reach a ‘synthesis’—whatever this term may mean—and I am convinced that specialization is the only basis of sound knowledge.… The common belief that we gain ‘historical perspective’ with increasing distance seems to me utterly to misrepresent the actual situation. What we gain is merely confidence in generalizations which we would never dare make if we had access to the real wealth of contemporary evidence." ], [ 1, "the inherent accuracy of the mathematical sciences will penetrate to some extent into purely historical problems." ], [ 4, "The Roman system is perhaps the most widespread method, historically speaking. The smallest numbers are simple repetitions of 1.… Fundamentally different from all these methods is the ‘place value notation’ of our present system, where neither 12 nor 21 represents 1 + 2 or 2 + 1 but 1 times ten plus 2, and 2 times ten plus 1 respectively. Here the position of a number symbol determines its value and consequently a limited number of symbols suffices to express numbers, however large, without the need for repetitions or creation of new higher symbols. The invention of this place value notation is undoubtedly one of the most fertile inventions of humanity. It can be properly compared with the invention of the alphabet as contrasted to the use of thousands of picture-signs intended to convey a direct representation of the concept in question." ], [ 15, "Counting of the vertical wedges leads directly to the readings 1, 2, 3, etc. up to 9. Then follows […] which must be 10, and consequently we can also read the remaining signs as 11, 12, 13. Using this exceedingly plausible hypothesis, we should also be able to read the right-hand column of signs. The first five look as follows […] Obviously we must read these signs as 10, 20, 30, 40, 50 if the first sign represents 10 as we have established in our first list." ], [ 16, "The notation […] is ‘sexagesimal’ in the sense that 60 units of one kind are written as 1 of the next higher order. This is exactly the same principle we found in Ptolemy's table of chords. The only difference consists in the fact that Old-Babylonian texts have not yet developed a special sign for ‘zero’." ], [ 17, "The other inconstistency of the modern astronomical notation, namely, to continue beyond the seconds with decimal fractions, is a recent invention. It is interesting to see that it took about 2000 years of migration of astronomical knowledge from Mesopotamia via Greeks, Hindus, and Arabs to arrive at a truly absurd numerical system.… The example of our present system of numeration for degrees, hours, measures and ordinary numbers should suffice totally to discredit the popular idea that a number system was ‘invented’ at a certain moment.… First of all, there exists a common misconception as to the gcmerality of the use of the sexagesimal system. The very same tablet which contains hundreds of sexagesimal numbers, column beside column, to compute the dates of the new moons for a given year, might end with a ‘colophon’ containing the name of the owner of the tablet, the name of the scribe, and the date of writing of the text, the year being expressed in the form 2 _me_ 25 ‘2 hundred 25’ where the main text would express the very same date sexagesimally as 3,45. In other words, it is only in strictly mathematical or astronomical contexts that the sexagesimal system is consistently applied. In all other matters (dates, measures of weight, areas, etc.), use was made of mixed systems which have their exact parallel in the chaos of 60-division, 24-division, 12-division, 10-division, 2-division which characterizes the units of our own civilization." ], [ 19, "Beside these basic elements, many modifications of number symbols were in use for different classes of objects, such as capacity measures, weights, areas, etc. Among these a clear decimal system has been recognized with signs for 1, 10, and 100. The numbers 1 and 10 we have already described. The 100 was written as a circular impression which looks like 10, but is made much bigger. Thus 100 is simply ‘big 10’. Another system proceeds sexagesimally, at least partially. Distinct units are 1 and 10 as before. A big 1 represents 60. Two big units written in opposing directions are combined into one sign to form 120. A 10-sign added in the middle gives 1200. A very big 10 sign stands for 3600. Variations of these systems, both decimal and more or less sexagesimal, can be established at different localities. The main facts, however, are common to all of them, namely, the existence of a decimal substratum and the use of bigger symbols to represent higher units. This latter fact is obviously the root for the development of the place value notation. When the script slowly became simplified and standardized, the distinction between bigger and smaller signs of the same type disappeared. Whereas originally one big unit, meaning 60, and one 10 symbol were written to denote 60 + 10, later a simple ‘1’ followed by a 10 was read 70, in contrast to a 10 followed by 1 meaning 11.