Part 2. The Modern Epoch, and the emergence of the Modern calculator

Early Modern 1550 - 1799 Political and economic dynamics¹ Trade, power and navigation Merchants vs Old order Clocks and Astronomy

The Problem of multiplication - grows as multiple results required, and quick results in situations from warfare, to commerce to navigation. As the mathematics and science become more sophisticated the mathematical entities that have to be computed become more challenging.

Mathematical Practioners² The Sector

Mathematics from the elite Description of the period - from Copernicus (note mathematical equivalence of earlier work), Keppler (note comments on Napier’s logs and the Shickard story) relationship of Galileo (and the new astronomy) to the French Revolution Napier, Gunter Logarithms Slide rule

Beyond ready reference tables. Schickard

Pascale

Court mathematics Moreland Thomas as bridge head

Late Modern 1800 - mid 2000 The triumphs of industrial capitalism and consumerism

Thomas Multiplication - etc.

From Thomas to HP35

Similarly in India there appears to have been an old but sophisticated civilisation contemporaneous with the time of the construction of the Egyptian pyramids. Boyer considers it likely that India also had its “rope stretchers” to assist in the construction of temples and the like, but notes that Hindu mathematics has even greater discontinuity than that in China. The first known Indian mathematical text (by Aryabhata) is placed much later, traditionally about 476 CE (the time of the fall of the Western Roman Empire).³ Once more we see texts on

From what is available it does seem that the questions, and their answers,in both Sumerian and Egyptian socieities, have been largely formed around practical problems (like determining the dimensions and areas of regular bodies, and in particular triangles). In the course of this Pythagorean relationships between the lengths of sides were tabulated. These questions, intermediate answers, and methods for using these to give final answers were clearly systematically developed, and perhaps were seen as building one upon the other, but not in the systematic and abstract way that is both required and celebrated in the way modern mathematics depends on methods of proof to establish its authority. The development of this particular desire, and methodology, is not evident until it emerged in the mathematical writings of an elite group of mathematical philosophers in Ancient Greece.

In both cases the surviving historical evidence is not sufficiently robust to confidently answer the question of how far back this work extended. There is reason to place the first Chinese empire as stretching back to 2750 BCE, although more conservative estimates assert a closer date of around 1000 BCE. There is no more agreement on the dating of the oldest chinese mathematical work, the Chou Pei with estimates ranging from 1200 BCE to 100 CE. Boyer in his imposing work opts for 300 BCE.⁶ This and subsequent works reveal a mixture of “accurate and inaccurate, primitive and sophisticated results”.⁷ Certainly the Chinese mathematical development appears to have largely been of Chinese origin, although early lessons may well have be drawn from interchange with Mesopotania. The areas of triangles, rectangles and trapezoids are correctly calculated in the context of problems, and the area of the circle (with pi approximated by 3). Magic squares, the solutions to problems which would now be considered simultaneous linear equations, and the like are also solved. The development was not continuous, with developments disrupted by the occasional burning of books and other interruptions, together with the sparse availability of the few known written works even though printing was developed in China as early as the eleventh century CE.[ibid p. 198–203] Two important treatises from 1299 CE by the great mathematician Chu Shih-chieh (1280–1303 CE) marked a peak in the development of Chinese algebra and revealed an understanding of how to solve some quite sophisticated problems (in modern terms involving variables up to x¹⁴).

Similarly in

Pragmatic mathematics - Egyptian and Mesopotanian foundations

In both cases the surviving historical evidence is not sufficiently robust to confidently answer the question of how far back this work extended. There is reason to place the first Chinese empire as stretching back to 2750 BCE, although more conservative estimates assert a closer date of around 1000 BCE. There is no more agreement on the dating of the oldest chinese mathematical work, the Chou Pei with estimates ranging from 1200 BCE to 100 CE. Boyer in his imposing work opts for 300 BCE.

     http://meta-studies.net/pmwiki/uploads/Evmath/BaboonBogoshi.jpg

  http://meta-studies.net/pmwiki/uploads/Evmath/RopeStretchers.jpg 

  http://meta-studies.net/pmwiki/uploads/Evmath/HanDynCountingRods.jpg

  http://meta-studies.net/pmwiki/uploads/Evmath/swanpanSmithp29.jpg 

  http://meta-studies.net/pmwiki/uploads/Evmath/TokensSchmandtBp41.jpg

  http://meta-studies.net/pmwiki/uploads/Evmath/EnvelopeTokensSchmandtp358.jpg

  http://meta-studies.net/pmwiki/uploads/Evmath/RecipOxfordp210.jpg

Charette notes that from the middle of the C10, the progressive disintegration of the Abbasid empire led to a proliferation of local dynasties and principalities, providing greater opportunity for a diversity of places in which patronage for astronomical and mathematical development might occur. Other centres such as Cairo and Damascus were also centers of such work over C13-C16 whilst over this time, whilst the emphasis of patronage of such work in Courts diminished, the practical usefulness of the work in religious observance was picked up, with a consequent diffusion (and to some extent rediscovery) of mathematical and astronomical knowledge and instrument making from the Courts to the Mosques. The boundary between Mosque and Public was in any case indistinct and these prized instruments, constructed in private shops, could been purchased by citizen and Mosque alike.¹⁵ By C15, the knowledge and possession of astronomical instruments such as the astrolabe, and the practical utility of it for determining time and direction, had become diffused more generally through Islamic society. However, the Islamic world was now shrinking under military pressure from Western Europe, forced out of Spain, and the Renaissance was under way in the rising powers of Europe, dominated in religion by Christianity, but drawing heavily nevertheless on the accumulated including the Islamic world’s trove of rediscovered ancient knowledge knowledge and the developments it had made on these in mathematical and scientific knowledge, and calculational and observational instruments.¹⁶

Description of the period - from Copernicus (note mathematical equivalence of earlier work), Keppler (note comments on Napier’s logs and the Shickard story) relationship of Galileo (and the new astronomy) to the French Revolution

Mathematics of the elite

Description of the period - from Copernicus and Gallileo (and the new astronomy) to the French Revolution

Merchants vs Old order

The Problem of multiplication

From what is available it does seem that the questions, and their answers,in both Sumerian and Egyptian socieities, have been largely formed around practical problems (like determining the dimensions and areas of regular bodies, and in particular triangles). In the course of this Pythagorean relationships between the lengths of sides were tabulated. These questions, intermediate answers, and methods for using these to give final answers were clearly systematically developed, and perhaps were seen as building one upon the other, but not in the systematic and abstract way that is both required and celebrated in the way modern mathematics depends on methods of proof to establish its authority. The development of this particular desire, and methodology, is not evident until it emerged in the mathematical writings of an elite group of mathematical philosophers in Ancient Greece.

Separate, but application focussed.

Right: is from a modern representation (from an “ipad app”) of an “astrolabe clock”²¹ for a location on Earth 38 degrees North and 75 degrees West on 12 February 2012 at 9.22 PM. Sun, moon and planets can be seen on the Zodiac (red circle). Below the horizon is shaded gray. Green arrow through the sun shows the time on the 24 hour clock on the perimeter. In this modern representation the blue grid marks the angles of objects in vertical (altitude) and horizontal (azimuth) planes. ²²

Right: is from a modern representation (from an “ipad app”) of an “astrolabe clock” for a location on Earth 38 degrees North and 75 degrees West on 12 February 2012 at 9.22 PM. Sun, moon and planets can be seen on the Zodiac (red circle). Below the horizon is shaded gray. Green arrow through the sun shows the time on the 24 hour clock on the perimeter. In this modern representation the blue grid marks the angles of objects in vertical (altitude) and horizontal (azimuth) planes. ²⁵

• role of authority and power. The driving force of systems of authority, from the Egyptian pharonic society to the Islamic Caliphates, to the replacement of the Inca system of accounting via the Kippu by the Spanish conquerers with European approaches,²⁸ the role of authority and, as already mentioned, patronage, in shaping calculational approaches and technologies, is evident.

• breadth of need and literacy. Where the need for calculation is confined to a small subsection of society (for example, to priests of the dominant religion) then the need may be able to be met by training in traditional methods. But where the need spreads (for example, with trade and intensifying markets) then calculational technologies may offset the need for intense training in numerical and mathematical literacy.

China

• extent of communicationwith other societies. Trade and other forms of communication with other societies enlarges the access to new knowledge, skills, and technological developments. The diffusion of Islamic knowledge (for example, about astronomy) to Europe and India, as well as Indian knowledge (and in particular the Islamic numeric script) to the Islamic world, provide examples of this.

• accessibility of calculational technology. Finally the utility of a calculating technology will be determined in part by its accessibility. This will be determined by the extent that the problems it can solve are seen to be widely experienced, the training and literacy required to use it, and its cost of production. The use of instruments like the astrolabe or Roman abacus, let alone the Antikythera, were clearly circumscribed by what was required to construct them, in contrast, for example, to the use of pebbles for reckoning.

• dynamic between mathematics and calculational technology. Developments in each of these depends on the developments in the other. As illustrated by the case of Roman numerals, even the form of numerals developed to write down quantities in part is shaped by the medium in which they are to be inscribed (for example, stone or parchment), the needs of the society to use them, and the technologies (such as pebbles or abacus) available to assist in manipulating them.

Enough has been said in this brief survey to suggest some issues that need to be brought into account when positioning the calculating instruments in this collection against the unfolding history of the Modern period. This suggests a number of factors which have affected the types, direction and extent of development of different calculating technologies. These include the related issues of the:

• value placed in on innovation and the existing stock of knowledge. The history of calculation has been in part one of rise and decline of the desire for innovation and the loss and rediscovery of past insights and instruments. At different times he value placed on innovation and even retaining existing knowledge has changed for different groups in society, and for societies as a whole. The “Dark Ages” in Europe following the collapse of the Western Roman Empire (C5-C15) is an example of a time when mathematical innovation apparently lost much of its lustre and prior technical knowledge was, for a time, lost.
• complexity of the social organisation and in particular the scale of hierarchical organisation and power and consequent organisation needs.
• relationship between perceived tasks required and the technologies of calculation that can be brought to bear. Of course this is not a one-way relationship, since the invention of new calculational approaches (for example of more efficient numeral systems, or of a system of geometry) also opened the possibility of new social needs becoming evident.
• overlap between mathematics and calculational technology. There is no sharp division between these and the form of each depends on the developments in the other. Even the form of numerals developed to write down quantities stands in a relationship to the medium available to inscribe them, the needs of the society to use them, and the technologies (such as pebbles) available to assist in manipulating them.

Reference to these factors is not only helpful in understanding why particular technologies were developed and used at different times in the pre-Modern period which reviewed above. As will be seen in Part 2, they have also been vital in shaping the directions of development in the dynamic Modern period which followed.

• value placed in on innovation and the existing stock of knowledge. The history of calculation has been in part one of rise and decline of the desire for innovation and the loss and rediscovery of past insights and instruments. At different times he value placed on innovation and even retaining existing knowledge has changed for different groups in society, and for societies as a whole.

• overlap between mathematics and calculational technology. Even the form of numerals developed to write down quantities stands in a relationship to the medium available to inscribe them, the needs of the society to use them, and the technologies (such as pebbles) available to assist in manipulating them.
• relationship between calculating technologies and those who use them. From the Classical Greek period on we see the division emerging between the roles of calculation in the Courts, or the elites, which can lead to a distancing of developments from practical application needs, and the needs and capacities in the broader community. Instruments which are part of the play for patronage, or part of the process of celebrating power and standing may take a very different form, and play a very different role to those used as part of every day social interchange.
• relationship with those who make calculating technologies, and their social positioning. Where artisans are required to develop such instruments they may play a very different social role to the users. The Classical Greek divergence between theoretical mathematics and pragmatic artisanal mathematics has been referred to. Artisans not only have a social role, but also an accumulated knowledge that may be in part be encapsulated in embodied skills, and may routinely be transferred in non-codified forms. In various periods of history the guild-apprenticeship model has been the method of development of transference of this knowledge, the bulk or all of this being handed on through a mixture of oral instruction, and learned skills. Developments in this process may be directed much more sharply to solving perceived practical problems and may lead to quite different emphases and outcomes from those in more abstract and theoretical environments.
• role of authority and power. The driving force of systems of authority, from the Egyptian pharonic society to the Islamic Caliphates, to the replacement of the Inca system of accounting via the Kippu by the Spanish conquerers with European approaches,³⁰ the role of authority in shaping calculational approaches and technologies, is evident.

• role of authority and power. The driving force of systems of authority, from the Egyptian pharonic society to the Arabic Caliphates, to the replacement of the Inca system of accounting via the Kippu by the Spanish conquerers with European approaches, the role of authority in shaping calculational approaches and technologies, is evident.

• cost of calculational technology. Finally the utility of a calculating technology will be determined in part by its accessibility. This will be determined by the extent that the problems it can solve are seen to be widely experienced, the training and literacy required to use it, and its cost of production. The use of instruments like the astrolabe or Roman abacus, let alone the Antikythera, were clearly circumscribed by what was required to construct them, in contrast, for example, to the use of pebbles for reckoning.

Enough has been said in this brief survey to suggest some issues that need to be brought into account when positioning the calculating instruments in this collection against the unfolding history of the Modern period. This suggests a number of factors which have affected the types, direction and extent of development of different calculating technologies. These include the:

• relationship between perceived tasks in societies with diverse cultures and levels of complexity and the technologies of calculation that can be brought to bear. Of course this was not a one-way relationship, since the invention of new calculational approaches (for example of more efficient numeral systems, or of a system of geometry) also opened the possibility of new social needs becoming evident.
• overlap between arithmetic and the other emerging branches of mathematics and calculational technology. Even the form of numerals developed to write down quantities stands in a relationship to the medium available to inscribe them, the needs of the society to use them, and the technologies (such as pebbles) available to assist in manipulating them.
• relationship between calculating technologies and those who use them. From the Classical Greek period on we see the division emerging between the roles of calculation in the Courts, or the elites, which can lead to a distancing of developments from practical application needs, and the needs and capacities in the broader community. Instruments which are part of the play for patronage, or part of the process of celebrating power and standing may take a very different form, and play a very different role to those used as part of every day social interchange.
• role of of those who make calculating technologies, and their social positioning. Where artisans are required to develop such instruments they may play a very different social role to the users. The Classical Greek divergence between theoretical mathematics and pragmatic artisanal mathematics has been referred to. Artisans not only have a social role, but also an accumulated knowledge that may be in part be encapsulated in embodied skills, and may routinely be transferred in non-codified forms. In various periods of history the guild-apprenticeship model has been the method of development of transference of this knowledge, the bulk or all of this being handed on through a mixture of oral instruction, and learned skills. Developments in this process may be directed much more sharply to solving perceived practical problems and may lead to quite different emphases and outcomes from those in more abstract and theoretical environments.
• role of authority and power. The driving force of systems of authority, from the Egyptian pharonic society to the Arabic Caliphates, to the replacement of the Inca system of accounting via the Kippu by the Spanish conquerers with European approaches, the role of authority in shaping calculational approaches and technologies, is evident.
• breadth of need and literacy. Where the need for calculation is confined to a small subsection of society (for example, to priests of the dominant religion) then the need may be able to be met by training in traditional methods. But where the need spreads (for example, with trade and intensifying markets) then calculational technologies may offset the need for intense training in numerical and mathematical literacy.
• specialised responses versus generalised responses to developing needs.
• related to the above, the cost of calculational technology. A counting tray or pen and paper is much cheaper than

abstract - applied theoretical - artisanal breadth of use (cost) - accessible - inaccessible special purpose - general purpose ceremonial - productive

A fourth factor is the role of specialisation versus generality. Numerical and other forms of literacy w

Enough has been said in this brief survey to suggest some issues that need to be brought into account when positioning the calculating instruments in this collection against the unfolding history of the Modern period. Clearly there was an important relationship between the tasks which different societies faced as they developed in complexity and the technologies of calculation that could be brought to bear. Of course this was not a one-way relationship, since the invention of new calculational approaches (for example of more efficient numeral systems, or of a system of geometry) also opened the possibility of new social needs becoming evident. It is also clear that there is no sharp division between arithmetic and the other emerging branches of mathematics and calculational technology.

Enough has been said in this brief survey to indicate that the development of calculational technologies occurred across a number of directions depending on a range of factors. it is

Charette notes that from the middle of the C10, the progressive disintegration of the Abbasid empire led to a proliferation of local dynasties and principalities, providing greater opportunity for a diversity of places in which patronage for astronomical and mathematical development might occur. Other centres such as Cairo and Damascus were also centers of such work over C13-C16 whilst over this time, whilst the emphasis of patronage of such work in Courts diminished, the practical usefulness of the work in religious observance was picked up, with a consequent diffusion (and to some extent rediscovery) of mathematical and astronomical knowledge and instrument making from the Courts to the Mosques. The boundary between Mosque and Public was in any case indistinct and these prized instruments, constructed in private shops, could been purchased by citizen and Mosque alike.³³ By C15, the knowledge and possession of astronomical instruments such as the astrolabe, and the practical utility of it for determining time and direction, had become diffused more generally through Islamic society. However, the Islamic world was now shrinking under military pressure from Western Europe, forced out of Spain, and the Renaissance was under way in the rising powers of Europe, dominated in religion by Christianity, but drawing heavily nevertheless on the accumulated including the Islamic world’s trove of rediscovered ancient knowledge knowledge and the developments it had made on these in mathematical and scientific knowledge, and calculational and observational instruments.³⁴

Charette notes that from the middle of the C10, the progressive disintegration of the Abbasid empire led to a proliferation of local dynasties and principalities, providing greater opportunity for a diversity of places in which patronage for astronomical and mathematical development might occur. Other centres such as Cairo and Damascus were also centers of such work over C13-C16 whilst over this time, whilst the emphasis of patronage of such work in Courts diminished, the practical usefulness of the work in religious observance was picked up, with a consequent diffusion (and to some extent rediscovery) of mathematical and astronomical knowledge and instrument making from the Courts to the Mosques. The boundary between Mosque and Public was in any case indistinct and these prized instruments, constructed in private shops, could been purchased by citizen and Mosque alike.³⁷ By C15-C16, the knowledge and possession of astronomical instruments such as the astrolabe, and the practical utility of it for determining time and direction, had become diffused not only more generally through Islamic society but also through increasing trade and other interchange to India and Europe.

