The following is my translation of an article called: Forschungs- und mathematikegeschichtliche Umschau von Dr Georg Biedenkapp, Frankfurt a.M. It appeared in a German popular science magazine, Kosmos Handweiser für Naturfreunde 1910, Heft 8, Seiten 281-285. Observant readers may notice helpful references to illustrations. As these don't accompany the translation, they're invited to provide their own alternatives. I'm not aware of any previous translation.
Trevor Dykes.

A review of the history of research and mathematics by Dr Georg Biedenkapp, Frankfurt a.M.
It is a curious fact that painters have distinguished themselves on a number of occasions in the history of natural science and technology. The widespread telegramme system comes from the painter Morse, and Fulton, who invented the first effective steamship, had arrived in England as a portrait painter before turning to technical things. From the eighteenth century we have an excellent and still valuable reference book for the study of insects from the painter, Rösel von Rosenhof. He presented, from his own press, regardless of the warnings and ridicule from the learned, four volumes of his researches into insects. The greatest of the fine artists active in natural science and technology was Leonardo da Vinci, and nobody is now astonished by his significance for natural science and technology. Six years ago, Marie Herzfeld produced a German book about the great painters as thinkers and researchers, and there was recently a French one by Pelladan about 'Leonardo as philosopher'. A century before Galileo's struggle against the blind belief in Aristotle's views, a century prior to the shining ascent of experimental research, the great painter was teaching the answering of questions on nature by considered experiments, making fun of appeals to authorities, and wrote in his diary, which grew to twenty large volumes of manuscripts, about the Sun being fixed and the Earth one star among stars, made a surprising number of geological observations, dissected corpses, which was previously unheard of, made explicable the movement of light and sound by comparisons with waves in water produced by a thrown in stone, and left behind a quantity of mechanical inventions. The painter, researcher and engineer taught that love of an object is awoken by an exact understanding of it. At the same time, however, Leonardo only wanted to be read by those who were mathematicians: "who understand the highest security of mathematics, approach confusion but never resort to sophist tendencies of silence which give rise to nothing other than eternal screaming." What Leonardo wrote in his many volumes of manuscripts, which have first been published in recent decades, was quite sufficient to see him dispatched to a bonfire, as happened to his contemporary Savonarola and, a hundred years later, Giordano Bruno. The growing astonishment about the significance of Leonardo as a natural scientist should therefore serve, to those interested in the history of natural science, as a spur to busy themselves with the literature about the great painter.

Slowly, but unstoppably, the literary interest in the heroes of research ascended. A new work about Galileo can also be found or, better said, a 644 page strong first volume of a new work by Wohlwill. In this, we learn that Galileo's father partly earned his living as a musician, wrote scripts upon the theory of music, and that it was due to this busy and thinking father that his son took up the study of mathematics. Galileo first studied Euclid when aged twenty; and it attracted him so strongly that he could not put it down. One knows that Galileo found his work on falling objects contradicted the thinking of Aristotle, and he taught accordingly, Wohlwill has now shown that a supporter of Aristotle from the sixth century, Philponus, had already demonstrated against Aristotle's laws as being false; the writing of Philopons, a commentary on Aristotle, was republished in 1536 and undoubtedly had an influence on Galileo. Nevertheless, it took from the 6th until the 16th century before a further and greater progress occurred.

So more and more light is rising over the forerunners of the discovery of laws about falling, and it shows a development; sensual presumptions are often the fatherhood of painting and music, and they can be found on the crib of experimental natural research. It was perhaps the Greeks who were the nation that brought mathematics to its highest blossom in antiquity because they were also the artistically most gifted. This is the sense in which the author of the 1909 work, Geschichte der Mathematik im Altertum, speaks as well, Prof. Max Simon, a book which is presently very popular, wherein is considered what occurred with natural research and mathematics, the backbone of all strict natural sciences, in the soils of Babylon and Egypt. Excavations have brought forth entire libraries from thousands of years ago, and many other surprises have come to light. We know that the earliest great Greek researchers, Pythagoras and Democritus, the latter counted as the founder of atomic theory, undertook journeys to such places. This begs the question as to what the natural researchers could have learnt from the learned of such areas, who were priests, in Babylon, India or Egypt. In his book about the history of mathematics in antiquity, Simon makes much use of material gained from recent excavations. He summarises the cultures of Egypt, Babylon and India, reveals the core of their writings and mathematical notation, and shows how mathematically advanced these peoples were.