… Thus the ‘sexagesimal’ order eventually became the main numerical system and with it the place value writing derived from the use of bigger and smaller signs. The decimal substratum, however, always remained visible for all numbers up to 60." ], [ 20, null, "A space sometimes marked a zero, and a period-like zero mark eventually appeared but was inconsistently applied, e.g. there were never trailing zeroes — magnitude had to be inferred from context" ], [ 20, null, "The sexagesimal system was also used (just the same!) for fractions." ], [ 21, null, "Scribes multiplied multiple-digit numbers nearly exactly the same as we would, by taking it one place at a time and adding. It was all there!" ], [ 22, null, "Ptolomy took after the Babylonian astronomers’ sexagesimal divisions, which survives today in our minutes and seconds. But he used them “exclusively for fractions but not for integers”, i.e. writing 365 (base 10) instead of 65 (base 60), whereas Copernicus actually often used sexagesimal straight through." ], [ 24, "It is a mistake to assume that the Islamic mathematicians and astronomers consistently used the ‘Hindu-Arabic’ numerals. By and large the Hindu-Arabic numerals are restricted to mathematical context, whereas astronomical tables use the alphabetic numerals. In Egypt the Greek or Coptic alphabetic numerals remained in use for centuries after the Arabic conquest." ], [ 28, "It must be said, however, that the number of errors in the texts is comparatively very small. I have had the experience that I committed many more errors in checking the ancient computations than there were in the original documents. Often errors in a text are very helpful because they constitute one of the main tools for establishing the details of a numerical procedure followed by the ancient computer." ], [ 30, "The total amount of Babylonian tablets which have reached museums might be estimated to be at least 500,000 tablets and this is certainly only a small fraction of the texts which are still buried in the ruins of Mesopotamian cities. Our task can therefore properly be compared with restoring the history of mathematics from a few torn pages which have accidentally survived the destruction of a great library." ], [ 33, "This system of tables alone, as it existed in 1800 B.C., would put the Babylonians ahead of all numerical computers in antiquity. Between 350 and 400 A.D., Theon Alexandrinus wrote pages of explanations in his commentaries to Ptolemy’s sexagesimal computations in the Almagest. A scribe of the administration of an estate of a Babylonian temple 2000 years before Theon would have rightly wondered about so many words for such a simple technique." ], [ 36, "The above example of the determination of the diagonal of the square from its side is sufficient proof that the ‘Pythagorean’ theorem was known more than a thousand years before Pythagoras." ], [ 43, "Investigating such series, one finds that they all have the same pair x = 30, y = 20 as solutions. This indicates that it was of no concern to the teacher that the result must have been known to the pupil. What he obviously had to learn was the method of transforming such horrible expressions into simpler ones and to arrive finally at the correct solutions. We have several tablets of the first class which solve one such example after another from corresponding collections of the second class. From actually computed examples it becomes obvious that it was the general procedure, not the numerical result, which was considercd important. If accidentally a factor has the value 1 the multiplication by 1 will be explicitly performed, obviously because this step is necessary in the general case." ], [ 44, "Compared with the algebraic and numerical component in Babylonian mathematics the role of ‘geometry’ is rather insignificant." ], [ 45, "It must also be underlined that we have not the faintest idea about anything amounting to a ‘proof’ concerning relations between geometrical magnitudes. Several tablets dealing with the division of areas show figures of trapezoids or triangles but without any attempt at being metrically correct. The description of geometry as the science of proving correct theorems from incorrect figures certainly fits Babylonian geometry so far as the figures are concerned and also with regard to the algebraic relations. But the real ‘geometric’ part often escapes us. It is, for instance, not at all certain whether the triangles and trapezoids are right-angle figures or not.… There are nevertheless cases where no reasonable doubt can arise as to the correct interpretation of geometrical relations. The concept of similarity is utilized in numerous examples." ], [ 48, "However incomplete our present knowledge of Babylonian mathematics may be, so much is established beyond any doubt: we are dealing with a level of mathematical development which can in many aspects be compared with the mathematics, say, of the early Renaissance. Yet one must not overestimate these achievements. In spite of the numerical and algebraic skill and in spite of the abstract interest which is conspicuous in so many examples, the contents of Babylonian mathematics remained profoundly elementary. In the utterly primitive framework of Egyptian mathematics the discovery of the