Charette notes that from the middle of the C10, the progressive disintegration of the Abbasid empire led to a proliferation of local dynasties and principalities, providing greater opportunity for a diversity of places in which patronage for astronomical and mathematical development might occur. Other centres such as Cairo and Damascus were also centers of such work over C13-C16 whilst over this time, whilst the emphasis of patronage of such work in Courts diminished, the practical usefulness of the work in religious observance was picked up, with a consequent diffusion (and to some extent rediscovery) of mathematical and astronomical knowledge and instrument making from the Courts to the Mosques. The boundary between Mosque and Public was in any case indistinct and these prized instruments, constructed in private shops, could been purchased by citizen and Mosque alike. By C15-C16, the knowledge and possession of astronomical instruments such as the astrolabe, and the practical utility of it for determining time and direction, had become diffused more generally through Islamic society.⁴⁰

Charette notes that from the middle of the C10, the progressive disintegration of the Abbasid empire led to a proliferation of local dynasties and principalities, providing greater opportunity for a diversity of places in which patronage for astronomical and mathematical development might occur. Other centres such as Cairo and Damascus were also centers of such work over C13-C16 whilst over this time, whilst the emphasis of patronage of such work in Courts diminished, the practical usefulness of the work in religious observance was picked up, with a consequent diffusion (and to some extent rediscovery) of mathematical and astronomical knowledge and instrument making from the Courts to the Mosques. The boundary between Mosque and Public was in any case indistinct and these prized instruments, constructed in private shops, could been purchased by citizen and Mosque alike. By the the fifteenth century, the knowledge and possession of astronomical instruments such as the astrolabe, and the practical utility of it for determining time and direction, had become diffused much more generally through Islamic society.⁴²

Further, as Charette points out, this discussion of usefulness tends to ignore another important use. That was the usefulness of science and scientific instruments for religious practice (as well as more mundane activities). Several leading astronomers (including the legendary al-Khwarizmi - member of al-Mamun’s scientific academy, and Bayt al-Hikma, the founder of “algebra”) applied astronomical tables to the determination of prayer times and for predicting the religiously important first sighting of the lunar crescent, and laid out methods for finding the direction of Mecca, the construction of water clocks, sundials and hororaries, and the designs for sine quadrants and instruments for predicting lunar eclipses.⁴⁴ Over

Similarly, mathematics or science developed, or instruments prepared for Court astronomy in the Arabic world, frequently had little of what might normally be considered practical application - the dynamic of development was either the hope of attracting royal patronage, or to confirm the glory of royal personages. Whether amongst the free elite in the slave society of Ancient Greece, or the royal courts of the Arabic world, the actual doing of constructive work was something left to others and from which the elite tended to distance themselves.

Of course the documentation of the history of mathematics and science is biased inevitably towards those areas that would have been recorded. Access to this would have been highly restricted, being most likely to be found in Courts and Mosques. Certainly royal patronage was crucial in the early stages of reconstruction of ancient knowledge and in supporting the development of instruments and facilities of astronomical observation. This led to a pattern reminiscent of the “two cultures” we have already referred to. As we noted earlier a division between “two cultures” of theoretical and pragmatic mathematics could be seen in the Athenian state of the Classical period. For the free citizenry forming the elite of this period the performance of work subordinated the arisan or other performer of it to the user. Thus for Aristotle and Plato the free man is a user, never a producer ideally using things correctly and never transforming them by work.⁴⁹ This explains in part the highly abstract presentation of mathematics by the Classical Greek mathematicians.

Of course the documentation of the history of mathematics and science is biased inevitably towards those areas that would have been recorded. Access to this would have been highly restricted, being most likely to be found in Courts and Mosques. Certainly royal patronage was crucial in the early stages of reconstruction of ancient knowledge and in supporting the development of instruments and facilities of astronomical observation. As Charette notes,

As already mentioned the measurement and prediction of the movement of celestial bodies has been central to many aspects of ancient life, from their astrological and religious implications, to the timing of ceremonies, to the prediction of seasonal changes for agricultural purposes. This was no less true of the Islamic world where celestial observation was required to time the call to prayer, to determine the direction of Mecca, and to inform the work of astrologers and assist in navigation as trade expanded. From the program supported by al-Mamun the Ptolmaic view of the universe became adopted everywhere in the Islamic world and the instruments named in the Almagest such as the gnomon, sun dial and quadrant, and armillary sphere were constructed and utilised. It was the C10 bilibliographer Ibn al Nadim who reported that al-Fazari was the first Muslim to have made an astrolabe.⁵² The astrolabe diffused through the Islamic world, with the most elegant crafted by leading scholars appearing, often with globes, in many of the royal libraries as well as the observatories which were constructed to systematise observations of the skies.

Right: is from a modern representation (from an “ipad app”) of an “astrolabe clock” for a location on Earth 38 degrees North and 75 degrees West on 12 February 2012 at 9.22 PM. Sun, moon and planets can be seen on the Zodiac (red circle). Below the horizon is shaded gray. Green arrow through the sun shows the time on the 24 hour clock on the perimeter. In this modern representation the blue grid marks the angles of objects in vertical (altitude) and horizontal (planes. ⁵⁴

• Right**: is from a modern representation (from an “ipad app”) of an “astrolabe clock” for a location on Earth 38 degrees North and 75 degrees West on 12 February 2012 at 9.22 PM. Sun, moon and planets can be seen on the Zodiac (red circle). Below the horizon is shaded gray. Green arrow through the sun shows the time on the 24 hour clock on the perimeter. In this modern representation the blue grid marks the angles of objects in vertical (altitude) and horizontal (planes. ⁵⁶

Left diagram is a schematic representation of the wire grid pattern placed in front of anaphoric clock disk. Concentric circles mark lattitudes, radial curves mark the hours, wide arc marks the horizon⁵⁸ Right picture is from a modern representation (from an “ipad app”) of an “astrolabe clock” for a location on Earth 38 degrees North and 75 degrees West on 12 February 2012 at 9.22 PM. Sun, moon and planets can be seen on the Zodiac (red circle). Below the horizon is shaded gray. Green arrow through the sun shows the time on the 24 hour clock on the perimeter. In this modern representation the blue grid marks the angles of objects in vertical (altitude) and horizontal (planes. ⁵⁹

illustration ~C2 CE79|| Early Astrolabe 

(~1400 CE)⁸⁰||

Perhaps the earliest description of an astrolabe like device appears once more in Vitruvius’s De architectura (~88–26 CE) where he describes an anaphoric clock.⁹⁰ This device consisted of a large vertical disc rotated by a water or other mechanism so that it turned through a complete revolution from sunrise to sunrise on the following day. On it are marked the stars (or constellations of stars) of the Northern Hemisphere (together with those south of the equator down to the tropic of Capricorn - whose projection forms the outer limit of the disk). A circle represents the Zodiac and thus the path taken by the Sun in its journey across the sky over the year. Along the Zodiac circle are 365 holes, one for each day, and each day a marker for the Sun is to be advanced one hole to take account of the changing lengths of the day with season. In front of this rotating disk is a fixed disk of wires. A vertical wire marks the meridian (which divides the Earth from East to West). Concentric circles mark out selected months. Radial curved wires represent the 24 hours of the day as the Sun on its disk rotates behind them. Made for a particular location, further curved wire arc lays out the horizon for that place (below which stars on the disc still rotate, but cannot be seen at that place. This device thus displays the positions of selected groups of stars in the sky as they appear to move from a fixed point on Earth with the passing hours.[⁹¹ (There is now a very nice free Apple ipad/iphone “app” called “astrolabe clock” which schematically illustrates just such an arrangement in its display.)⁹²

Right picture is from a modern representation of an “astrolabe clock” for a location on Earth 38 degrees North and 75 degrees West on 12 February 2012 at 9.22 PM. Sun, moon and planets can be seen on the Zodiac (red circle). Below the horizon is shaded gray. Green arrow through the sun shows the time on the 24 hour clock on the perimeter. In this modern representation the blue grid marks the angles of objects in vertical (altitude) and horizontal (planes. ⁹⁵

Left diagram is a schematic representation of the wire grid pattern placed in front of anaphoric clock disk. Concentric circles mark lattitudes, radial curves mark the hours, wide arc marks the horizon⁹⁷ Right picture is from a modern representation of an “astrolabe clock” for a location on Earth 38 degrees North and 75 degrees West on 12 February 2012 at 9.22 PM. Sun, moon and planets can be seen on the Zodiac (red circle). Below the horizon is shaded gray. Green arrow through the sun shows the time on the 24 hour clock on the perimeter. In this modern representation the blue grid marks the angles of objects in vertical (altitude) and horizontal (planes. ⁹⁸

Left diagram is a schematic representation of the wire grid pattern placed in front of anaphoric clock disk. Concentric circles mark lattitudes, radial curves mark the hours, wide arc marks the horizon¹⁰¹

Right picture is from a modern representation of an “astrolabe clock” for a location on Earth 38 degrees North and 75 degrees West on 12 February 2012 at 9.22 PM. Sun, moon and planets can be seen on the Zodiac (red circle). Below the horizon is shaded gray. Green arrow through the sun shows the time on the 24 hour clock on the perimeter. In this modern representation the blue grid marks the angles of objects in vertical (altitude) and horizontal (planes. ¹⁰²

Perhaps the earliest description of an astrolabe like device appears once more in Vitruvius’s De architectura (~88–26 CE) where he describes an anaphoric clock.¹⁰⁹ This device consisted of a large vertical disc rotated by a water or other mechanism so that it turned through a complete revolution from sunrise to sunrise on the following day. On it are drawn the projected paths of stars of the Northern Hemisphere together with the curve taken by the Sun in its journey across the sky over the year (“the zodiac”). Along the Sun’s path are 365 holes, one for each day, in which for a particular day a marker can be inserted so the clock can be adjusted to take account of the changing lengths of the day with season. This sky map rotates behind a disk of wires with concentric circles for selected months and radial curved wires representing the 24 hours of the day. Made for a particular location, an arc across the display represents the horizon for the place for which it was constructed (below which no stars can be seen). With this device it is possible to represent the positions of selected groups of stars in the sky at the particular hour.[¹¹⁰

As already mentioned the measurement and prediction of the movement of celestial bodies has been central to many aspects of ancient life, from their astrological and religious implications, to the timing of ceremonies, to the prediction of seasonal changes for agricultural purposes. This was no less true of the Islamic world where celestial observation was required to time the call to prayer, to determine the direction of Mecca, and to inform the work of astrologers and assist in navigation as trade expanded. From the program supported by al-Mamun the Ptolmaic view of the universe became adopted everywhere in the Islamic world and the instruments named in the Almagest such as the gnomon, sun dial and quadrant, and armillary sphere were constructed and utilised. It was the C10 bilibliographer Ibn al Nadim who reported that al-Fazari was the first Muslim to have made an astrolabe.¹¹⁶ The astrolabe diffused through the Islamic world, with the most elegant crafted by leading scholars appearing, often with globes, in many of the royal libraries as well as the observatories which were constructed to systematise observations of the skies.

As already mentioned the measurement and prediction of the movement of celestial bodies has been central to many aspects of ancient life, from their astrological and religious implications, to the timing of ceremonies, to the prediction of seasonal changes for agricultural purposes. This was no less true of the Islamic world where celestial observation was required to time the call to prayer, to determine the direction of Mecca, and to inform the work of astrologers and assist in navigation as trade expanded. From the program supported by al-Mamun the Ptolmaic view of the universe became adopted everywhere in the Islamic world and the instruments named in the Almagest such as the gnomon, sun dial and quadrant, and armillary sphere were constructed and utilised. It was the C10 bilibliographer Ibn al Nadim who reported that al-Fazari was the first Muslim to have made an astrolabe.¹¹⁸ The astrolabe diffused through the Islamic world, with the most elegant crafted by leading scholars appearing, often with globes, in many of the royal libraries as well as the observatories which were constructed to systematise observations of the skies.

As already mentioned the measurement and prediction of the movement of celestial bodies has been central to many aspects of ancient life, from their astrological and religious implications, to the timing of ceremonies, to the prediction of seasonal changes for agricultural purposes. This was no less true of the Islamic world where celestial observation was required to time the call to prayer, to determine the direction of Mecca, and to inform the work of astrologers and assist in navigation as trade expanded. With the rediscovery of prior knowledge of existing instruments such as the gnomon, sun dial and quadrant, and the Ptolemic understanding of the universe as materialised in the armillary sphere became available for these purposes.

As already noted, mathematical development from the Ancient world had ceased around 500 CE with the fall of the Western Roman empire. From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state and which by 641 CE had included in its conquest much of Mesopotania and the former center of mathematical learning at Alexandria. The conquerers had little interest in mathematics and to the extent that they enjoyed collective unity it was primarily through economic exchange and shared tenets of the Islamic religion. The following century was characterised by internal warfare and political turmoil which began to subside around 750 CE. Following that there commenced a re-discovery of lost mathematical and scientific knowledge. Informally a process of diffusion of Indian, Greek and Persian knowledge had begun during the preceding century.¹²¹ A more formal and systematic process now began focussed on Baghdad under the three powerful patrons of learning - the caliphs al-Mansur (754–75 CE), Haroun al-Raschid (763–809 CE) and al-Mamun (812–33 CE). There al-Mamun established a “House of Wisdom” not unlike the prior Library of Alexandria. Once more the detail of what was rediscovered and further developments upon that base is described in detail elsewhere.¹²² It is sufficient to note here that Baghdad became a cosmopolitan centre of learning. But following Baghdad, over time the Arab world - comprising as it did a wide range of communities with diverse histories, backgrounds and cultures (including the Egyptian and Greek peoples) including also not only Muslims, but also Jews and Christians - provided a fertile background against which the desire to recapture and build on lost knowledge could be realised.

As already mentioned the measurement and prediction of the movement of celestial bodies has been central to many aspects of ancient life, from their astrological and religious implications, to the timing of ceremonies, to the prediction of seasonal changes for agricultural purposes. This was no less true of the Islamic world where celestial observation was required to time the call to prayer, to determine the direction of Mecca, and to inform the work of astrologers and assist in navigation as trade expanded. With the rediscovery of prior knowledge of existing instruments such as the gnomon, sun dial and quadrant, and the Ptolemic understanding of the universe as materialised in the armillary sphere became available for these purposes. Religion and the skies Astronomy/Astrology/Religion Astrolabe

Astrology in particular is a central theme, based of course around different conventions for describing celestial configurations and events, and with different interpretations, ranging from Ancient Babylonia through even to Modern societies. Evidence of astrological considerations can be found in ancient Egyptian, Greek and Roman temples, and beliefs and practices in magic, religion, philosophy, mathematics, and astronomical observation can be found mixed together in a variety of forms. Numerous written astrological works can be found written in Greek and Latin (generally incorporating earlier Egyptian and other astrological concepts), with the earliest systematic treatise being Manilius’s Astronomica written in Latin verse in the C1 CE. ¹²⁶ Early astrological instruments not surprisingly included variants of technologies used in other fields, from the counting board to the use of diagrams, inscriptions and calculations on papyrus, clay, stone and cloth to record and convey horoscopes and other astrological information. In particular, astrologers developed boards on which different configurations of the planets and stars could be illustrated and moved to show transformations as they occur over time.¹²⁷

Ivory astrological board from Grand (Vosges) C2 CE¹³⁰

Envelope and tokens from Susa, Iran, 3200–3100 BCE132

Rope stretchers measure the land for agriculture. Picture in
 the Tomb Chapel of Menna, Luxor (Thebes) ~1200 BCE134

Metal counting rods of the Western Han Dynasty, unearthed in
 Xi’an of Shaanxi Province  ~0–200 BCE136  

Using a swan-pan board for calculation138  

Rhind Papyrus ~1650–1550 BCE 140  

Plimpton 322 Tablet ~1800 BCE (held by Yale University) -
 bearing what is believed to be a table of numerical relationships
 between the sides of triangles and associated squares142  

Roman calculator(s) at work. ~0–100 CE**¹⁴⁴

Small piece of fibula of a Baboon marked with 29 well defined
 notches ~35,000 BCE146

Khipu of 322 strands said to be from Nosca, Peru148

Set of complex tokens from Susa, Iran, from ~3350–3100
 BCE150

Envelope and tokens from Susa, Iran, 3200–3100
 BCE152

Babylonian scribal school tablet showing list of reciprocals
 ~1700 BCE154

Rope stretchers measure the land for agriculture. Picture in
 the Tomb Chapel of Menna, Luxor (Thebes) ~1200 BCE156

Metal counting rods of the Western Han Dynasty, unearthed in
 Xi’an of Shaanxi Province  ~0–200 BCE 158  

Using a swan-pan board for calculation160  

Roman calculator(s) at work. ~0–100 CE166

**Ivory astrological board from Grand (Vosges) C2 CE¹⁶⁸

**Ivory astrological board from Grand (Vosges) C2 CE170

Astrology in particular is a central theme, based of course around different conventions for describing celestial configurations and events, and with different interpretations, ranging from Ancient Babylonia through even to Modern societies. Numerous written astrological works can be found written in Greek and Latin (generally incorporating earlier Egyptian and other astrological concepts), with the earliest systematic treatise being Manilius’s Astronomica written in Latin verse in the C1 CE. Early astrological instruments included ¹⁷³

The relationship between religion, astrology and astronomical measurement should also be stressed. Many ancient structures and devices can be identified which served to measure astronomical events, and to seek to predict future movements of the sun, moon, planets and stars, the seasons, and religiously significant events shaped by these. Many such devices have been found ranging from possible astronomical implications of the sarsen circular stone monument at Stonehenge¹⁷⁷ erected in about 2200 BCE to devices explicitly constructed to measure the angle subtended above the horizon by a star.