While most people leave school with the ability to add, subtract, to multiply and to divide, this is seen simply as something unremarkable, and it's what schools are for. But few teachers make it clear to them what an arduous, long route was required, in order to reach the elemental arts of arithmetic and geometry from simple numbers. How few people realise that our school mathematics, which is a long way removed from higher mathematics, also has a very long history and presents a great triumph of intellect. If a reader of these lines has all the works mentioned available, then they could read of the poetic epic 'Nala and Damajanti', translated from the Indian, which can, by the way, be found as a translation by Franz Bopp. In this old Indian poem, King Nala loses his kingdom in a game of dice. With this game, he obviously did not have the art to understand quick and secure counting as he one day learns, after becoming a wagon driver for a king, from the following experience. He travels with his master on a rapid journey across the land, and there is a tree with fruit along the way, and Nala's master decides to instantaneously count how many fruits are on the tree as they drive by. The number is in the thousands and it is given precisely. Nala asks, highly astonished, how his lord did this; if his master were to teach him the art of counting, then he would teach the art of steering horses in return, as he knew this skill better than anyone else. He now learns that his master firstly counted the branches of the tree and then counted the number of fruits on a single branch. As all branches bore the same number of fruits -this is naturally from a fairy tale- Nala's lord only needed to multiply the number of branches with the fruits found, and that gave the number of fruits on the whole tree quickly. Here we have an illustration that multiplication is nothing other than a massively shortened form of addition; nevertheless, it is the gift of kings to recognise the number of branches and fruits on one branch so quickly, and that is made obvious, although not necessarily implausible, as there really have been mathematical Wunderkinder able to give the exact number of a majority of things from a single glance, whereas, to other eyes, there appeared only a confusion of many with no particular number calculable. After Nala has learned this art, he returns to his homeland and wins his kingdom back in a new game of dice. While this poem actually relates that he had only originally lost as an evil spirit had gotten into the dice, it is nevertheless unmistakable that the good Nala was deficient in the art of numbers, and the story offers an illustration that even straightforward multiplication, which appears self-explanatory to us, first had to be invented. It seems to me, that this episode from the old Indian poem is suitable to give some small pleasure to pupils, as it shows the value of their simple arithmetic. One should, however, immediately go further and also show how much nicer and pleasant it is that they are working with Indian numbers, introduced to us by the Arabs, rather than with Roman, Greek or even Egyptian ones. Let us compare the number 212,635 and see how complex it appears in Egyptian and Roman symbols as opposed to Indian ones! (Illustration 1).

(Added note: Draw your own illustration! The Ancient Egyptian version essentially involves two rows with eleven symbols in each. Nero's accountant may have written, unless he was fiddling as well, CCXIIMDCXXXV.)

In the Egyptian number, the five strokes are ones, the three horseshoe-like shapes are easily recognisable as tens, the six hundreds are six coiled ropes of a hundred cubits in length, the symbol for a thousand is the lotus flower which grew in the water courses and many artificial canals, and tadpoles also occurred in vast numbers as soon as the dried out canals and runnels were refilled by the flood of the Nile; therefore, the tadpole is the symbol for a hundred thousand. (Additional note: If you're now preparing to draw your Ancient Egyptian number, these tadpoles are well on their way to frogginess, as they're equipped with legs. I hope the Egyptians pronounced that lot 4 lotus flowers and 5 coiled rope years ago, but fear it unlikely.)

The Papyrus Rhind has been known as the oldest mathematics book for a number of decades. It comes from the time of the Hyklos dynasty, which was during the seventeenth or eighteenth century BC. Recent researchers are of the opinion that it is a school book with mathematical questions. The Ancient Egyptians had specialist schools for builders, surveyors, administrators, merchants and so on, also for agriculture, and one such agricultural school has provided this papyrus filled with questions, and corrections have been added in the red by the hand of a teacher. It has a mass of errors which were not all corrected: most examples are for the direct usage of the worker. For arithmetic, the Egyptians developed a system based on counting fingers transposed to counting boards and stones, and they were aware of four simple forms for calculations with whole numbers and fractions, and this allowed them to tackle equations of the first and second grades, and apply these to arithmetical and geometrical rows, use methods for approximations, and work out the roots of square numbers. Concerning the geometry of the Egyptians, Simon writes that they had developed arts of the construction and draughtsmanship. Democritus, the contemporary of Plato, maintained that the draughtsmanship of the Egyptians had not been bettered. They had a "very respectable quadrature for the circle, knew of symmetry and proportions, confident at dividing circles, had rules for recognising similarities, the beginnings of trigonometry and elements of spatial geometry".

Today, we know that the Ancient Egyptian calendar, which remained in use until Roman times, must have been introduced in the year 4241BC. The New Year's Day at the time of introduction fell on a day when the flood star Sirius was again visible from Memphis before sunrise, which would be July 19th, and that is also the day upon which the Nile began to rise. As the Egyptians did not allow for a correcting quarter of a day, and imprecisely gave a year 365 days, after only four years the New Year's Day occurred a day prior to the correct point of ascent for Sirius, two days after eight years until, after four times 365 years, New Year's Day again corresponded with the required position of Sirius. This actual configuration occurred in the year of 2781BC, which means the calendar must have been adopted at least 4 x 365 years earlier, and that equates with the year 4241BC in our scale of time. This would have required very careful observations of the stars. Is it only fantasy to maintain that the measurements of the four thousand year old pyramid of Cheops, especially the proportions of numbers and dimensions, involved a great deal of astonishing mathematical knowledge and an appreciation of physics?