irrationality of √2 would be a strange miracle. But all the foundations were laid which could have given this result to a Babylonian mathematician, exactly in the same arithmetical form in which it was obviously discovered so much later by the Greeks. And even if it were only due to our incomplete knowledge of the sources that we assume that the Babylonians did not know that p^2 = 2*q^2 had no solution in integer numbers p and q, even then the fact remains that the consequences of this result were not realized. In other words Babylonian mathematics never transgressed the threshold of pre-scientific thought. It is only in the last three centuries of Babylonian history and in the field of mathematical astronomy that the Babylonian mathematicians or astronomers reached parity with their Greek contemporaries." ], [ 53, "There are many forces which cooperate in the destruction of source material, none more powerful than continuous peaceful life. The lack of interest in the far remote past will invariably change and eventually destroy what remains from earlier generations. Without violent catastrophies there would hardly be any archeology. If Mesopotamian cities had not been turned into desert hills we would have no chance of finding the hundreds of thousands of documents from which Babylonian history is written. It would be pointless to describe in any detail the unending sequence of disaster which provided us with the material for our studies. We shall devote our discussion exclusively to the modern attempts at reviving the past and penetrating into the intellectual life of previous generations." ], [ 55, "Early Greek astronomy from its beginnings about 400 B.C. to Ptolemy (about 150 A.D.) is almost completely destroyed, except for a few very elementary works which survived for teaching purposes. But the rest was obliterated by Ptolemy’s outstanding work, which relegated his predecessors to merely historically interesting figures." ], [ 56, "To quote only one work, I mention Cumont’s ‘L’Egypte des astrologues’ (1937). Here, stripped of all astrology, the background of the daily life and of the contemporary institutions of Hellenistic Egypt is depicted as it becomes visible from the mishaps and fortunes predicted for the men and women who consulted the astrologer." ], [ 59, "Especially well preserved texts are valuable for the antiquities trade and are therefore most exposed to rapid destruction." ], [ 59, "While the field work has been perfected to a very high standard during the last half century, the second part, the publication, has been neglected to such a degree that many excavations of Mesopotamian sites resulted only in a scientifically executed destruction of what was left still undestroyed after a few thousand years." ], [ 60, "Until 1951 not for a single astronomical or mathematical text was its provenance established by excavation.… Consequently it is, e. g., completely impossible to find out whether these texts came from a temple, a palace, a private house, etc. Not even the stratum is known to give us a more accurate date of the texts.… Texts which for more than 50 years were lying in the basement of a great museum could be relatively dated from the newspaper in which they were wrapped." ], [ 61, "The Mesopotamian soil has preserved tablets for thousands of years. This will not be the case in our climate." ], [63, null, "Example of reconstructing a fragment of equations"], [ 66, "And it is clear that the determination of the meaning of a text is generally the easier the more complicated the mathematical context is because this leaves fewer possibilities for the interpretation of the procedure. A context which contains only a few numbers combined by addition or subtraction is of almost no help in the determination of terminology. The advanced algebraic level of Babylonian mathematics was of the greatest help in its being decoded." ], [ 71, "One must simply realize that mathematics and astronomy had practically no effect on the realities of life in the ancient civilizations.", "The mathematical requirements of commerce were very basic; the mathematical requirements of engineering were far beyond them." ], [ 73, "[T]he whole procedure of Egyptian mathematics is essentially additive.… It simply consists in the proper collection and counting of the marks for units, tens, hundreds, etc., of which Egyptian number signs are composed. But also multiplication and division are reduced to the same process by breaking up any higher multiple into a sum of consecutive duplications. And each duplication is nothing but the addition of a number to itself.