It is not surprising that great attention was paid to the celestial realm, given its mysterious motions and in

Titled variously Μαθηματικὴ Σύνταξις (Mathēmatikē Syntaxis) in Greek, Syntaxis mathematica in Latin, or The Almagest (the “Greatest”) by arab translators (which became the English title) and then printed in England (in Latin) as Almagestum (in 1515 CE)[Claudius Ptolemaeus, Almagestum: Opus ingens ac nobile omnes Celorum motus continens. Felicibus Astris eat in lucem, 1515. Copy in the Institut für Astronomie, Universität Wien, Türkenschanzstraße 17, 1180 Wien reproduced at http://www.univie.ac.at/hwastro/books/1515_ptole_BWLow.pdf (viewed 31 Jan 2012)^], the book shows how with the Earth as stationary frame of reference it is possible to build a model of the motion of the solar system as seen from the Earth in which the Sun moves daily in a circle around the Earth, and each planet moves in a combination of two circles, the major circle (the deferent) slightly offset from the Earth (by half the distance from a point for each planet known as the Equant), and the minor circle (the epicycle) whose centre moves around the circumference of the major circle. The model was remarkably accurate in relation to observed motions of the planets, providing a clear explanation of why the planets have apparently “wandering” paths in relation to the fixed stars, and his discovery that offsetting the centres of the deferents by the use of the Equant gave a greatly improved correspondence between theory and observation.¹⁷⁹ So good was the result that Ptolemy’s model was to serve for some 1400 years until displaced, after a long ideological struggle by the heliocentric theory of Nicolaus Copernicus (1473–1543),¹⁸⁰.

Titled variously Μαθηματικὴ Σύνταξις (Mathēmatikē Syntaxis) in Greek, Syntaxis mathematica in Latin, or The Almagest by arab translators (which became the English title) the book shows how with the Earth as stationary frame of reference it is possible to build a model of the motion of the solar system as seen from the Earth in which the Sun moves daily in a circle around the Earth, and each planet moves in a combination of two circles, the major circle (the deferent) slightly offset from the Earth (by half the distance from a point for each planet known as the Equant), and the minor circle (the epicycle) whose centre moves around the circumference of the major circle. The model was remarkably accurate in relation to observed motions of the planets, providing a clear explanation of why the planets have apparently “wandering” paths in relation to the fixed stars, and his discovery that offsetting the centres of the deferents by the use of the Equant gave a greatly improved correspondence between theory and observation.¹⁸⁴ So good was the result that Ptolemy’s model was to serve for some 1400 years until displaced, after a long ideological struggle by the heliocentric theory of Nicolaus Copernicus (1473–1543),¹⁸⁵.

Titled variously Μαθηματικὴ Σύνταξις (Mathēmatikē Syntaxis) in Greek, Syntaxis mathematica in Latin, and later The Almagest by arab translators (which became the English title) the book shows how with the Earth as stationary frame of reference it is possible to build a model of the motion of the solar system as seen from the Earth in which the Sun moves daily in a circle around the Earth, and each planet moves in a combination of two circles, the major circle (the deferent) slightly offset from the Earth (by half the distance from a point for each planet known as the Equant), and the minor circle (the epicycle) whose centre moves around the circumference of the major circle. The model was remarkably accurate in relation to observed motions of the planets, providing a clear explanation of why the planets have apparently “wandering” paths in relation to the fixed stars, and his discovery that offsetting the centres of the deferents by the use of the Equant gave a greatly improved correspondence between theory and observation.¹⁸⁸ So good was the result that Ptolemy’s model was to serve for some 1400 years until displaced, after a long ideological struggle by the heliocentric theory of Nicolaus Copernicus (1473–1543),¹⁸⁹.

The relationship between religion, astrology and astronomical measurement should also be stressed. Many ancient structures and devices can be identified which served to measure astronomical events, and to seek to predict future movements of the sun, moon, planets and stars, the seasons, and religiously significant events shaped by these. Whilst there are many such devices ranging from possible astronomical implications of the sarsen circular stone monument at Stonehenge¹⁹² erected in about 2200 BCE to devices explicitly constructed to measure the angle subtended above the horizon by a star

By the time of the first century CE increasingly methodical observations of the sun, moon, known planets, and background stars had been undertaken, and it was with this heritage that Claudius Ptolemy (90 CE - 168 CE) in his book Almagestum set about writing his magisterial work on astronomy which achieved in that field the standing achieved by Euklid in his Elements in relation to Geometry. Ptolemy was a Roman citizen, writing in Greek, and living and working in Alexandria, which was by then the capital city of the Roman province of Egypt. By now he had at his disposal not only the full power of Greek mathematics now at its zenith, but also Arisotle’s philosophically constructed picture of the earth as stationary centre of a cosmos around which the stars and planets moved in circles, together with a history of recorded Greek astronomical observation (probably all of which works were available through the Library of Alexandria) - including specific references to Meton of Athens ( C5 BCE), Callipus of Cyzicus (C4 BCE), Aristarchus of Samos (C4–3 BCE), Eratosthenes of Cyrene (C3–2 BCE), and most notably Hipparchus of Nicaea (162–127 BCE).

The Almagest shows how with the Earth as stationary frame of reference it is possible to build a model of the motion of the solar system as seen from the Earth in which the Sun moves daily in a circle around the Earth, and each planet moves in a combination of two circles, the major circle (the deferent) slightly offset from the Earth (by half the distance from a point for each planet known as the Equant), and the minor circle (the epicycle) whose centre moves around the circumference of the major circle. The model was remarkably accurate in relation to observed motions of the planets, providing a clear explanation of why the planets have apparently “wandering” paths in relation to the fixed stars, and his discovery that offsetting the centres of the deferents by the use of the Equant gave a greatly improved correspondence between theory and observation.¹⁹³ So good was the result that Ptolemy’s model was to serve for some 1400 years until displaced, after a long ideological struggle by the heliocentric theory of Nicolaus Copernicus (1473–1543),¹⁹⁴.

Perhaps the earliest description of an astrolabe like device appears once more in Vitruvius’s De architectura (~88–26 CE) where he describes an anaphoric clock.¹⁹⁸ This device consisted of a large vertical disc rotated by a water or other mechanism so that it turned through a complete revolution from sunrise to sunrise on the following day. On it are drawn the projected paths of stars of the Northern Hemisphere together with the curve taken by the Sun in its journey across the sky over the year (“the zodiac”). Along the Sun’s path are 365 holes, one for each day, in which for a particular day a marker can be inserted so the clock can be adjusted to take account of the changing lengths of the day with season. This sky map rotates behind a disk of wires with concentric circles for selected months and radial curved wires representing the 24 hours of the day. Made for a particular location, an arc is scribed across the star map representing the horizon for the place for which it was constructed (below which no stars can be seen). With this device it is possible to represent the positions of selected groups of stars in the sky at the particular hour.[¹⁹⁹

As the above suggests, the confluence of Greek geometric developments, philosophic inclinations, and systematic observation (albeit with the aid of what to Modern eyes appear comparatively simple instruments) was able to give rise to a mathematical description of the cosmos that was to serve for more than a millenium. The armillary was a simple static representation of some of these basic ideas. The anaphoric clock and astrolabe used more sophisticated geometric ideas to provide a more sophisticated representation of the changing movements of constellations but with careful settings having to be made for day and either time or position of stars. However, extraordinarily, it appears that contemporaneously with this work, a mechanical mechanism had been devised which could model movements of the stars through time in a much more fluid and autonomous way enabling important and highly valued astronomical predictions to be made. This device, the most elaborate known to have existed in antiquity (and the most reminiscent of a Modern mechanical calculator or clock), is the Antikythera mechanism.

Perhaps the earliest description of an astrolabe like device appears once more in Vitruvius’s De architectura (~88–26 CE) where he describes an anaphoric clock.²⁰²

Immediately below are a modern representation of an anaphoric clock constructed by Prof . Kostas Kotsanas and his students, an illustration from the 1886 Hoffmann’s catalogue for the sale of one of two fragments of such clocks discovered in the 19th century, in this case at Grand (Vosges) France. The sun marker is shown set at the second hour on the seasonally varying hour lines. Finally the front of an early astrolabe made by Jean Fusoris of Giraumont in the Ardennes region of France (1365–1415) and now held in the Adler Astronomy Planetarium and Museum in Chicago is shown.

As the above suggests, the confluence of Greek geometric developments, philosophic inclinations, and systematic observation (albeit with the aid of what to Modern eyes appear extremely simple instruments) was able to give rise to a mathematical description of the cosmos that was to serve for more than a millenium. The armillary was a simple static representation of some of these basic ideas. However, extraordinarily, it appears that contemporaneously with this work, a dynamic mechanical mechanism had been devised which could also make important and highly valued astronomical predictions. This device, the most elaborate known to have existed in antiquity (and the most reminiscent of a Modern mechanical calculator or clock), is the Antikythera mechanism.

Perhaps the earliest description of an astrolabe like device appears once more in Vitruvius’s De architectura (~88–26 CE) where he describes an anaphoric clock.²²⁴

The astrolabe itself is based on a similar idea but instead of the clock has a scale from which time can be read if the projected paths of the stars and hour is aligned to match the observed position of a selected star or stars. It was described by the Greek and mathematician scholar Theon of Alexandria (335–405 CE). (Theon was the father of daughter Hypatia (350–415 CE), whose unpleasant death at the hands of a Christian mob effectively ended the age of mathematical development in Alexandria.²²⁵.) Theon’s description later would become a frequent reference for rediscovery of the device in later eras.

Perhaps the earliest description of an astrolabe like device appears once more in Vitruvius’s De architectura (~88–26 CE) where he describes an anaphoric clock. This device consisted of a large vertical disc rotated by a water or other mechanism so that it turned through a complete revolution from sunrise to sunrise on the following day. On it are projected the paths of stars of the Northern Hemisphere (a “planispheric projection”). Also across it lies the projected curve taken by the son as it moves across the sky over the year (“the zodiac”) and into this are holes in which a symbol for the sun can be inserted at sunrise, thereby adjusting it for the shortening day. In front of it a disk of wires represents the meridians, tropics and equator as well as circles for some key months.The clock must be made for a particular location, and across the circles is constructed an arc representing the horizon for the place for which it was constructed. The circles are also divided into twenty-four hours for each day. With this device it was possible to represent the positions of selected stars in the sky at the particular hour.[²³⁰

As already noted, mathematical development from the Ancient world had ceased around 500 CE with the fall of the Western Roman empire. From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state and which by 641 CE had included in its conquest much of Mesopotania and the former center of mathematical learning at Alexandria. The conquerers had little interest in mathematics and to the extent that they enjoyed collective unity it was primarily through economic exchange and shared tenets of the Islamic religion. The following century was characterised by internal warfare and political turmoil which began to subside around 750 CE. Following that there commenced a re-discovery of lost mathematical and scientific knowledge. Informally a process of diffusion of Indian, Greek and Persian knowledge had begun during the preceding century.²³⁴ A more formal and systematic process now began focussed on Baghdad under the three powerful patrons of learning - the caliphs al-Mansur (754–75 CE), Haroun al-Raschid (763–809 CE) and al-Mamun (812–33 CE). There al-Mamun established a “House of Wisdom” not unlike the prior Library of Alexandria. Once more the detail of what was rediscovered and further developments upon that base is described in detail elsewhere.²³⁵ It is sufficient to note here that Baghdad became a cosmopolitan centre of learning. But following Baghdad, over time the Arab world - comprising as it did a wide range of communities with diverse histories, backgrounds and cultures (including the Egyptian and Greek peoples) including also not only Muslims, but also Jews and Christians - provided a fertile background against which the desire to recapture and build on lost knowledge could be realised.

As already mentioned the measurement and prediction of the movement of celestial bodies has been central to many aspects of life, from their astrological and religious implications, to the timing of ceremonies, to the prediction of seasonal changes for agricultural purposes. The use of the gnomon (a vertical stick from which the shadow of the sun as it moves can be measured) is probably of very early origin. Such was its importance that in the

As already noted, mathematical development from the Ancient world had ceased around 500 CE with the fall of the Western Roman empire. From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state and which by 641 CE had included in its conquest much of Mesopotania and the former center of mathematical learning at Alexandria. The conquerers had little interest in mathematics and to the extent that they enjoyed collective unity it was primarily through economic exchange and shared tenets of the Islamic religion. The following century was characterised by internal warfare and political turmoil which began to subside around 750 CE. Following that there commenced a re-discovery of lost mathematical and scientific knowledge focussed on Baghdad under the three powerful patrons of learning - the caliphs al-Mansur (754–75 CE), Haroun al-Raschid (763–809 CE) and al-Mamun (812–33 CE). There al-Mamun established a “House of Wisdom” not unlike the prior Library of Alexandria. Once more the detail of what was rediscovered and further developments upon that base is described in detail elsewhere.²³⁸ It is sufficient to note here that Baghdad became a cosmopolitan centre of learning. But following Baghdad, over time the Arab world - comprising as it did a wide range of communities with diverse histories, backgrounds and cultures (including the Egyptian and Greek peoples) including also not only Muslims, but also Jews and Christians - provided a fertile background against which the desire to recapture and build on lost knowledge could be realised.

The Almagest was thus the most authoratitive exposition of astronomy to be produced in the pre-modern period. In it, as well as laying out the theory of the motions of the celestial bodies, Ptolemy lists the key measuring devices used to observe them. For example, he refers to the gnomon (a vertical stick from which the shadow of the sun as it moves can be measured) and armillary spheres (an astronomical device showing the concentric rings representing the major circles of the celestial sphere),²⁴³ and a mural quadrant (through which the elevation of celestial bodies could be measured).²⁴⁴

, the most elaborate device from antiquity yet to be found (and the most reminiscent of a Modern mechanical calculator or clock), has been the Antikythera mechanism.

By the time of the first century CE increasingly methodical observations of the sun, moon, known planets, and background stars had been undertaken, and it was with this heritage that Claudius Ptolemy (90 CE - 168 CE) in his book Almagestum set about writing his magisterial work on astronomy which achieved in that field the standing achieved by Euklid in his Elements in relation to Geometry. Ptolemy was a Roman citizen, writing in Greek, and living and working in Alexandria, which was by then the capital city of the Roman province of Egypt. By now he had at his disposal not only the full power of Greek mathematics now at its zenith, but also Arisotle’s philosophically constructed picture of the earth as stationary centre of a cosmos around which the stars and planets moved in circles, together with a history of recorded Greek astronomical observation (probably all of which works were available through the Library of Alexandria) - including Meton of Athens ( C5 BCE), Callipus of Cyzicus (C4 BCE), Aristarchus of Samos (C4–3 BCE), Eratosthenes of Cyrene (C3–2 BCE), and Menelaus of Alexandria (C1 CE).²⁴⁷

The Almagest shows how with the Earth as stationary frame of reference it is possible to build a model of the motion of the solar system as seen from the Earth in which the Sun moves daily in a circle around the Earth, and each planet moves in a combination of two circles, the major circle (the deferent) slightly offset from the Earth (by half the distance from a point for each planet known as the Equant), and the minor circle (the epicycle) whose centre moves around the circumference of the major circle. The model was remarkably accurate in relation to observed motions of the planets, providing a clear explanation of why the planets have apparently “wandering” paths in relation to the fixed stars, and his discovery that offsetting the centres of the deferents by the use of the Equant gave a greatly improved correspondence between theory and observation. So good was the result that Ptolemy’s model was to serve for some 1400 years until displaced, after a long ideological struggle by the heliocentric theory of Nicolaus Copernicus (1473–1543),²⁴⁸.