Various proportions of certain internal parts of the pyramids are said to show that, four thousand years ago, the Egyptians knew the number of pi, the diameter of the Earth, the distance of the Earth from the Sun and even the specific weight of the Earth. When one hears of the Babylonians who, in the middle of the third millennium BC, had already adopted as a unit of measurement the length of a pendulum swing in a second as proposed by Hugens in the 1650s, all that sounds less improbable. If a cube has equal squares of a tenth of a metre, and the volume is filled with water, then the weight will be a kilogramme, and the ancient Babylonians had already used the tenth of a Babylonian double cubit as the basis of a unit of volume with a weight in terms of water of the mine (very nearly the same weight as our kilogramme). Babylonian weights received artistic forms so we find lions, boar heads and ducks of bronze or iron have been preserved, and these served as weights. The average weight of the heavy mine is 982.4g. The square root of that is the tenth of a double cubit which, when in turn multiplied by ten, gives the double cubit a length of 992.35mm, and that number is exactly the length of a pendulum second at 31° latitude! From this Lehmann concluded, in agreement with Helmholz, that the Babylonians must have converted the pendulum second into, as said, a measurement.

The basis of the Babylonian counting system is grounded on sixty (rather than ten, the basis of ours), and it was not actually necessary to write a symbol for 60, but only to indicate the position on which the number stood in order to decide whether to read x times sixty or sixty times sixty or sixty times sixty times sixty -but, although this advance was known to the Babylonians, their numbers are still much more cumbersome than the Indian. The Babylonians counted with large calculation tables using multiplication, division and square numbers. They are said to have already been capable in the third millennium, as Hilbrecht maintains on the grounds of a table dating from the third millennium, of multiplication as practised in 1350. Many such tables are also equipped with a fourth power of 60, and this was recognised as a holy number by Plato; it was certainly holy to the Babylonians.

If there was much the Greeks could have learnt from the Egyptians and Babylonians, then it has not yet been mentioned that they really developed their knowledge far beyond the levels of the Egyptians and Greeks, and beyond all the peoples from whom they learned. Essentially, as Simon also concludes, mathematics is a creation of the Greeks which, even in its early form, already incorporated one of the few devices which, until recently, was seen as an advance added by recent history: using differentials and integers as was already done by old Archimedes, as shown only a few years ago by a newly discovered script from Archimedes.

Knowledge of Pythagoras' theorem is not evidenced for the Egyptians and Babylonians. The Indians, however, possessed this in the eighth century BC and, as Pythagoras is said to have been in India, held views on the wandering of souls otherwise only common in India, and certain numerical games of Pythagoras are reminiscent of Indian mathematical manipulations, it has been concluded that Pythagoras brought his famous theory back from India, with which the author of the most recent book on the history of ancient mathematics also agrees. This is possible but, when one thinks, especially when it comes to religious matters -and the Pythagoras theory possessed by the Indians was intimately connected with religious things- that the stranger finds it very hard to appreciate, as in our days European academics find it extremely difficult to get used to strange Indian customs of sacrifice, and Pythagoras is said to have taken a number of ideas about things by studying the lengths of lyres and recognising the simple proportions of numbers as being insignificant, and when one thinks of the usually right handed harps of his time and that 'hypotenuse' means 'to stretch under' (namely referring to the harp string), so it is also possible that Pythagoras arrived at his remarkable realisations about the relationships of sides of right angled triangles by the occasion of a harp being played, and not from a journey to India at some early time. But it is interesting that the Indians already possessed Pythagoras' theorem in the eighth century, long before the wise Greek, and used it in connection with religious rites. There exists a gigantic body of specifications for laying out altars constructed from rectangular baked tiles, and each of these made up figures which had to be fitted into an area of a similar size in a particular manner. In the instructions for measuring cord (Meßschnurleitfaden), the Sulvasutras, instructions are given for the construction of right angled altars composed entirely of right angled triangles. The tiles of skilfully made fire altars had an overall shape formed from a number of squares, and these should be reminiscent of a falcon in flight. In order to fulfil particular objectives, there were a great number of variations for this relatively simple figure; one could construct an altar showing a falcon with its wings drawn in and a spread tail, or the shape of an isosceles triangle or a double triangle or the wheel of a wagon, but all these various shapes had to conform to the given layout of a common norm and standard area. Should a larger area have been required, then there were instructions governing how large an increase was permitted. In order to resolve such challenges, the use of Pythagoras' theorem was often unavoidable; the Indians had this in their language and understood geometric construction based upon the roots of two and three, and that is also not possible without Pythagoras' theorem. If one observes the depiction of the falcon-shaped fire altar (Illustration 2) with the crooked wings, -the sacrificial fire was held to flap falcon-like to the sky- it will be apparent at the first glance that thought here was playing with squares and triangles, and was brought to bear on geometric discoveries.

An index of more of my translations of old Kosmos articles can be found at:

Kosmos Translations Archive

A number of Mesozoic (and post-Mesozoic) location summaries can be found at Localities.