… it is amusing to see that modern computing machines have again made use of this ‘dyadic’ principle of multiplication." ], [ 78, "Corresponding formulae for the elementary volumes were known, including a correct numerical computation for the volume of a truncated pyramid.", "Natch" ], [ 80, "But, by and large, one has to distinguish two widely separate types of ‘Greek’ mathematics. One is represented by the strictly logical approach of Euclid, Archimedes, Apollonius, etc.; the other group is only a part of general Hellenistic mathematics, the roots of which lie in the Babylonian and Egyptian procedures." ], [ 80, "Only in one point does the Egyptian tradition show a very beneficial influence, that is, in the use of the Egyptian calendar by the Hellenistic astronomers. This calendar is, indeed, the only intelligent calendar which ever existed in human history.", "Twelve months, 30 days each, 5 extra days at the end of the year, no intercalations." ], [ 81, "A second Egyptian contribution to astronomy is the division of the day into 24 hours, though these ‘hours’ were originally not of even length but were dependent on the seasons. These ‘seasonal hours’, twelve for daylight, twelve for night, were replaced by ‘equinoctial hours’ of constant length only in theoretical works of Hellenistic astronomy.", "I’ve always wondered about that sort of variable hour!" ], [ 91, "In summary, from the almost three millenia of Egyptian writing, the only texts which have come down to us and deal with a numerical prediction of astronomical phenomena belong to the Hellenistic or Roman period. None of the earlier astronomical documents contains mathematical elements; they are crude observational schemes, partly religious, partly practical in purpose. Ancient science was the product of a very few men; and these few happened not to be Egyptians." ], [ 151, "the hypothesis that Babylonian science had reached Egypt before Greece seems to me only to substitute one name of an unknown quantity for another." ], [ 152, "I think that it is evident that Plato's role has been widely exaggerated. His own direct contributions to mathematical knowledge were obviously nil." ], [ 153, "[T]he initial conditions of our planetary system were chosen in such a way that all the satellites of the sun-and our own satellite as well-behave with great modesty. Their orbits can be closely approximated by circles such that the simplest possible model of a circular motion with constant speed leads immediately to very reasonable results for the description of the solar and lunar phenomena. On the other hand, the deviations from the trivial circular orbits are just great enough to be observed and to challenge an explanation, but small enough such that again comparatively simple modifications of the trivial solution give satisfactory results." ], [ 158, "In Hellenistic and medieval geography, one frequently finds the latitude of a locality expressed by means of the ratio of the longest to the shortest daylight for the region in question. Alexandria, e. g., falls in the zone for which this ratio is 7 : 5; that is to say, the longest daylight is 14 hours and the shortest night, assumed to be equal in length to the shortest day, is 10 hours.", "This was called ‘clima’, hence ‘climate’!" ], [ 165, "I intended to investigate the transmission of Ptolemy’s theory of precession to the Arabs, and it was only natural to include here the Hindu sources. This led me to the Pañca Siddhāntikā of Varāha Mihira, written about 550 A.D.… the Pañca Siddhāntikā contains certain rules for the computation of the lunar motion based on the processes now know to us from Greek sources. Thibaut, who edited the Pañca Siddhāntikā in 1889, found it very difficult to understand these passages. He eventually found the key to the problem in the book Kāla Sankalita by J. Warren. The latter had traveled extensively in Southern India and had recorded the astronomical lore of the natives in the book mentioned, published in Madras in 1825. In this book, he describes a method followed by the Tamil inhabitants of the Coromandel coast for the computation of the lunar motion. His informants no longer had any idea about the reasons for the single steps which they performed according to their rules. The numbers themselves were not written down but were represented by groups of shells placed on the ground." ], [ 173, "This explains also the replacement of ‘degrees’ by ‘days’; the Hindu ‘days’ are simply sexagesimal fractions of a sidereal year, or, expressed differently, the mean solar motion is assumed to be 1° per day." ], [ 175, "the ‘Romans’ are, of course, the Greeks of the Roman or Byzantine empire" ], [ 176, "Al-Khwārizmī’s astronomical tables have been preserved through Latin translations; they show a curious mixture of Hindu and Greek methods. The relation between his mathematical writings and the Hellenistic tradition has already been mentioned." ], [ 177, "In the ‘Cloisters’ of the Metropolitan Museum in New York there hangs a magnificent tapestry which tells the tale of the Unicorn. At the end we see the miraculous animal captured, gracefully resigned to his fate, standing in an enclosure surrounded by a neat little fence. This picture may serve as a simile for what we have attempted here. We have artfully erected from small bits of evidence the fence inside which we hope to have enclosed what may appear as a possible, living creature. Reality, however, may be vastly different from the product of our imagination; perhaps it is vain to hope for anything more than a picture which is pleasing to the constructive mind when we try to restore the past." ], [ 183, "The transformation from the geocentric to the heliocentric system