The Almagest was thus the most authoratitive exposition of astronomy to be produced in the pre-modern period. In it, as well as laying out the theory of the motions of the celestial bodies, Ptolemy lists the key measuring devices used to observe them. These included a mural quadrant (with an interior dimension of twenty cubits - 10 m) from which the elevation of celestial bodies could be measured, and a ten cubit high iron gnomon (a vertical stick from which the shadow of the sun as it moves can be observed). The gnomon in particular had probably been used from the time that it was first understood that a shadow tracked the sun. Indeed, in the

The Almagest was thus the most authoratitive exposition of astronomy to be produced in the pre-modern period. In it, as well as laying out the theory of the motions of the celestial bodies, Ptolemy lists the key measuring devices used to observe them. These included the

The Almagest shows how with the Earth as stationary frame of reference it is possible to build a model of the motion of the solar system as seen from the Earth in which the Sun moves daily in a circle around the Earth, and each planet moves in a combination of two circles, the major circle (the deferent) slightly offset from the Earth (by half the distance from a point for each planet known as the Equant), and the minor circle (the epicycle) whose centre moves around the circumference of the major circle. The model was remarkably accurate in relation to observed motions of the planets, providing a clear explanation of why the planets have apparently “wandering” paths in relation to the fixed stars, and his discovery that offsetting

By the time of the first century CE increasingly methodical observations of the sun, moon, known planets, and background stars had been undertaken, these being brought together in the magesterial writing of Claudius Ptolemy (90 CE - 168 CE) in his book Almagestum which had in relation to astronomy the standing of Euklid’s Elements in relation to Geometry. Ptolemy was a Roman citizen, writing in Greek, and living and working in Alexandria, which was by then the capital city of the Roman province of Egypt. By now he had at his disposal not only the full power of Greek mathematics now at its zenith, but also Arisotle’s philosophically constructed picture of the earth as stationary centre of a cosmos around which the stars and planets moved in circles, together with a history of recorded Greek astronomical observation - including Meton of Athens ( C5 BCE), Callipus of Cyzicus (C4 BCE), Aristarchus of Samos (C4–3 BCE), Eratosthenes of Cyrene (C3–2 BCE), and Menelaus of Alexandria (C1 CE).²⁵⁴

By the time of the first century CE increasingly methodical observations of the sun, moon, known planets, and background stars had been undertaken, these being brought together in the magesterial writing of Claudius Ptolemy (90 CE - 168 CE) in his book Almagestum which had in relation to astronomy the standing of Euklid’s Elements in relation to Geometry. Ptolemy was a Roman citizen, writing in Greek, and living and working in Alexandria, which was by then the capital city of the Roman province of Egypt. With the full power of Greek mathematics now at its zenith, the philosophically derived picture of the

Measuring, timing, calculating and predicting.

The relationship between religion, astrology and astronomical measurement should also be stressed. Many ancient structures and devices can be identified which served to measure astronomical events, and to seek to predict future movements of the sun, moon, planets and stars, the seasons, and religiously significant events shaped by these. Whilst there are many such devices ranging from possible astronomical implications of the sarsen circular stone monument at Stonehenge²⁵⁷ erected in about 2200 BCE to devices explicitly constructed to measure the angle subtended above the horizon by a star, the most elaborate device from antiquity yet to be found (and the most reminiscent of a Modern mechanical calculator or clock), has been the Antikythera mechanism.

Damascus mathematician Abu al-Hasan Ahmad ibn Ibrahim al-Uqlidisi (“the Euclidian,” fl. ca. 953 CE) further advanced the Indian mode of calculation. The Indian system had used a dustboard to perform and erase a series of calculations. Al-Uqlidisi adapted the Indian system for pen and paper. Mathematicians could now “show their work,” sharing problems, equations, and methods for solving them across time and space. Mathematics advanced rapidly as a result of recording and publication.²⁵⁹

As already noted, mathematical development from the Ancient world had ceased around 500 CE with the fall of the Western Roman empire. From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state and which by 641 CE had included in its conquest much of Mesopotania and the former center of mathematical learning at Alexandria. The conquerers had little interest in mathematics and to the extent that they enjoyed collective unity it was primarily through economic exchange and shared tenets of the Islamic religion. The following century was characterised by internal warfare and political turmoil which began to subside around 750 CE. Following that there commenced a re-discovery of lost mathematical and scientific knowledge focussed on Baghdad under the three powerful patrons of learning (the caliphs al-Mansur, Haroun al-Raschid and al-Mamun). There al-Mamun established a “House of Wisdom” not unlike the prior Library of Alexandria. Once more the detail of what was rediscovered and further developments upon that base is described in detail elsewhere.²⁶¹ It is sufficient to note here that Baghdad became a cosmopolitan centre of learning. But following Baghdad, over time the Arab world - comprising as it did a wide range of communities with diverse histories, backgrounds and cultures (including the Egyptian and Greek peoples) including also not only Muslims, but also Jews and Christians - provided a fertile background against which the desire to recapture and build on lost knowledge could be realised.

• algebra derived from Greek, Hindu and Mesopotanian sources but assuming a new and systematic form²⁶³ albeit still not expressed in the economical symbolic form of the Modern era, the concept of which was first established much later by Rene Descartes.

The last of a line of great Muslim mathematicians of this period (who died in 1436), Al-Kashi, amongst his various achievements, calculated pi to 14 decimal places (in both sexagesimal and decimal forms), a feat not matched again in the late sixteenth century after the re-emergence of mathematical development in Europe.²⁶⁴

• algebra derived from Greek, Hindu and Mesopotanian sources but assuming a new and systematic form²⁶⁸ albeit still not expressed in the economical symbolic form of the Modern era, which was later established by Rene Descartes.

• algebra derived from Greek, Hindu and Mesopotanian sources but assuming a new and systematic form²⁷⁰

The last of a line of great Muslim mathematicians of this period (who died in 1436), Al-Kashi, amongst his various achievements, calculated pi to 14 decimal places (in both sexagesimal and decimal forms), a feat not matched again in the late sixteenth century after the re-emergence of mathematical development in Europe.

This civilisation took as necessary from the prior received knowledge. In relation to numerals the Indian system of discrete Hindu numerals, ordered in their places according to powers of ten, and with the zero included, slowly gained precedence giving rise by the 16th Century to the Modern European system of “Indian-Arabic” numerals described earlier.²⁷³ As Boyer and Merzbach describe it, an arabic mathematics developed which included:

• (i) an arithmetic

As already noted, mathematical development from the Ancient world had ceased around 500 CE with the fall of the Western Roman empire. From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state and which by 641 CE had included in its conquest much of Mesopotania and the former center of mathematical learning at Alexandria. The conquerers had little interest in mathematics and to the extent that they enjoyed collective unity it was primarily through economic exchange and shared tenets of the Islamic religion. The following century was characterised by internal warfare and political turmoil which began to subside around 750 CE. Following that there commenced a re-discovery of lost mathematical and scientific knowledge focussed on Baghdad under the three powerful patrons of learning (the caliphs al-Mansur, Haroun al-Raschid and al-Mamun). There al-Mamun established a “House of Wisdom” not unlike the prior Library of Alexandria. Once more the detail of what was rediscovered and further developments upon that base is described in detail elsewhere.²⁷⁶ It is sufficient to note here that Baghdad became a cosmopolitan centre housing with reasonable tolerance Muslims, Jews and Christians, and developed as an important centre of trade providing a fertile background against which the desire to recapture and build on lost knowledge could be realised.

As already noted, mathematical development from the Ancient world had ceased around 500 CE with the fall of the Western Roman empire. From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state and which by 641 CE had included in its conquest much of Mesopotania and the former center of mathematical learning at Alexandria.

From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state and which by 641 CE had included in its conquest much of Mesopotania and the center of mathematical learning at Alexandria.

It is not necessary here to dwell for long on the expansive achievements of the Ancient Greeks in the development of abstract geometry and mathematical reasoning. All this is described both in great detail, but also accessibly in summary elsewhere.²⁸⁰ There is a shadowy history of mathematics in Ancient Greece, beginning with the entry into the area in the second millenium BCE by invaders from the north with no known capacities in literacy or numeracy, to the likelihood of trade and other interchanges with Egypt and Mesopotania, and then to the sophisticated Greek literature, already evident by the first Olympic Games in 776 BCE, and then to the beginnings of the formal abstract mathematics for which they have been so celebrated, traced to the illusive figures of Thales of Miletus (around 585 BCE) and his use of geometry to solve practical problems, and Pythagoras of Samos (around 580–500 BCE).²⁸¹ The development of mathematics from these sources (and the oral mathematical knowledge which they may have formalised) flowered until the destruction of the Academy of Athens in 529 ADE. That period saw the emergence of “schools” of mathematical philosophers including the magisterial Pythagorean School and its emphasis on proof (including, so it is believed, the proof of “Pythagoras’s Theorem”), Plato’s Academy in Athens (which became a centre of mathematics in the 4th century BCE), and associated achievements including an iterative method used to determine the areas and volumes of complex curved and other objects. The achievements were famously brought together by Euclid in his Elements in the 3rd century BCE in which the formalisation of what we now understand as ‘mathematical’ rigour and its use for “proof” was systematically displayed. From thereon, the “Golden Age” of Greek mathematics began to decline although there were nevertheless a series of significant analytic developments, especially in algebra.

Importantly for the discussion here, which is after all directed at providing part of the context for the development of calculating technology, it should be noted that this abstract, and thereafter much celebrated invention of formally and systematically written abstract mathematical (and other applications) of reasoning, does not represent the only mathematics that was done. Indeed there was at least what Marcus Asper has styled “the two cultures of mathematics” in Ancient Greece.²⁸² To be more precise he stresses that recently “a consensus has emerged that Greek mathematics was heterogenous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices.”²⁸³ This picture of a tapestry of mathematical practices being in play by different participants in the society must be closer to the reality than two cultures (the old joke should be borne in mind that ‘there are two classes of people: those who divide the world into two classes of people, and those who don’t’). Nevertheless, with that caveat it is useful to reflect on the fact that at least two practices were in play: the abstractions being developed by a relatively small philosophically inclined elite (what Asper calls “theoretical mathematics”), and the continuation of the practical mathematics in the style of useful recipes for practical purposes in everyday activities, which was the legacy of Mesopotania and Egypt, to name just two major centres (which will be referred to here as “artisanal” or “pragmatic” mathematics and which Asper refers to as “practical mathematics”²⁸⁴).

The Roman Republic (510–44 BCE) gave rise to the Roman Empire which at its height stretched from England to the Persian Gulf, and ended with the fall of its Western half in 476 AD, Whilst it was a powerful military and organised society, it was not much attracted to the power of mathematical investigation, contributed little to what is known about mathematics, and gained much of its practical knowledge of it from the civilisations it conquered and with which it traded, not the least from the Ancient Greeks.²⁹² The development of Roman numerals, and the importance of the use of the abacus in Ancient Rome in manipulating them efficiently has already been discussed. Beyond this only one further aspect of the use of mathematics in Rome will be considered here as a useful illustration of a more general conclusion.

Building on multiple pasts - the “Arabic hegemony”²⁹⁵

From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state sweeping before it much of Mesopotania, and which by 641 CE had included in its conquest the mathematical center of Alexandria.

From the death of the prophet Mohammed at Medina in 632 CE there was a rapid expansion of the Islamic state sweeping before it much of Mesopotania, and by 641 marked by the conquest of the mathematical center of Alexandria.

The relationship between religion, astrology and astronomical measurement should also be stressed. Many ancient structures and devices can be identified which served to measure astronomical events, and to seek to predict future movements of the sun, moon, planets and stars, the seasons, and religiously significant events shaped by these. Whilst there are many such devices ranging from possible astronomical implications of the sarsen circular stone monument at Stonehenge²⁹⁷ erected in about 2200 BCE to devices explicitly constructed to measure the angle subtended above the horizon by a star, the most elaborate device from antiquity yet to be found (and the most reminiscent of a Modern mechanical calculator or clock), has been the Antikythera mechanism.

The relationship between religion, astrology and astronomical measurement should also be stressed. Many ancient structures and devices can be identified which served to measure astronomical events, and to seek to predict future movements of the sun, moon, planets and stars, the seasons, and religiously significant events shaped by these. Whilst there are many such devices ranging from possible astronomical uses of the circular stone ring at Stonehenge erected in about 2200 BCE to devices to measure the angle subtended above the horizon by a star, the most elaborate device from antiquity yet to be found (and the most reminiscent of a Modern mechanical calculator or clock), has been the Antikythera mechanism.

Building on multiple pasts - the “Arabic hegemony”³⁰⁰

This relationship between perceived need and innovation is strikingly illustrated by the fact that the principle of the steam engine was demonstrated in Ancient Greece, , in the form of the aeolipile (or “Hero’s engine”), more than two thousand years before the industrial revolution. As described by Hero of Alexandria, a ball was caused to spin by steam blown out through two counterposed jets (for a diagram and explanation click here).³⁰² It may well have been possible with the known technology at the time to extend this method to create a working steam powered machine. But it is likely there would have been little incentive to explore this possibility since there was plenty of slave labour available to the free citizens of Athens. Indeed in the same tract in which the steam device is described there is serious attention paid to the perceived value of applying practical applications of pneumatic devices to producing mysterious effects for use in religious ceremonies.³⁰³

Building on multiple pasts - the “Islamic hegemony”

As the above suggests, apparently obvious developments may be ignored where no need for them is perceived. However, where power, authority or ceremony is seen to require a state of the art development, and sufficient resources are made available, then the best minds and most skilled talent may be brought to the task. It is reasonable to speculate that as with the success of the Manhattan and Moon Landing projects of the Twentieth Century (CE), some two thousand years before there were some of high authority also able to martial the best talents available to create an object of technological achievement that would be a source of wonder for all privileged to view it. Surviving evidence for that lies in the eroded but still comprehensible remains of the Antikythera mechanism.

As the above suggests, apparently obvious developments may be ignored where no need for them is perceived. However, where power, authority or ceremony is seen to require a state of the art development, and sufficient resources are made available, then the best minds and most skilled talent may be brought to the task. It is reasonable to speculate that as with the success of the Manhattan and Moon Landing projects of the Modern era, some two thousand years before there were some of high authority able to martial the best talents available to create an object of technological achievement that would be a source of wonder for all privileged to view it. Surviving evidence for that lies in the eroded but still comprehensible remains of the Antikythera mechanism.

As the above suggests, apparently obvious developments may be ignored where no need for them is perceived. However, where power, authority or ceremony is seen to require a state of the art development, and sufficient resources are made available, then the best minds and most skilled talent may be brought to the task. It is reasonable to speculate that as with the success of such a strategy in the Manhattan and Moon Landing projects of the Modern era, there existed a similar capacity by those of high authority to martial resources to meet a perceived social need for a pioneering technological development (albeit at a suitable scale for the time) several thousand years ago as exemplified in the eroded but still comprehensible remains of the Antikythera mechanism.

This relationship between perceived need and innovation is strikingly illustrated by the fact that the principle of the steam engine was demonstrated in Ancient Greece, , in the form of the aeolipile (or “Hero’s engine”), more than two thousand years before the industrial revolution. As described by Hero of Alexandria, a ball was caused to spin by steam blown out through two counterposed jets (for a diagram and explanation click here).³⁰⁷ It may well have been possible with the known technology at the time to extend this method to create a working steam powered machine. But it is likely there would have been little incentive to explore this possibility since there was plenty of slave labour available to the free citizens of Athens. Indeed in the same tract in which the steam device is described there is serious attention paid to the perceived value of applying practical applications of pneumatic devices to producing magical effects for use in religious ceremonies.³⁰⁸

Religion and the skies Astronomy/Astrology/Religion Antikythera Astrolabe

Copernicus

Trade, power and navigation Clocks and Astronomy

Aristocracy and merchants Merchants vs Old order

The relationship between religion, astrology and astronomical measurement should also be stressed. Many ancient structures and devices can be identified which served to measure astronomical events, and to seek to predict future movements of the sun, moon, planets and stars, the seasons, and religiously significant events shaped by these. Whilst there are many such devices ranging from the Druid circular stone ring at Stonehenge erected in about 2200 BCE to devices to measure the angle subtended above the horizon by a star, the most elaborate device from antiquity yet to be found (and the most reminiscent of a Modern mechanical calculator or clock), has been the Antikythera mechanism.

By turning the wheels on this device with a handle the user was able to determine the relative positions of Sun and Moon. The lower back dial predicted luna eclipses whilst the upper dial was a Metonic calendar (based on the 19 year cycle the Moon takes, seen from a particular place on Earth, to return to the same place in the sky)³¹¹ and all 12 months have now been identified. The device was of Corinthian origin, suggesting a heritage going back to Archimedes (who died in 212 BCE). In support of this contention in 1974 Derek J De Solla Price pointed out that Cicero (106–43 BCE) wrote that Archimedes built a planetarium which showed the Moon rise following the Sun above the Earth, together with eclipses and the motions of the five known planets, suggesting that the Antikythera had a similar purpose. However, the extent to which the Planetarium could have performed these claimed tasks has been open to controversy .³¹²

By turning the wheels on this device with a handle the user was able to determine the relative positions of Sun and Moon. The lower back dial predicted luna eclipses whilst the upper dial was a Metonic calendar (based on the 19 year cycle the Moon takes, seen from a particular place on Earth, to return to the same place in the sky)³¹⁵ and all 12 months have now been identified. The device was of Corinthian origin, suggesting a heritage going back to Archimedes (who died in 212 BCE). In support of this contention in 1974 Derek J De Solla Price pointed out that Cicero (106–43 BCE) wrote that Archimedes built a planetarium which showed the Moon rise following the Sun above the Earth, together with eclipses and the motions of the five known planets, suggesting that the Antikythera had a similar purpose. However, that connection has been the subject of controversy.³¹⁶

By turning the wheels on this device with a handle the user was able to determine the relative positions of Sun and Moon. The lower back dial predicted luna eclipses whilst the upper dial was a Metonic calendar (based on the 19 year cycle the Moon takes, seen from a particular place on Earth, to return to the same place in the sky)³¹⁹ and all 12 months have now been identified. The device was of Corinthian origin, suggesting a heritage going back to Archimedes (who died in 212 BCE). According to Cicero, Archimedes invented a planetarium and wrote a book on astronomical mechanisms (which remains unrecovered). An upper minor dial follows the four-year cycle of the Olympiad - the most important of the associated Panhellic games. The device, identified as being from about 100 BCE, was thus “not simply an instrument of abstract science, but exhibited astronomical phenomena in relation to Greek social institutions.” ³²⁰

As the above suggests, apparently obvious developments may be ignored where no need for them is perceived. However, where power, authority or ceremony is seen to require a state of the art development, and sufficient resources are made available, then the best minds and most skilled talent may be brought to the task. It is reasonable to speculate that as with the success of such a strategy in the Manhattan and Moon Landing projects of the Modern era, there existed a similar capacity by those of high authority to martial resources to meet a perceived social need for a pioneering technological development (albeit at a suitable scale for the time) several thousand years before as exemplified in the eroded but still comprehensible remains of the Antikythera mechanism.

As the above suggests, apparently obvious developments may be ignored where no need for them is perceived. However, where power, authority or ceremony is seen to require a state of the art development, and sufficient resources are made available, then the best minds and most skilled talent may be brought to the task. It is reasonable to speculate that as with the Manhattan and Moon Landing projects of the Modern era, there was a similar capcity to martial resources to such an end several thousand years before as exemplified in the eroded but still comprehensible remains of the Antikythera mechanism.

Whilst this discussion is focussed on calculation the need for this is of course is only one aspect of the developing needs of a complex society. As already mentioned, measurement has been equally crucial, and measurement and calculation form parts of a bigger whole. In all the evolving social settings mentioned so far, measurement has played an important role. This ranges from the work of the rope stretchers of Egypt (and Athens) to the early methodologies for measuring time (for example with sand and water flows, and the burning of graduated candles) in Ancient Rome.

This relationship between perceived need and innovation is strikingly illustrated by the fact that the principle of the steam engine was demonstrated in Ancient Greece, , in the form of the aeolipile (or “Hero’s engine”), more than two thousand years before the industrial revolution. As described by Hero of Alexandria, a ball was caused to spin by steam blown out through two counterposed jets (for a diagram and explanation click here).³²⁵

It would be possible with the known technology at the time to extend the method used above to create a working steam powered machine. But it is likely there would have been little incentive to explore this possibility since there was plenty of slave labour available to the free citizens of Athens. Indeed in the same tract in which the steam device is described there is serious attention paid to the perceived value of applying practical applications of pneumatic devices to producing magical effects for use in religious ceremonies.³²⁷

This relationship between perceived need and innovation is strikingly illustrated by the fact that the principle of the steam engine was demonstrated in Ancient Greece, , in the form of the aeolipile (or “Hero’s engine”), more than two thousand years before the industrial revolution. As described by Hero of Alexandria, a ball was caused to spin by steam blown out through two counterposed jets (see drawing below).³³⁰

As the above suggests, whilst apparently obvious developments may be ignored where no need for them is perceived, where power, authority or ceremony is seen to require a state of the art development, and sufficient resources are made available, then the best minds and most skilled talent may be brought to the task. It is reasonable to speculate that as with the Manhattan and Moon Landing projects of the Modern era, there was a similar capcity to martial resources to such an end several thousand years before as exemplified in the eroded but still comprehensible remains of the Antikythera mechanism.

It would be possible with the known technology at the time to extend this principle to a working steam powered machine. But it is likely there would have been little incentive to explore this possibility since there was plenty of slave labour available to the free citizens of Athens. Indeed in the same tract in which the steam device is described there is serious attention paid to the perceived value of applying practical applications of pneumatic devices to producing magical effects for use in religious ceremonies.³³²

||Graphic representation of Hero’s Engine ~200–100
 BCE|| Associated description||340

  http://meta-studies.net/pmwiki/uploads/Evmath/HeroEnginejpg.jpg
Graphic representation of Hero’s Engine ~200–100
 BCE342

This relationship between perceived need and innovation is strikingly illustrated by the fact that the principle of the steam engine was demonstrated in Ancient Greece, , in the form of the aeolipile (or “Hero’s engine”), more than two thousand years before the industrial revolution. As described by Hero of Alexandria, a ball was caused to spin by steam blown out through two counterposed jets.³⁴⁴

Hero’s Engine ~200–100 BCE346

It would have been possible to extend this principle to a working steam powered machine. But it can be argued this possibility was not vigorously pursued since there was plenty of slave labour available to the free citizens of Athens. Indeed in the same tract in which the steam device is described there is much attention to the practical applications of pneumatic devices to producing magical effects for use in religious ceremonies.³⁴⁸

  http://meta-studies.net/pmwiki/uploads/Evmath/HeroEngine.jpg

This relationship between perceived need and innovation is strikingly illustrated by the fact that the principle of the steam engine was demonstrated in Ancient Greece, , in the form of the aeolipile (or “Hero’s engine”), more than two thousand years before the industrial revolution. As described by Hero of Alexandria, a ball was caused to spin by steam blown out through two counterposed jets.³⁵⁰ It would have been possible to extend this principle to a working steam powered machine. But it can be argued this possibility was not vigorously pursued since there was plenty of slave labour available to the free citizens of Athens. Indeed in the same tract in which the steam device is described there is much attention to the practical applications of pneumatic devices to producing magical effects for use in religious ceremonies.³⁵¹

This relationship between perceived need and innovation is strikingly illustrated by the fact that the principle of the steam engine was demonstrated in Ancient Greece, more than two thousand years before the industrial revolution, in the form of the aeolipile (or “Hero’s engine”). Attributed to Hero of Alexandria, a ball was caused to spin by steam blown out through two counterposed jets.³⁵⁵ It would have been possible to extend this principle to a working steam powered machine. But it can be argued this possibility was not vigorously pursued since there was plenty of slave labour available to the free citizens of Athens. Indeed in the same tract in which the steam device is described there is much attention to the practical applications of pneumatic devices to producing magical effects for use in religious ceremonies.³⁵⁶

This relationship between perceived need and innovation is strikingly illustrated by the fact that the principle of the steam engine was demonstrated in Ancient Greece, more than two thousand years before the industrial revolution, in the form of the aeolipile (or “Hero’s engine”) where a ball was caused to spin by steam blown out through two counterposed jets.[^First described by the Roman architect Vitruvius in De architectura, ~15 BCE It would have been possible to extend this principle to a working steam powered machine. But it can be argued this was never done since there was plenty of slave labour available to the free citizens of Athens.

Where power, authority, ceremony is seen to require a state of the art development, then the best minds and most skilled tallents can be brought to the task. It is reasonable to speculate that as with the Manhattan and Moon Landing projects of the Modern era, there was a similar capcity to martial resources to such an end several thousand years before as exemplified in the eroded but still comprehensible remains of the Antikythera mechanism.

A chasm exists in sophistication between this high-precision geared calculating device and the pebble accounting described earlier. Whilst much is unknown about it it is clear that the development and production of the Antikythera machine would have required high mathematical understanding, and the most sophisticated artisanal skills available. It is likely it was produced for a person or institution of the highest standing in Greek society who was able to call upon the most advanced talents of the day. Apart from the extraordinary accomplishments embodied in it, and all this has to say about what could be achieved in the sophisticated society of Ancient Greece in the later part of the “Golden Era” of Greek mathematical invention, this device also illustrates the dangers of assuming a sharp division between pragmatic and abstract mathematical cultures, or useful versus theoretical outcomes, since this accomplishment clearly drew on high skills in both areas. Above all it is a reminder between perceived social need, and innovation. Where pebble accounting will do, why use anything more complicated and expensive? Where power, authority, ceremony is seen to require a state of the art development, then the best minds and most skilled tallents can be brought to the task. It is reasonable to speculate that as with the Manhattan and Moon Landing projects of the Modern era, there was a similar capcity to martial resources to such an end several thousand years before as exemplified in the eroded but still comprehensible remains of the Antikythera mechanism.

A chasm exists in sophistication between this device and the pebble accounting described earlier. Whilst much is unknown about it it is clear that the development and production of the Antikythera machine would have required high mathematical understanding, and the most sophisticated artisanal skills available. It is likely it was produced for a person or institution of the highest standing in Greek society who was able to call upon the most advanced talents of the day. Apart from the extraordinary accomplishments embodied in it, and all this has to say about what could be achieved in the sophisticated society of Ancient Greece in the later part of the “Golden Era” of Greek mathematical invention, this device also illustrates the dangers of assuming a sharp division between pragmatic and abstract mathematical cultures, or useful versus theoretical outcomes, since this accomplishment clearly drew on high skills in both areas. Above all it is a reminder between perceived social need, and innovation. Where pebble accounting will do, why use anything more complicated and expensive? Where power, authority, ceremony is seen to require a state of the art development, then the best minds and most skilled tallents can be brought to the task. It is reasonable to speculate that as with the Manhattan and Moon Landing projects of the Modern era, there was a similar capcity to martial resources to such an end several thousand years before as exemplified in the eroded but still comprehensible remains of the Antikythera mechanism.

Building on multiple pasts - Islamic mathematics

A chasm exists in sophistication between this device and the pebble accounting described earlier. Whilst much is unknown about it it is clear that its production would have required high mathematical understanding, and the most sophisticated artisanal skills available. It is likely it was produced for a person or institution of the highest standing in Greek society who was able to call upon the most advanced talents of the day. Apart from the extraordinary accomplishments embodied in it, and all this has to say about what could be achieved in the sophisticated society of Ancient Greece in the later part of the “Golden Era” of Greek mathematical invention, this device also illustrates the dangers of assuming a sharp division between pragmatic and abstract mathematical cultures, since this accomplishment clearly drew on high skills in both areas. Above all it is a reminder between perceived social need, and innovation. Where pebble accounting will do, why use anything more complicated and expensive? Where power, authority, ceremony is seen to require a state of the art development, then the best minds and most skilled tallents can be brought to the task. It is reasonable to speculate that as with the Manhattan and Moon Landing projects of the Modern era, there was a similar capcity to martial resources to such an end several thousand years before as exemplified in the eroded but still comprehensible remains of the Antikythera mechanism.

The Roman Republic (510–44 BCE) gave rise to the Roman Empire which at its height stretched from England to the Persian Gulf, and ended with the fall of its Western half in 476 AD, Whilst it was a powerful military and organised society, it was not much attracted to the power of mathematical investigation, contributed little to what is known about mathematics, and gained much of its practical knowledge of it from the civilisations it conquered and with which it traded, not the least from the Ancient Greeks. The development of Roman numerals, and the importance of the use of the abacus in Ancient Rome in manipulating them efficiently has already been discussed. Beyond this only one further aspect of the use of mathematics in Rome will be considered here as a useful illustration of a more general conclusion.

A chasm exists in sophistication between this device and the pebble accounting described earlier. Whilst much is unknown about it it is clear that its production would have required high mathematical understanding, and the most sophisticated artisanal skills available. It is likely it was produced for a person or institution of the highest standing in Greek society who was able to call upon the most advanced talents of the day. Apart from the extraordinary accomplishments embodied in it, and all this has to say about what could be achieved in the sophisticated society of Ancient Greece as it moved into

The Roman Republic (510–44 BCE) gave rise to the Roman Empire which at its height stretched from England to the Persian Gulf, and ended with the fall of its Western half in 476 AD, Whilst it was a powerful military and organised society, it gained much of its practical knowledge from the civilisations it conquered and with which it traded, not the least in mathematics, from the Ancient Greeks. The development of Roman numerals, and the importance of the use of the abacus in Ancient Rome in manipulating them efficiently has already been discussed. Beyond this only one further aspect of the use of mathematics in Rome will be considered here as a useful illustration of a more general conclusion.

By turning the wheels on this device with a handle the user was able to determine the relative positions of Sun and Moon. The lower back dial predicted luna eclipses whilst the upper dial was a Metonic calendar for which all 12 months have now been identified. The device was of Corinthian origin, suggesting a heritage going back to Archimedes (who died in 212 BCE). According to Cicero, Archimedes invented a planetarium and wrote a book on astronomical mechanisms (which remains unrecovered). An upper minor dial follows the four-year cycle of the Olympiad - the most important of the associated Panhellic games. The device, identified as being from about 100 BCE, was thus “not simply an instrument of abstract science, but exhibited astronomical phenomena in relation to Greek social institutions.” ³⁶²

This heavily corroded and encrusted set of 82 remenant fragments was discovered by Greek sponge divers in 1901 amongst a range of treasures from C2-C1 BCE in the wreck of an ancient Roman galleon at a depth of 60 metres off Point Glyphadia on the Greek island Antikythera. Whilst there was much speculation on its mechanism the fragments have now yielded much greater detail to modern imaging technology, revealing a highly complex ancient Greek mechanism complete with inscriptions which have been translated. In 2008 the mechanism was identified in an article in Nature as consisting of a device with a bronze system of interlocking cogs, and front and back output dials with the following significance: the device, originally housed in a wooden frame, had some 30 intermeshed cogs which represented calendar cycles.³⁶⁵ The accuracy with which the mechanism was constructed is considered to be perhaps unrivalled until clockwork mechanisms developed in the Middle Ages a thousand years later.³⁶⁶

By turning the wheels on this device with a handle the user could determine the relative positions of Sun and Moon. The the lower back dial predicts luna eclipses whilst the upper dial is a Metonic calendar with all 12 months identified, and of Corinthian origin, suggesting a heritage going back to Archimedes (who died in 212 BCE) and who, according to Cicero, invented a planetarium and wrote a book on astronomical mechanisms (which remains unrecovered). An upper minor dial follows the four-year cycle of the Olympic and associated Panhellic games. The device is identified as being from about 100 BCE and “not simply an instrument of abstract science, but exhibited astronomical phenomena in relation to Greek social institutions.” ³⁶⁹

Measurement and Calculation.

This heavily corroded and encrusted set of 82 remenant fragments was discovered by Greek sponge divers in 1901 amongst a range of treasures from C2-C1 BCE in the wreck of an ancient Roman galleon at a depth of 60 metres off Point Glyphadia on the Greek island Antikythera. Whilst there was much speculation on its mechanism the fragments have now yielded much greater detail to modern imaging technology, revealing a highly complex ancient Greek mechanism complete with inscriptions which have been translated. In 2008 the mechanism was identified in an article in Nature as consisting of a device with a bronze system of interlocking cogs, and front and back output dials with the following significance: the device had some 30 intermeshed cogs which represented calendar cycles.³⁷¹ The accuracy with which the mechanism was constructed is considered to be perhaps unrivalled until clockwork mechanisms developed in the Middle Ages.³⁷²

By turning the wheels on this device with a handle the user could determine the relative positions of Sun and Moon. The the lower back dial predicts luna eclipses whilst the upper dial is a Metonic calendar with all 12 months identified, and of Corinthian origin, suggesting a heritage going back to Archimedes (who died in 212 BCE) and who, according to Cicero, invented a planetarium and wrote a book on astronomical mechanisms (which remains unrecovered). An upper minor dial follows the four-year cycle of the Olympic and associated Panhellic games. The device is identified as being from about 100 BCE and “not simply an instrument of abstract science, but exhibited astronomical phenomena in relation to Greek social institutions.” ³⁷⁵

This heavily corroded and encrusted set of 82 remenant fragments was discovered by Greek sponge divers in 1901 amongst a range of treasures from C2-C1 BCE in the wreck of an ancient Roman galleon at a depth of 60 metres off Point Glyphadia on the Greek island Antikythera. Whilst there was much speculation on its mechanism the fragments have now yielded much greater detail to modern imaging technology, revealing a highly complex ancient Greek mechanism complete with inscriptions which have been translated. In 2008 the mechanism was identified in an article in Nature as consisting of a device with a bronze system of interlocking cogs, and front and back output dials with the following significance: the device had some 30 intermeshed cogs which represented calendar cycles.³⁷⁷ The accuracy with which the mechanism was constructed is considered to be perhaps unrivalled until clockwork mechanisms developed in the Middle Ages.[^ T. Freeth, Y. Bitsakis, X. Moussas, J.H. Seiradakis, A.Tselikas, E. Magkou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M.G. Edmunds, “Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism”, Nature vol 444, 2006, pp. 587–591]^

This heavily corroded and encrusted set of 82 remenant fragments was discovered by Greek sponge divers in 1901 amongst a range of treasures from C2-C1 BCE in the wreck of an ancient Roman galleon at a depth of 60 metres off Point Glyphadia on the Greek island Antikythera. Whilst there was much speculation on its mechanism the fragments have now yielded much greater detail to modern imaging technology, revealing a highly complex ancient Greek mechanism complete with inscriptions which have been translated. In 2008 the mechanism was identified in an article in Nature as consisting of a device with a bronze system of interlocking cogs, and front and back output dials with the following significance: the device had some 30 intermeshed cogs which represented calendar cycles.³⁸⁰

By turning the wheels on this device with a handle the user could determine the relative positions of Sun and Moon. The the lower back dial predicts luna eclipses whilst the upper dial is a Metonic calendar with all 12 months identified, and of Corinthian origin, suggesting a heritage going back to Archimedes (who died in 212 BCE) who according to Cicero invented a planetarium and wrote a book on astronomical mechanisms (which remains unrecovered). An upper minor dial follows the four-year cycle of the Olympic and associated Panhellic games. The device is identified as being from about 100 BCE and “not simply an instrument of abstract science, but exhibited astronomical phenomena in relation to Greek social institutions.” ³⁸¹

This heavily corroded and encrusted remenant was discovered by Greek sponge divers in 1901 amongst a range of treasures from C2-C1 BCE in the wreck of an ancient Roman galleon at a depth of 60 metres off Point Glyphadia on the Greek island Antikythera. Whilst there was much speculation on its mechanism the fragments have now yielded much greater detail to modern imaging technology, revealing a highly complex ancient Greek mechanism. The mechanism has recently been formally identified as consisting of a ³⁸⁴

The Antikythera mechanism[^Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the

Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617.^] This heavily corroded and encrusted remenant was discovered by Greek sponge divers in 1901 amongst a range of treasures from C2-C1 BCE in the wreck of an ancient Roman galleon at a depth of 60 metres off Point Glyphadia on the Greek island Antikythera. Whilst there was much speculation on its mechanism the fragments have now yielded much greater detail to modern imaging technology, revealing a highly complex ancient Greek mechanism. The mechanism has recently been formally identified as consisting of a

The Antikythera mechanism³⁸⁸

[^ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. ^]

The Antikythera mechanism

[^Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617.^]

This heavily corroded and encrusted remenant was discovered by Greek sponge divers in 1901 amongst a range of treasures from C2-C1 BCE in the wreck of an ancient Roman galleon at a depth of 60 metres off Point Glyphadia on the Greek island Antikythera. Whilst there was much speculation on its mechanism the fragments have now yielded much greater detail to modern imaging technology, revealing a highly complex ancient Greek mechanism. The mechanism has recently been formally identified as consisting of a [^Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617.^]

The relationship between religion, astrology and astronomical measurement should also be stressed. Many ancient structures and devices can be identified which served to measure astronomical events, and to seek to predict future movements of the sun, moon, planets and stars, the seasons, and religiously significant events shaped by these.

  http://meta-studies.net/pmwiki/uploads/Evmath/antikythera.jpg
Antikythera mechanism fragments ~150–80 BCE398

Antikythera - who made it? What for?