is often hailed as one of the greatest discoveries of modern science, though foreshadowed by the Greek genius. In fact, however, the equivalence of these two modes of description of the observable phenomena had scarcely been forgotten by the astronomers of the Middle Ages. Aryabhata (about 500 A.D.) argued for a movable and rotating earth (Aryabhatiya IV. 8; trsl. Clark p. 64 fl.) and al-Bīrūnī (1030 A.D.) remarks in his ‘India’ (trsl. Sachau I p. 276 fl.) rather casually, ‘Besides, the rotation of the earth does in no way impair the value of astronomy, as all appearances of an astronomic character can quite as well be explained according to this theory as to the other.’" ], [ 185, "The ancient astronomers rightly had greater confidence in the accuracy of their mathematical theory than in their instruments.", "Never let experiment get in the way of theory, ha!" ], [ 189, "Datta and Singh in their ‘History of Hindu Mathematics’, vol. I (1935), quote (p. 59) the commentary of Bhattotpala to the Brihat Samhita of Varāha Mihira for an excerpt from the original Paulisa Siddhānta in which a huge number, ending in ..7800, is expressed by number words in opposite arrangement in the form ‘zero, zero, eight, seven, ....’. It seems to me rather plausible to explain the decimal place value notation as a modification of the sexagesimal place value notation with which the Hindus became familiar through Hellenistic astronomy." ], [ 190, "In our discussions we have frequently used the word ‘Greek’ with no further qualification. It may be useful to remark that we use this term only as a convenient geographical or linguistic notation. A concept like ‘Greek mathematics’, however, seems to me more misleading than helpful. We are fairly well acquainted with three mathematicians—Euclid, Archimedes, and Apollonius—who represent one consistent tradition. We know only one astronomer, Ptolemy. And we are familiar with about equally many minor figures who more or less follow their great masters. Thus what is usually called ‘Greek’ mathematics consists of the fragments of writings of about 10 or 20 persons scattered over a period of 600 years. It seems to me a dangerous generalization to abstract from this material a common type and then to establish some mysterious deeper principle which supposedly connects a mathematical document with some other work of art." ], [ 191, null, "Appendix on the Almagest, which N. highly recommends trying to understand!" ], [ 197, "This obvious advantage of the use of secondary epicycles induced Copernicus to apply the same construction also to the planetary motion and thus to initiate complications which destroyed the inherent elegance and simplicity of the Ptolemaic model." ], [ 200, "“Nulle planète n’a demandé aux astronomes plus de soins et de peines que Mercure, et ne leur a donne en recompense tant d’inquiétudes, tant de contrariétes” says Leverrier.", "We should be very grateful! Imagine what planets that would falsify and correct our models we’re missing!" ], [ 204, "The popular belief that Copernicus’s heliocentric system constitutes a significant simplification of the Ptolemaic system is obviously wrong. The choice of the reference system has no effect whatever on the structure of the model, and the Copernican models themselves require about twice as many circles as the Ptolemaic models and are far less elegant and adaptable. In fact the importance of Copernicus's work lies in a totally different direction than generally announced. One may enumerate his main contributions as follows: (a) The return to a strictly Ptolemaic methodology which made all steps from the empirical data to the parameters of the model perfectly clear and opened the way to a refinement of the basic observations which eventually led to the proper generalization of the Ptolemaic methods, discarding the iterated epicycles of Copernicus. (b) The insight that we can obtain information about the actual planetary distances if we assume that all planetary orbits have essentially the same center, namely, the sun. Then the radii of the epicycles of the inner planets directly give their distances from the sun in terms of R; for the outer planets the reciprocals of the radii of the epicycles measure the heliocentric distances. Again the question which body is supposedly at rest is of no interest whatever and therefore Tycho Brahe’s system was exactly as good as the Copernican system, and equally complicated because of the same doctrine of circularity of admissible motions. (c) The assumption of a common center of the planetary orbits suggested also the proper solution of the problem of latitudes, namely, that the inclined planes of the planetary orbits pass through that common center. Unfortunately the postulate of circularity induced Copernicus to use the mean sun and not the real sun as common center and thus resulted in a theory of latitudes which labored under exactly the same complications as Ptolemy’s theory. Nevertheless, this modification of the ancient theory of latitudes helped Kepler to find the real solution, which then permitted the computation of heliocentric coordinates in a uniform fashion and