Rome

There is a subtle issue to be careful with in the above argument. It is generally an oversimplification to suggest that a social change led to a technological change or vice-versa. After all, it is certainly true that more complex forms of social organisation created the impetus to develop better forms of calculation, but it is also true that the invention of more sophisticated forms of calculation allowed these more complex forms of organisation to develop and flourish. What we can say is that the two - more complex society (in the sense of greater interdepenency and interaction across larger numbers of people and institutions), and corresponding governance systems (of a variety of types) - co-evolved with more sophisticated forms of mathematical capacity (which, as argued elsewhere, is part of a broader capacity for reflexivity).⁴⁰⁷

Whilst pebble counting technology may lie at the pragmatic end of the pragmatic-abstract dimension it does not follow that all calculational technologies can be properly placed there. Indeed, there is at least one technology surviving from Ancient Greece that illustrates this point graphically.

The Antikythera

In

What can we make of this? Asper argues that the abstract or theotetical “culture” of mathematics was developed by a small group of elite Greek citizens with an impetus that was partly aesthetic, partly a ‘game’ in which successful players, already Athenian ‘gentlemen of means’ who could afford such pursuits without the need to gain financial return,.⁴⁰⁹ gained prestige (and no doubt associated authority), and which was focussed on communicating through a developing orthodox written form of discourse, general theorems about ideal geometric forms. The work emerged in the sixth to fifth century from practical roots, but the thrust was sharply against the ‘vulgarity’ of meeting practical needs and indeed in its mode of presentation and areas of work was in part directed to maintaining a sharp distinction between crude application, and the intellectual abstract search for proved knowledge, which lay at its core.⁴¹⁰

Even for the limited synoptic history described above the relationship between the emergence of these mathematical systems and the needs and organisation of the societies is quite striking. In each case we have touched on, the evolution of counting and the technologies to facilitate that, has been shaped, and shaped the types of society which could be constructed.

For the Egyptian and Sumerian civilisations which could settle in relatively climatically stable and fertile valleys, the consequent emergence of agricultural practices, and the correspondingly more settled agricultural societies led to the possibility and growth of a more complex urbanised social organisation. In these circumstances roles could more richly diversify with trade and barter developing along with some form of broader organisation of ruler and ruled, with accompanying trappings of allocation of land and other resources.

The geometry of forms becomes the paradigm physical embodiment of the abstract end, whilst numbers, and simple calculational devices for manipulating them, most clearly the use of “pebbles” (whose origins are already shrouded at the dawn of humanity) is situated at the other. Pebble arithmetic, together with the means of measuring, weighing, sighting angles, and the like never appears in the theoretical accounts, has left sufficient shadow through sporadic references (such as in Aristophanes quoted earlier) to indicate that their use as counters, often on a marked board, was widely used.⁴¹³ And the manipulation of pebbles, in patterns, and then stacked into volumes, is enough to provide a wide range of mathematical insights. So whilst the abstract mathematical game playing of the elite who developed that may never have mentioned it, an unknown number of insights no doubt translated from the more practical pursuits of the pragmatic world of calculation, and its pebble counting technologies.

Whilst pebble counting technology may lie at the pragmatic end of the pragmatic-abstract dimension it does not follow that all calculational technologies can be properly placed there. Indeed, there is at least one technology surviving from Ancient Greece that illustrates this point graphically.

The Antikythera

In

Antikythera - who made it? What for?

Building on multiple pasts - Islamic mathematics

Society and Mathematics - a dynamic relationship

The geometry of forms becomes the paradigm physical embodiment of the abstract end, whilst numbers, and simple calculational devices for manipulating them, most clearly the use of “pebbles” (whose origins are already shrouded at the dawn of humanity) is situated at the other. Pebble arithmetic, together with the means of measuring, weighing, sighting angles, and the like never appears in the theoretical accounts, has left sufficient shadow through sporadic references (such as in Aristophanes quoted earlier) to indicate that their use as counters, often on a marked board, was widely used. And the manipulation of pebbles, in patterns, and then stacked into volumes, is enough to provide a wide range of mathematical insights. So whilst the abstract mathematical game playing of the elite who developed that may never have mentioned it, an unknown number of insights no doubt translated from the more practical pursuits of the pragmatic world of calculation, and its pebble counting technologies.

The geometry of forms becomes the paradigm physical embodiment of the abstract end, whilst numbers, and simple calculational devices for manipulating them, most clearly the use of “pebbles” (whose origins are already shrouded at the dawn of humanity) is situated at the other. Indeed pebbles

What can we make of this? Asper argues that the abstract or theotetical “culture” of mathematics was developed by a small group of elite Greek citizens with an impetus that was partly aesthetic, partly a ‘game’ in which successful players gained prestige (and no doubt associated authority), and which was focussed on communicating through a developing orthodox written form of discourse, general theorems about ideal geometric forms. The work emerged in the sixth to fifth century from practical roots, but the thrust was sharply against the ‘vulgarity’ of meeting practical needs and indeed in its mode of presentation and areas of work was in part directed to maintaining a sharp distinction between crude application, and the intellectual abstract search for proved knowledge, which lay at its core.⁴¹⁷

Thus in the Greek story, the evolution of calculation merges into an evolution of mathematics, but divided along a spectrum marked by the pragmatic and practical at one end, and the theoretical and abstract at the other. The geometry of forms becomes the paradigm physical embodiment of the abstract end, whilst numbers, and simple calculational devices for manipulating them, most clearly “pebbles” is situated at the other. Indeed pebbles

What can we make of this? Asper argues that the abstract or theotetical “culture” of mathematics was developed by a small group of elite Greek citizens with an impetus that was partly aesthetic, partly a ‘game’ in which succesful players gained prestige (and no doubt associated authority), and which was focussed on communicating through a developing orthodox written form of discourse, general theorems about ideal geometric forms. The work emerged in the sixth to fifth century from practical roots, but the thrust was sharply against the vulgarity of meeting practical needs and indeed in its mode of presentation and areas of work was in part directed to maintaining a sharp distinction between crude application, and the intellectual abstract search for proved knowledge, which lay at its core.⁴²⁰

The

What can we make of this?

It is not necessary here to dwell for long on the expansive achievements of the Ancient Greeks in the development of abstract geometry and mathematical reasoning. All this is described both in great detail, but also accessibly in summary elsewhere.⁴²⁴ It is sufficient to note that this source flowered from around 600 BCE (Thales of Miletus and his use of geometry to solve practical problems) to the destruction of the Academy of Athens in 529 ADE. That period saw the emergence of “schools” of mathematical philosophers including the magisterial Pythagorean School and its emphasis on proof (including, so it is believed, the proof of “Pythagoras’s Theorem”), Plato’s Academy in Athens (which became a centre of mathematics in the 4th century BCE), and associated achievements including an iterative method used to determine the areas and volumes of complex curved and other objects. The achievements were famously brought together by Euclid in his Elements in the 3rd century BCE in which the formalisation of what we now understand as ‘mathematical’ rigour and its use for “proof” was systematically displayed. From thereon, the “Golden Age” of Greek mathematics began to decline although there were nevertheless a series of significant analytic developments, especially in algebra.

What can we make of this? There is a shadowy history of mathematics in Ancient Greece, beginning with the entry into the area in the second millenium BCE by invaders from the north with no known capacities in literacy or numeracy, to the likelihood of trade and other interchanges with Egypt and Mesopotania, and then to the sophisticated Greek literature, already evident by the first Olympic Games in 776 BCE, and then to the beginnings of the formal abstract mathematics for which they have been so celebrated, traced to the illusive figures of Thales of Miletus (around 585 BCE) and Pythagoras of Samos (around 580–500 BCE).⁴²⁷

Even in these couple of lines we seen enough to establish the gulf between the practical problems to be dealt with in day to day life, and the sophisticated highly abstract geometric reasoning of the classical Greek mathematicians. On the one hand, for example, is the highly abstract question of how to develop a pure geometrical or system of harmonic ratios, for understanding musical harmony, or to prove a theorem about the volume of a geometric object in terms of its sides, which was perhaps seen as a good intellectual preparation for the more important art of philosophical reasoning wherein the really big questions could be tackled.⁴²⁸ On the other was the practical question of how to keep track of resources from multiple sources, aided, notably by calculational aids (of which the use of pebbles as counters was a frequent candidate). The former development in abstract mathematics was characteristically Greek. The latter, was common to all mathematically literate and complex societies (including those of ancient China, Mesopotania and Egypt).

As

Whilst there are only glimpses of this pragmatic mathematics we get a taste of it in Aristophanes play The Wasps from 422 BCE:

As Lloyd points out, this distinction between the abstract and the pragmatically applied, can be found in Plato’s remarks on the quite different types of usefulness embodied in on the one hand meeting the needs of everyday life, and on the other of training the intellect. But the search for proof is the most striking feature of the abstract work of the Greek philosopher-mathematicians. And here we may see a political driver. For the world of the Athenian free citizens was governed through the law courts and assemblies, and in these, as Plato stresses, mere rhetorical skill may be sufficient to sway the participants, whatever the actual truth. The task of philosophers was to find a method of demonstrating truth, not through rhetoric but through a higher and absolutely authoritative process of reasoning.

As Lloyd points out, this distinction between the abstract and the pragmatically applied, can be found in Plato’s remarks on the quite different types of usefulness embodied in on the one hand meeting the needs of everyday life, and on the other of training the intellect. But the search for proof is the most striking feature of the abstract work of the Greek philosopher-mathematicians, and here again it can be found in the claim by the philosophers that whilst rhetorical skill may persuade

Even in these couple of lines we seen enough to establish the gulf between the practical problems to be dealt with in day to day life, and the sophisticated highly abstract geometric reasoning of the classical Greek mathematicians. On the one hand, for example, is the highly abstract question of how to develop a pure geometrical or system of harmonic ratios, for understanding musical harmony, or to prove a theorem about the volume of a geometric object in terms of its sides, which was perhaps seen as a good intellectual preparation for the more important art of philosophical reasoning wherein the really big questions could be tackled.⁴³² On the other was the practical question of how to keep track of resources from multiple sources, aided, notably by calculational aids (of which the use of pebbles as counters was a frequent candidate). The former development in abstract mathematics was characteristically Greek. The latter, common to all mathematically literate and complex societies (including those of ancient China, Mesopotania and Egypt).

Thus on the one hand we have the aesthetically pleasing and analytic developments carried through a series of schools

Thus on the one hand we have the

It may be useful to think of the emergence of counting in an unusual way (at least in terms of the literature). First we may recall that in the approach to ‘artificial intelligence’ in which networks are set up using computers to mimic the neurological networks in the brain, these prove remarkably effective in pattern recognition.⁴³⁶ The distinction between counting and pattern recognition can be fine indeed in experiments carried out with animals (for instance chicks who when imprinted with five objects as constituting their ‘mother’ then search for her when two of the objects are removed)⁴³⁷ Similarly, recently it was reported that a neural network which had not be programmed with the concept of number was able to develop a capacity to determine between different patterns which had more dots. Said the leader of the research, Marco Zorzi (University of Padua) “It answers the question of how numerosity emerges without teaching anything about numbers in the first place.”⁴³⁸

Plimpton 322 Tablet (Yale University) - bearing what is
 believed to be a table of numerical relationships between the sides of
 triangles and associated squares443  

It may be useful to think of the emergence of counting in an unusual way (at least in terms of the literature). First we may recall that in the approach to ‘artificial intelligence’ in which networks are set up using computers to mimic the neurological networks in the brain, these prove remarkably effective in pattern recognition.⁴⁴⁵ The distinction between counting and pattern recognition can be fine indeed in experiments carried out with animals (for instance chicks who when imprinted with five objects as constituting their ‘mother’ then search for her when two of the objects are removed)⁴⁴⁶ It is possible therefore that the act of counting is one where an evolutionarily endowed physical capacity for pattern recognition is complemented by the capacity to manipulate objects (whether fingers, marks on bones, or counters) thus constructing abstract formulations of the pattern in relation to observed patterns. From this the invention of words to associate with the abstraction is but one additional step, and symbols as shorthand for those words, another. This raises the interesting question of whether the usual assumption that calculators were invented to assist counting should be reversed, with the possibility that the (very rudimentary) calculator was a necessary step towards developing counting.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: “The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?”⁴⁵⁰ Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be ‘smutted with the Black Art’.⁴⁵¹ However, despite their negative connotations, practioners of “the dark arts” played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of “star signs” as they moved across the celestial sphere.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: “The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?”⁴⁵⁴ Nevertheless, practioners of “the dark arts” played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of “star signs” as they moved across the celestial sphere.

Artisanal mathematics - Egyptian and Mesopotanian foundations

Mathematics has many expressions. But the evidence suggests, not surprisingly, that the beginnings were built by the people who needed to answer practical problems demanded by an increasingly sophisticated society - for example, how to build a regular shape and how many people would be needed, how much food would they need, how much tribute would be required to keep the administration in operation, what would be the crop production and how much could each person give? The people who needed this sort of mathematics were scribes and architects, builders and those who supervised the payment of tributes, and others, who we will refer to here (not quite comfortably) as “artisans” (in the sense of skilled worker). The beginnings of mathematics can be found in ancient Egypt and Mesopotania, derived pragmatically from experience and shaped and written down precisely by and for such artisans (including officials).

Greece - philosopher and artisan - a “two” track mathematics.

Even in these couple of lines we seen enough to establish the gulf between the practical problems to be dealt with in day to day life, and the sophisticated highly abstract geometric reasoning of the classical Greek mathematicians. On the one hand, for example, is the highly abstract question of how to develop a pure geometrical or system of harmonic ratios, for understanding musical harmony, or to prove a theorem about the volume of a geometric object in terms of its sides, which was perhaps seen as a good intellectual preparation for the more important art of philosophical reasoning wherein the really big questions could be tackled.⁴⁵⁸ On the other was the practical question of how to keep track of resources from multiple sources, aided, notably by calculational aids (of which the use of pebbles as counters was a frequent candidate). The former development in abstract mathematics was characteristically Greek. The latter, common to all mathematically literate and complex societies.

Even in these couple of lines we seen enough to establish the gulf between the practical problems to be dealt with in day to day life, and the sophisticated highly abstract geometric reasoning of the classical Greek mathematicians.

There was tension over whether importance of this sort of reasoning
 was for practical applications (for example in following the heavenly
 bodies, or performing engineering works, or whether in its abstract
 purity - for example in developing a pure geometrical or system of
 harmonic ratios, for understanding musical harmony, it should be seen
 as a good intellectual preparation for the more important art of
 philosophical reasoning wherein the really big questions could be
 tackled.460

Whilst there are only glimpses of this pragmatic mathematics we get a taste of it in Aristophanes play Wasps from 422 BCE:

Mathematics has many expressions. But the evidence suggests, not surprisingly, that the beginnings were built by the people who needed to answer practical problems demanded by an increasingly sophisticated society - for example, how to build a regular shape and how many people would be needed, how much food would they need, how much tribute would be required to keep the administration in operation, what would be the crop production and how much could each person give? The people who needed this sort of mathematics were scribes and architects, builders and those who supervised the payment of tributes, and others, who we will refer to here (not quite comfortably) as “artisans” (in the sense of skilled worker). The beginnings of mathematics can be found in ancient Egypt and Mesopotania, shaped and written down precisely by and for such artisans (including officials).