the finding of the geocentric coordinates through an independent procedure." ], [ 208, "I see the main purpose of historical studies in the unfolding of the stupendous wealth of phenomena which are connected with any phase of human history and thus to counteract the natural tendency toward over-simplification and philosophical constructions which are the faithful companions of ignorance." ], [ 224, "The history of mathematics provides good illustrations for the fact that continuity of tradition alone is not sufficient to keep a scientific field alive. The ‘Elements’ of Euclid formed for centuries the basis of mathematical instruction. Nevertheless the significance of its axiomatic structure was not understood until the problems connected with the foundations of analysis led the mathematicians of the 19th century to similar methods. A particular case in point is the theory of proportions in Book V which was only seen in its proper role within Greek mathematics through the theory of irrational numbers and continuity developed by R. Dedekind since 1858." ], [ 225, "[Aristotle’s logic’s] introduction of variables, represented by letters into logic would seem to have constituted the essential step to a formulation of mathematical rules which we would call ‘Algebra’. Such a development would have been still more natural since the axiomatic approach to mathematics originated in the same time and among the same circle of men. Yet nothing of this type happened and the origin of algebra is totally independent of the existence of an algebraic notation in one of the most famous philosophical works of antiquity. This is a good illustration for the futility of any attempt to reconstruct ‘reasons’ for the incidents of historical events. Similarly the absence of algebraic notations should not have prevented the Greek geometers from developing what was called in the 19th century ‘synthetic’ and ‘projective’ geometry since many of the basic concepts were ready at hand in the works of Apollonius. Again such a ‘natural’ development did not take place and all that we may ever hope to establish in historical research is facts and conditions but never causes." ] ] }, { key: "mactutorBabylonian", date: 2000, authors: [ { name: "John J. O’Connor" }, { name: "Edmund F. Robertson", life: [1943] } ], title: "Babylonian numerals", urls: [ "http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html" ], quotes: [ [ null, "Finally we should look at the question of why the Babylonians had a number system with a base of 60. The easy answer is that they inherited the base of 60 from the Sumerians but that is no answer at all. It only leads us to ask why the Sumerians used base 60. Theon of Alexandria tried to answer this question in the fourth century AD and many historians of mathematics have offered an opinion since then without any coming up with a really convincing answer. I feel that all of these reasons are really not worth considering seriously. I just do not believe that anyone ever chose a number base for any civilisation." ] ] } ]36:[] 31:{"id":"71a38c9046b7f0ef","slug":"babylonian-numerals","title":"Babylonian numerals \n DRAFT\n ","thumbnail":"ac11fe5acb9cd32c3463bb765db6e2a960930acdb59540d73fdc95f21b2fc7df","access_level":"public","create_time":"$D2019-06-24T02:00:48.748Z","update_time":"$D2025-04-13T22:55:26.460Z","view_time":null,"first_public_time":"$D2019-06-24T07:18:34.121Z","fork":false,"stars":9,"starred":false,"owner":{"type":"team","id":"97290e936c339d8e","login":"tophtucker","name":"Toph Tucker","avatar_url":"https://avatars-next.observableusercontent.com/3706c31a8f9c59bb716c022c04b9c073144cd1368b78ac85f49139be6c93c34e","tier":"pro_2024"},"version":889,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"0bb2842b01e0c652","login":"tophtucker","name":"Toph Tucker","avatar_url":"https://avatars-next.observableusercontent.com/3706c31a8f9c59bb716c022c04b9c073144cd1368b78ac85f49139be6c93c34e","tier":null}],"members":[{"type":"individual","id":"0bb2842b01e0c652","login":"tophtucker","name":"Toph Tucker","avatar_url":"https://avatars-next.observableusercontent.com/3706c31a8f9c59bb716c022c04b9c073144cd1368b78ac85f49139be6c93c34e","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"d40cc4f153f404b4","slug":"how-we-do-math","title":"How We Do Math","owner":{"id":"97290e936c339d8e","login":"tophtucker","name":"Toph Tucker"}},"type":"notebook","body":{"title":"Babylonian numerals \n DRAFT\n ","cells":[{"id":0,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":408,"value":"md`Digits go from 1 to 59:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":177,"value":"viewof n = slider({\n min: 1,\n max: 59,\n step: 1,\n value: 1\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":187,"value":"{\n const node = svg``\n \n if (animate) {\n d3.select(node).selectAll(\"path\")\n .attr(\"transform\",\"scale(0)\")\n .transition()\n .duration(300)\n .delay((d, i) => i * 300)\n .attr(\"transform\",\"scale(1)\")\n }\n \n return node\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":426,"value":"md`Beyond that it’s sexagesimal, with a blank space for zero. 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