Importantly for the discussion here, which is after all directed at providing part of the context for the development of calculating technology, it should be noted that this abstract, and thereafter much celebrated invention of formally and systematically written abstract mathematical (and other applications) of reasoning, does not represent the only mathematics that was done. Indeed there was at least what Marcus Asper has styled “the two cultures of mathematics” in Ancient Greece.⁴⁶⁹ To be more precise he stresses that recently “a consensus has emerged that Greek mathematics was heterogenous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices.”⁴⁷⁰ This picture of a tapestry of mathematical practices being in play by different participants in the society must be closer to the reality than two cultures (the old joke should be borne in mind that ‘there are two classes of people: those who divide the world into two classes of people, and those who don’t’). Nevertheless, with that caveat it is useful to reflect on the fact that at least two practices were in play: the abstractions being developed by a relatively small philosophically inclined elite (what Asper calls “theoretical mathematics”), and the continuation of the practical mathematics in the style of useful recipes for practical purposes in everyday activities, which was the legacy of Mesopotania and Egypt, to name just two major centres (which has been referred to here as “artisanal mathematics” and which Asper refers to as “practical mathematics”⁴⁷¹).

Greece - philosopher and artisan - a two track mathematics.

Importantly for the discussion here, which is after all directed at providing part of the context for the development of calculating technology, it should be noted that this abstract, and thereafter much celebrated invention of formally and systematically written abstract mathematical (and other applications) of reasoning, does not represent the only mathematics that was done. Indeed there was at least what Marcus Asper has styled “the two cultures of mathematics” in Ancient Greece.⁴⁷⁸ To be more precise he stresses that recently “a consensus has emerged that Greek mathematics was heterogenous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices.”⁴⁷⁹ This picture of a tapestry of mathematical practices being in play by different participants in the society must be closer to the reality than two cultures (the old joke should be borne in mind that ‘there are two classes of people: those who divide the world into two classes of people, and those who don’t’). Nevertheless, with that caveat it is useful to reflect on the fact that at least two practices were in play: the abstractions being developed by a relatively small philosophically inclined elite, and the continuation of the practical mathematics in the style of useful recipes for practical purposes in everyday activities, which was the legacy of Mesopotania and Egypt, to name just two major centres.

Importantly for the discussion here, which is after all directed at providing part of the context for the development of calculating technology, it should be noted that this abstract, and thereafter much celebrated invention of formally and systematically written abstract mathematical (and other applications) of reasoning, does not represent the only mathematics that was done. Indeed there was at least what Marcus Asper has styled “the two cultures of mathematics” in Ancient Greece.⁴⁸³

There was tension over whether importance of this sort of reasoning
 was for practical applications (for example in following the heavenly
 bodies, or performing engineering works, or whether in its abstract
 purity - for example in developing a pure geometrical or system of
 harmonic ratios, for understanding musical harmony, it should be seen
 as a good intellectual preparation for the more important art of
 philosophical reasoning wherein the really big questions could be
 tackled.484 

Importantly for the discussion here, which is after all directed at providing part of the context for the development of calculating technology, it should be noted that this abstract, and thereafter much celebrated invention of formally and systematically written abstract mathematical (and other applications) of reasoning, does not represent the only mathematics that was done. As

In Ancient Greece we see evidence of different sectors of the society
 picking up and working through different forms forming and advocating
 their claims of truth, not only quite broadly, but also in
 mathematics. The much celebrated development was that of written
 abstract reasoning, which was developed and reinforced by a small but
 powerful elite, and given power also in the form of rhetoric which
 became the language of power within the structures of Greek politics.

There was tension over whether importance of this sort of reasoning
 was for practical applications (for example in following the heavenly
 bodies, or performing engineering works, or whether in its abstract
 purity - for example in developing a pure geometrical or system of
 harmonic ratios, for understanding musical harmony, it should be seen
 as a good intellectual preparation for the more important art of
 philosophical reasoning wherein the really big questions could be
 tackled.488  It is not unreasonable to suggest that
 the role of artifacts which simplify (and vulgarise) the need for
 certain forms of mathematical reasoning, has some reflections of this
 same debate.

As shown in the table below, other early representations similarly drew directly on patterns representing counters (or fingers - even the Roman system can be seen as counting to five on the one hand, reserving the thumb and forefinger for the V to represent five, and the X representing a V on each hand). The Egyptian Hieratic and then the Greek system replaced combination numerals with single characters, and finally, in the ninth century, the familiar symbols of the modern (arabic-Indian) place-based system (complete with the numeral 0 to replace earlier spaces for “place holders” finally emerged.

A multi-stranded development - mathematics in China

Building on the past - Islamic mathematics

Islam

Stages of development Ancient. Islamic. The Renaissance Renaissance C14-C17 - so Gallileo etc - birth of the new astronomy

C17 Schickard, Pascale, Napier, Gunter, …..

Description of the period - from Gallileo to the French Revolution

Greece - philosopher and artisan - a two track mathematics.

It is not necessary here to dwell for long on the extraordinary achievements of the Ancient Greeks in the development of abstract geometry and mathematical reasoning. All this is described both in great detail, but also accessibly in summary elsewhere.⁵⁰⁰ It is sufficient to note that this source flowered from around 600 BCE (Thales of Miletus and his use of geometry to solve practical problems) to the destruction of the Academy of Athens in 529 ADE. That period saw the emergence of “schools” of mathematical philosophers including the magisterial Pythagorean School and its emphasis on proof (including, so it is believed, the proof of “Pythagoras’s Theorem”), Plato’s Academy in Athens (which became a centre of mathematics in the 4th century BCE), and associated achievements including an iterative method used to determine the areas and volumes of complex curved and other objects. The achievements were famously brought together by Euclid in his Elements in the 3rd century BCE in which the formalisation of what we now understand as ‘mathematical’ rigour and its use for “proof” was systematically displayed. From thereon, the “Golden Age” of Greek mathematics began to decline although there were nevertheless a series of significant analytic developments, especially in algebra.

Importantly for the discussion here, which is after all directed at providing part of the context for the development of calculating technology, it should be noted that this abstract, and thereafter much celebrated invention of formally and systematically written abstract mathematical (and other applications) of reasoning, does not represent the only mathematics that was done. There was as well

In Ancient Greece we see evidence of different sectors of the society picking up and working through different forms forming and advocating their claims of truth, not only quite broadly, but also in mathematics. The much celebrated development was that of written abstract reasoning, which was developed and reinforced by a small but powerful elite, and given power also in the form of rhetoric which became the language of power within the structures of Greek politics.

Other stands

China

Artisans and Philosophers

There was tension over whether importance of this sort of reasoning was for practical applications (for example in following the heavenly bodies, or performing engineering works, or whether in its abstract purity - for example in developing a pure geometrical or system of harmonic ratios, for understanding musical harmony, it should be seen as a good intellectual preparation for the more important art of philosophical reasoning wherein the really big questions could be tackled.⁵⁰³ It is not unreasonable to suggest that the role of artifacts which simplify (and vulgarise) the need for certain forms of mathematical reasoning, has some reflections of this same debate.

The approach of developing mathematics as deductions from an axiomatic structure found in early form in Euklid’s Elements in ~300 BCE, and perhaps also in the work of earlier Greek figures such as Hippocrates of Chios, reflected trends in Ancient Greek society, philosophy (for example Aristotle 384–322 BCE), and (perhaps) theology (at least in its astrological aspects). Euklid brought together in a systematic form most of the mathematics known to the Greeks at that time, which already encompassed an impressive array of approaches, especially to geometry which had the required sense of demonstrable truth, elegance, and power. For example, the Greeks had not only determined how to calculate the volumes of quite complicated objects but had also determined a way to calculate the area of a circle by a method successive approximations using ever more many sided inscribed polygons.⁵⁰⁴

In contrast, Chinese mathematics developed with an emphasis on finding useful solutions

In Egypt the focus of the mathematics, even though at times producing quite significant results, was on addition.

¹ E. K. Hunt and Howard J. Sherman, Economics: Introduction to Traditional and Radical Views, Second Edition, Harper & Row, New York, 1972. (↑)

² E. G. R. Taylor, The Mathematical Practioners of Tudor & Stuart England 1485–1714, Cambridge University Press, Cambridge, UK, 1954. (↑)

³ ibid pp. 206–7. (↑)

⁴ ibid pp. 206–11. (↑)

⁵ ibid p. 211–3 (↑)

⁶ Boyer and Merzbach /A History of Mathematics//, pp. 195–7. (↑)

⁷ ibid p. 196 (↑)

⁸ Boyer and Merzbach /A History of Mathematics//, pp. 195–7. (↑)

⁹ ibid p. 196 (↑)

¹⁰ ibid p. 198–203 (↑)

¹¹ ibid p. 204 (↑)

¹² ibid pp. 206–7. (↑)

¹³ Boyer and Merzbach /A History of Mathematics//, pp. 195–7. (↑)

¹⁴ ibid p. 196 (↑)

¹⁵ ibid. pp. 128–132 (↑)

¹⁶ Alison Abbott, “Islamic Science: Rebuilding the Past”, Nature, Vol 432, 16 Dec 2004, pp. 795–5. (↑)

¹⁷ ibid. pp. 128–132 (↑)

¹⁸ Alison Abbott, “Islamic Science: Rebuilding the Past”, Nature, Vol 432, 16 Dec 2004, pp. 795–5. (↑)

¹⁹ E. K. Hunt and Howard J. Sherman, Economics: Introduction to Traditional and Radical Views, Second Edition, Harper & Row, New York, 1972. (↑)

²⁰ E. G. R. Taylor, The Mathematical Practioners of Tudor & Stuart England 1485–1714, Cambridge University Press, Cambridge, UK, 1954. (↑)

²¹ For a nice description of the progressive development of the astrolabe clock see George Burnett-Stuart, “Astronomical Clocks of the Middle Ages: A Guided Tour”, http://www.almagest.co.uk/middle/astclk.htm (viewed 2 Feb 2012) (↑)

²² http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

²³ For a nice description of the progressive development of the astrolabe clock see George Burnett-Stuart, “Astronomical Clocks of the Middle Ages: A Guided Tour”, http://www.almagest.co.uk/middle/astclk.htm (viewed 2 Feb 2012) (↑)

²⁴ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

²⁵ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

²⁶ For a nice description of the progressive development of the astrolabe clock see George Burnett-Stuart, “Astronomical Clocks of the Middle Ages: A Guided Tour”, http://www.almagest.co.uk/middle/astclk.htm (viewed 2 Feb 2012) (↑)

²⁷ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

²⁸ See Robsan and Stedall, The Oxford Handbook, Chapter 1.2, pp. 27–55. (↑)

²⁹ See Robsan and Stedall, The Oxford Handbook, Chapter 1.2, pp. 27–55. (↑)

³⁰ See Robsan and Stedall, The Oxford Handbook, Chapter 1.2, pp. 27–55. (↑)

³¹ See Robsan and Stedall, The Oxford Handbook, Chapter 1.2, pp. 27–55. (↑)

³² See Robsan and Stedall, The Oxford Handbook, Chapter 1.2, pp. 27–55. (↑)

³³ ibid. pp. 128–132 (↑)

³⁴ Alison Abbott, “Rebuilding the Past”, Nature, Vol 432, 16 Dec 2004, pp. 795–5. (↑)

³⁵ ibid. pp. 128–132 (↑)

³⁶ Alison Abbott, “Islamic Science: Rebuilding the Past”, Nature, Vol 432, 16 Dec 2004, pp. 795–5. (↑)

³⁷ ibid. pp. 128–132 (↑)

³⁸ ibid. pp. 128–132 (↑)

³⁹ Alison Abbott, “Rebuilding the Past”, Nature, Vol 432, 16 Dec 2004, pp. 795–5. (↑)

⁴⁰ ibid. pp. 128–132 (↑)

⁴¹ ibid. pp. 128–132 (↑)

⁴² ibid. pp. 128–132 (↑)

⁴³ ibid. pp. 128–132 (↑)

⁴⁴ ibid, p. 126. (↑)

⁴⁵ ibid, p. 126. (↑)

⁴⁶ ibid. pp. 128–132 (↑)

⁴⁷ Charette, “The Locales”, p. 125. (↑)

⁴⁸ ibid, p. 126. (↑)

⁴⁹ Herbert A. Applebaum, The Concept of Work: Ancient, Medieval and Modern, State University of New York Press, Albany, 1992, p. 31 (↑)

⁵⁰ Herbert A. Applebaum, The Concept of Work: Ancient, Medieval and Modern, State University of New York Press, Albany, 1992, p. 31 (↑)

⁵¹ Herbert A. Applebaum, The Concept of Work: Ancient, Medieval and Modern, State University of New York Press, Albany, 1992, p. 31 (↑)

⁵² Ibn al-Nadim (ed. by R. Tajaddud), Ktab al-Fihrist, Tehran, 1971, p. 332 (cited in Charette, Locales, p.124). (↑)

⁵³ Ibn al-Nadim (ed. by R. Tajaddud), Ktab al-Fihrist, Tehran, 1971, p. 332 (cited in Charette, Locales, p.124). (↑)

⁵⁴ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁵⁵ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁵⁶ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁵⁷ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁵⁸ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

⁵⁹ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁶⁰ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

⁶¹ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁶² Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

⁶³ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁶⁴ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁶⁵ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

⁶⁶ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁶⁷ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁶⁸ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁶⁹ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁷⁰ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁷¹ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁷² Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

⁷³ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁷⁴ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁷⁵ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

⁷⁶ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁷⁷ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁷⁸ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

⁷⁹ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁸⁰ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁸¹ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

⁸² Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁸³ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁸⁴ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

⁸⁵ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁸⁶ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁸⁷ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

⁸⁸ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

⁸⁹ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

⁹⁰ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012) (↑)

⁹¹ A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

⁹² http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁹³ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012) (↑)

⁹⁴ A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

⁹⁵ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁹⁶ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁹⁷ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

⁹⁸ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

⁹⁹ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

¹⁰⁰ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

¹⁰¹ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

¹⁰² http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

¹⁰³ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

¹⁰⁴ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

¹⁰⁵ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

¹⁰⁶ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

¹⁰⁷ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

¹⁰⁸ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

¹⁰⁹ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012) (↑)

¹¹⁰ A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

¹¹¹ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012) (↑)

¹¹² A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

¹¹³ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

¹¹⁴ Drachmann, “The Plane Astrolobe”, Fig. 1, p. 184. (↑)

¹¹⁵ http://itunes.apple.com/au/app/astrolabe-clock/id421777015?mt=8 (viewed 1 Feb 2012) (↑)

¹¹⁶ Ibn al-Nadim (ed. by R. Tajaddud), Ktab al-Fihrist, Tehran, 1971, p. 332 (cited in Charette, Locales, p.124). (↑)

¹¹⁷ Ibn al-Nadim (ed. by R. Tajaddud), Ktab al-Fihrist, Tehran, 1971, p. 332 (cited in Charette, Locales, p.124). (↑)

¹¹⁸ Ibn al-Nadim (ed. by R. Tajaddud), Ktab al-Fihrist, Tehran, 1971, p. 332 (cited in Charette, Locales, ibid, p.124). (↑)

¹¹⁹ Ibn al-Nadim (ed. by R. Tajaddud), Ktab al-Fihrist, Tehran, 1971, p. 332 (cited in Charette, Locales, p.124). (↑)

¹²⁰ Ibn al-Nadim (ed. by R. Tajaddud), Ktab al-Fihrist, Tehran, 1971, p. 332 (cited in Charette, Locales, ibid, p.124). (↑)

¹²¹ François Charette, “The Locales of Islamic Astronomical Instrumentation”, History of Science, xliv, 2006, p. 124 (↑)

¹²² Boyer and Merzbach A History of Mathematics, pp. 225–45. (↑)

¹²³ François Charette, “The Locales of Islamic Astronomical Instrumentation”, History of Science, xliv, 2006, p. 124 (↑)

¹²⁴ ibid, p. 124 (↑)

¹²⁵ Boyer and Merzbach A History of Mathematics, pp. 225–45. (↑)

¹²⁶ James E Evans, “The Astrologer’s Apparatus: A Picture of Professional Practice in Greco-Roman Egypt”, Journal for the History of Astronomy, Vol 35, Part 1, No 118, February 2004, pp. 1–44. (↑)

¹²⁷ ibid (↑)

¹²⁸ James E Evans, “The Astrologer’s Apparatus: A Picture of Professional Practice in Greco-Roman Egypt”, Journal for the History of Astronomy, Vol 35, Part 1, No 118, February 2004, pp. 1–44. (↑)

¹²⁹ ibid (↑)

¹³⁰ ibid, Fig 1., p. 6. (↑)

¹³¹ ibid, Fig 1., p. 6. (↑)

¹³² Denise Schmandt-Besserat, “The Envelopes That Bear the First Writing”, Technology and Culture, Vol. 21, No. 3, July 1980, Figure 1, p. 358. (↑)

¹³³ Denise Schmandt-Besserat, “The Envelopes That Bear the First Writing”, Technology and Culture, Vol. 21, No. 3, July 1980, Figure 1, p. 358. (↑)

¹³⁴ Measuring the land using rope, from the Tomb Chapel of Menna, New Kingdom, Valley of the Nobles, Thebes, Egypt, contained in the Bridgeman Art Library, reproduced at various sites on the web including http://www.civilization.ca/civil/egypt/images/fback3b.jpg (viewed 26 Dec 2011) and with fuller description at http://www.gettyimages.com.au/detail/illustration/measuring-the-land-using-rope-from-the-tomb-chapel-of-stock-graphic/55995332 (↑)

¹³⁵ Measuring the land using rope, from the Tomb Chapel of Menna, New Kingdom, Valley of the Nobles, Thebes, Egypt, contained in the Bridgeman Art Library, reproduced at various sites on the web including http://www.civilization.ca/civil/egypt/images/fback3b.jpg (viewed 26 Dec 2011) and with fuller description at http://www.gettyimages.com.au/detail/illustration/measuring-the-land-using-rope-from-the-tomb-chapel-of-stock-graphic/55995332 (↑)

¹³⁶ Cultural China (viewed 28 Dec 2011l (↑)

¹³⁷ Cultural China (viewed 28 Dec 2011l (↑)

¹³⁸ From Mayake Kenryu’s work of 1795, reproduced in Smith and Mikami, A History of Japanese Mathematics, p. 29. (↑)

¹³⁹ From Mayake Kenryu’s work of 1795, reproduced in Smith and Mikami, A History of Japanese Mathematics, p. 29. (↑)

¹⁴⁰ source http://3.bp.blogspot.com/_Om5WdRNbuEE/RhR7fCaoGfI/AAAAAAAAABs/4K8uUPqpLPc/s320/rhind.jpg (viewed 30 Dec 2011) (↑)

¹⁴¹ source http://3.bp.blogspot.com/_Om5WdRNbuEE/RhR7fCaoGfI/AAAAAAAAABs/4K8uUPqpLPc/s320/rhind.jpg (viewed 30 Dec 2011) (↑)

¹⁴² From Boyer and Merzbach /A History of Mathematics//, p. 34. (↑)

¹⁴³ From Boyer and Merzbach /A History of Mathematics//, p. 34. (↑)

¹⁴⁴ Menninger, A Cultural History of Numbers, Fig 183, p. 306. (↑)

¹⁴⁵ Menninger, A Cultural History of Numbers, Fig 183, p. 306. (↑)

¹⁴⁶ Jonas Bogoshi, Kevin Naidoo, John Webb, The Oldest Mathematical Artefact, The Mathematical Gazette, Vol. 71, No. 458 (Dec., 1987), p. 294. (↑)

¹⁴⁷ Jonas Bogoshi, Kevin Naidoo, John Webb, The Oldest Mathematical Artefact, The Mathematical Gazette, Vol. 71, No. 458 (Dec., 1987), p. 294. (↑)

¹⁴⁸ From the Museum for World Culture, Göteborg, Sweden. Image retained in the Harvard University Khipu Database http://khipukamayuq.fas.harvard.edu/images/KhipuGallery/MiscAlbum/images/UR113%20Valhalla_jpg.jpg (viewed 27 Dec 2011) (↑)

¹⁴⁹ From the Museum for World Culture, Göteborg, Sweden. Image retained in the Harvard University Khipu Database http://khipukamayuq.fas.harvard.edu/images/KhipuGallery/MiscAlbum/images/UR113%20Valhalla_jpg.jpg (viewed 27 Dec 2011) (↑)

¹⁵⁰ Denise Schmandt-Besserat, “The Origins of Writing : An Archaeologist’s Perspective”, Written Communication Vol 3, No1, January, 1986, Figure 3, p. 41. (↑)

¹⁵¹ Denise Schmandt-Besserat, “The Origins of Writing : An Archaeologist’s Perspective”, Written Communication Vol 3, No1, January, 1986, Figure 3, p. 41. (↑)

¹⁵² Denise Schmandt-Besserat, “The Envelopes That Bear the First Writing”, Technology and Culture, Vol. 21, No. 3, July 1980, Figure 1, p. 358. (↑)

¹⁵³ Denise Schmandt-Besserat, “The Envelopes That Bear the First Writing”, Technology and Culture, Vol. 21, No. 3, July 1980, Figure 1, p. 358. (↑)

¹⁵⁴ Eleanor Robson, “Mathematical Education in an Old Babylonian Scribal School”, in Robson and Stedall, The Oxford handbook, Figure 3.1.4, p. 210. (↑)

¹⁵⁵ Eleanor Robson, “Mathematical Education in an Old Babylonian Scribal School”, in Robson and Stedall, The Oxford handbook, Figure 3.1.4, p. 210. (↑)

¹⁵⁶ Measuring the land using rope, from the Tomb Chapel of Menna, New Kingdom, Valley of the Nobles, Thebes, Egypt, contained in the Bridgeman Art Library, reproduced at various sites on the web including http://www.civilization.ca/civil/egypt/images/fback3b.jpg (viewed 26 Dec 2011) and with fuller description at http://www.gettyimages.com.au/detail/illustration/measuring-the-land-using-rope-from-the-tomb-chapel-of-stock-graphic/55995332 (↑)

¹⁵⁷ Measuring the land using rope, from the Tomb Chapel of Menna, New Kingdom, Valley of the Nobles, Thebes, Egypt, contained in the Bridgeman Art Library, reproduced at various sites on the web including http://www.civilization.ca/civil/egypt/images/fback3b.jpg (viewed 26 Dec 2011) and with fuller description at http://www.gettyimages.com.au/detail/illustration/measuring-the-land-using-rope-from-the-tomb-chapel-of-stock-graphic/55995332 (↑)

¹⁵⁸ Cultural China (viewed 28 Dec 2011l (↑)

¹⁵⁹ Cultural China (viewed 28 Dec 2011l (↑)

¹⁶⁰ From Mayake Kenryu’s work of 1795, reproduced in Smith and Mikami, A History of Japanese Mathematics, p. 29. (↑)

¹⁶¹ From Mayake Kenryu’s work of 1795, reproduced in Smith and Mikami, A History of Japanese Mathematics, p. 29. (↑)

¹⁶² The oldest surviving counting board, made of marble. Photo from the National Museum of Epigraphy, Athens, reproduced from Menninger, A Cultural History of Numbers, Fig. 128, p. 300. (↑)

¹⁶³ Image is from Museo Nazionale Ramano at Piazzi delle Terme, Rome. Made of brass plate it is approximately 115 x 99 mm with nine long slots (equipped with four sliders for 1–4) and eight short slots each with one slider (for 5). The eight left most columns run in powers of ten from units on the right to millions on the left. The ninth slot is for a fraction of ounces. [reproduced from http://www.ee.ryerson.ca/~elf/abacus/roman-hand-abacus.html. Further details in Cesare Rossi, Flavio Russo and Ferrucio Russo, ”Ancient Engineers and Inventions, History of Mechanism and Machine Science, Springer, Vol. 8, 2009, p. 42 (↑)

¹⁶⁴ The oldest surviving counting board, made of marble. Photo from the National Museum of Epigraphy, Athens, reproduced from Menninger, A Cultural History of Numbers, Fig. 128, p. 300. (↑)

¹⁶⁵ Image is from Museo Nazionale Ramano at Piazzi delle Terme, Rome. Made of brass plate it is approximately 115 x 99 mm with nine long slots (equipped with four sliders for 1–4) and eight short slots each with one slider (for 5). The eight left most columns run in powers of ten from units on the right to millions on the left. The ninth slot is for a fraction of ounces. [reproduced from http://www.ee.ryerson.ca/~elf/abacus/roman-hand-abacus.html. Further details in Cesare Rossi, Flavio Russo and Ferrucio Russo, ”Ancient Engineers and Inventions, History of Mechanism and Machine Science, Springer, Vol. 8, 2009, p. 42 (↑)

¹⁶⁶ Menninger, A Cultural History of Numbers, Fig 183, p. 306. (↑)

¹⁶⁷ Menninger, A Cultural History of Numbers, Fig 183, p. 306. (↑)

¹⁶⁸ ibid, Fig 1., p. 6. (↑)

¹⁶⁹ ibid, Fig 1., p. 6. (↑)

¹⁷⁰ ibid, Fig 1., p. 6. (↑)

¹⁷¹ ibid, Fig 1., p. 6. (↑)

¹⁷² ibid, Fig 1., p. 6. (↑)

¹⁷³ James E Evans, “The Astrologer’s Apparatus: A Picture of Professional Practice in Greco-Roman Egypt”, Journal for the History of Astronomy, Vol 35, Part 1, No 118, February 2004, pp. 1–44. (↑)

¹⁷⁴ James E Evans, “The Astrologer’s Apparatus: A Picture of Professional Practice in Greco-Roman Egypt”, Journal for the History of Astronomy, Vol 35, Part 1, No 118, February 2004, pp. 1–44. (↑)

¹⁷⁵ ibid (↑)

¹⁷⁶ James E Evans, “The Astrologer’s Apparatus: A Picture of Professional Practice in Greco-Roman Egypt”, Journal for the History of Astronomy, Vol 35, Part 1, No 118, February 2004, pp. 1–44. (↑)

¹⁷⁷ For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

¹⁷⁸ For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

¹⁷⁹ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

¹⁸⁰ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

¹⁸¹ Claudius Ptolemaeus, Almagestum: Opus ingens ac nobile omnes Celorum motus continens. Felicibus Astris eat in lucem, 1515. Copy in the Institut für Astronomie, Universität Wien, Türkenschanzstraße 17, 1180 Wien reproduced at http://www.univie.ac.at/hwastro/books/1515_ptole_BWLow.pdf (viewed 31 Jan 2012) (↑)

¹⁸² Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

¹⁸³ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

¹⁸⁴ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

¹⁸⁵ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

¹⁸⁶ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

¹⁸⁷ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

¹⁸⁸ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

¹⁸⁹ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

¹⁹⁰ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

¹⁹¹ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

¹⁹² For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

¹⁹³ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

¹⁹⁴ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

¹⁹⁵ For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

¹⁹⁶ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

¹⁹⁷ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

¹⁹⁸ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012) (↑)

¹⁹⁹ A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²⁰⁰ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012) (↑)

²⁰¹ A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²⁰² Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012)] This device consisted of a large vertical disc rotated by a water or other mechanism so that it turned through a complete revolution from sunrise to sunrise on the following day. On it are drawn the projected paths of stars of the Northern Hemisphere together with the curve taken by the Sun in its journey across the sky over the year (“the zodiac”). Along the Sun’s path are 365 holes, one for each day, in which for a particular day a marker can be inserted so the clock can be adjusted to take account of the changing lengths of the day with season. This sky map rotates behind a disk of wires with concentric circles for selected months and radial curved wires representing the 24 hours of the day. Made for a particular location, an arc is scribed across the star map representing the horizon for the place for which it was constructed (below which no stars can be seen). With this device it is possible to represent the positions of selected groups of stars in the sky at the particular hour.[[^A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²⁰³ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012) (↑)

²⁰⁴ A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²⁰⁵ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²⁰⁶ Illustration from 1886 Hoffman sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

²⁰⁷ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²⁰⁸ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²⁰⁹ Illustration from 1886 Hoffmann sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

²¹⁰ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²¹¹ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²¹² Illustration from 1886 Hoffman sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152.|| Early Astrolabe (~1400 CE)[^ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²¹³ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²¹⁴ Illustration from 1886 Hoffman sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152. (↑)

²¹⁵ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²¹⁶ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²¹⁷ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²¹⁸ Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²¹⁹ Illustration from 1886 Hoffman sale catalogue showing Grand (Vosges) fragment of anasphoric clock now in the museum in Saint-Germain-en-Laye. Shown in John North, God’s Clockmaker: Richard of Wallingford and the Invention of Time, Continuum, Londong, 2006 (first published 2005), p. 152.|| Early Astrolabe (~1400 CE)[^ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²²⁰ extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²²¹ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²²² Portion of a replica constructed by Prof . Kostas Kotsanas and his students; extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²²³ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²²⁴ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012)] This device consisted of a large vertical disc rotated by a water or other mechanism so that it turned through a complete revolution from sunrise to sunrise on the following day. On it are projected the paths of stars of the Northern Hemisphere (a “planispheric projection”). Also across it lies the projected curve taken by the son as it moves across the sky over the year (“the zodiac”) and into this are holes in which a symbol for the sun can be inserted at sunrise, thereby adjusting it for the shortening day. In front of it a disk of wires represents the meridians, tropics and equator as well as circles for some key months.The clock must be made for a particular location, and across the circles is constructed an arc representing the horizon for the place for which it was constructed. The circles are also divided into twenty-four hours for each day. With this device it was possible to represent the positions of selected stars in the sky at the particular hour.[[^A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²²⁵ Boyer and Merzbach, History of Mathematics, p. 192 (↑)

²²⁶ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012)] This device consisted of a large vertical disc rotated by a water or other mechanism so that it turned through a complete revolution from sunrise to sunrise on the following day. On it are drawn the projected paths of stars of the Northern Hemisphere together with the curve taken by the Sun in its journey across the sky over the year (“the zodiac”). Along the Sun’s path are 365 holes, one for each day, in which for a particular day a marker can be inserted so the clock can be adjusted to take account of the changing lengths of the day with season. This sky map rotates behind a disk of wires with concentric circles for selected months and radial curved wires representing the 24 hours of the day. Made for a particular location, an arc is scribed across the star map representing the horizon for the place for which it was constructed (below which no stars can be seen). With this device it is possible to represent the positions of selected groups of stars in the sky at the particular hour.[[^A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²²⁷ Boyer and Merzbach, History of Mathematics, p. 192 (↑)

²²⁸ extracted from a photo by Augusta Stylianou, http://www.mlahanas.de/Greeks/Technology/AncientGreekTechnology001.html (viewed 29 Jan 2012) (↑)

²²⁹ Front of Fusoris astrolabe (Photo from the Adler Planetarium and Astronomy Museum, reproduced at http://astrolabes.org/pages/fusoris.htm (viewed 29 Jan 2012) (↑)

²³⁰ A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²³¹ Vitruvius, De architectura, Book VIII, “Sundials and Water Clocks”, sections 8–15^ http://www.mlahanas.de/Greeks/Texts/Vitruvius/Book9.html (viewed 29 Jan 2012)] This device consisted of a large vertical disc rotated by a water or other mechanism so that it turned through a complete revolution from sunrise to sunrise on the following day. On it are projected the paths of stars of the Northern Hemisphere (a “planispheric projection”). Also across it lies the projected curve taken by the son as it moves across the sky over the year (“the zodiac”) and into this are holes in which a symbol for the sun can be inserted at sunrise, thereby adjusting it for the shortening day. In front of it a disk of wires represents the meridians, tropics and equator as well as circles for some key months.The clock must be made for a particular location, and across the circles is constructed an arc representing the horizon for the place for which it was constructed. The circles are also divided into twenty-four hours for each day. With this device it was possible to represent the positions of selected stars in the sky at the particular hour.[[^A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²³² A. G. Drachmann, “The Plane Astrolabe and the Anaphoric Clock”, Centaurus, vol 3, issue 1, 1970, pp. 184–9 (↑)

²³³ Boyer and Merzbach, History of Mathematics, p. 192 (↑)

²³⁴ François Charette, “The Locales of Islamic Astronomical Instrumentation”, History of Science, xliv, 2006, p. 124 (↑)

²³⁵ Boyer and Merzbach A History of Mathematics, pp. 225–45. (↑)

²³⁶ François Charette, “The Locales of Islamic Astronomical Instrumentation”, History of Science, xliv, 2006, p. 124 (↑)

²³⁷ Boyer and Merzbach A History of Mathematics, pp. 225–45. (↑)

²³⁸ ibid (↑)

²³⁹ François Charette, “The Locales of Islamic Astronomical Instrumentation”, History of Science, xliv, 2006, p. 124 (↑)

²⁴⁰ Boyer and Merzbach A History of Mathematics, pp. 225–45. (↑)

²⁴¹ annon, Ptolemy from Clavdio Tolomeo Principe De Gli Astrologi, et De Geografi, Giordano Ziletti, Venezia, 1564 http://www.er.uqam.ca/nobel/r14310/Ptolemy/Ziletti.html (viewed 28 Jan 2012) (↑)

²⁴² museo galileo Institute and Museum of the History of Science, VII.36 Model of the solar orb http://brunelleschi.imss.fi.it/museum/esim.asp?c=407036 (viewed 28 Jan 2012) (↑)

²⁴³ Ptolemy, Almagest, Book 1 section 6 or . For a translation see http://www.brycecorkins.com/wp-content/uploads/documents/PtolemysAlmagest.pdf (viewed 28 Jan 2012) (↑)

²⁴⁴ William Cecil Dampier, A history of science and its relations with philosophy & religion, University Press, Cambridge, 1929, p. 49. (↑)

²⁴⁵ Ptolemy, Almagest, Book 1 section 6 or . For a translation see http://www.brycecorkins.com/wp-content/uploads/documents/PtolemysAlmagest.pdf (viewed 28 Jan 2012) (↑)

²⁴⁶ William Cecil Dampier, A history of science and its relations with philosophy & religion, University Press, Cambridge, 1929, p. 49. (↑)

²⁴⁷ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

²⁴⁸ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

²⁴⁹ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

²⁵⁰ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

²⁵¹ Ptolemy, Almagest, Book 1 section 6 or . For a translation see http://www.brycecorkins.com/wp-content/uploads/documents/PtolemysAlmagest.pdf (viewed 28 Jan 2012) (↑)

²⁵² William Cecil Dampier, A history of science and its relations with philosophy & religion, University Press, Cambridge, 1929, p. 49. (↑)

²⁵³ Nicolaus Copernicus, De revolutionibus orbium coelestium, Holy Roman Empire of the German Nation, Nuremberg, 1543 (↑)

²⁵⁴ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

²⁵⁵ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

²⁵⁶ Richard Fitzpatrick, A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System, University of Texas at Austin, http://farside.ph.utexas.edu/syntaxis/Almagest.pdf, (viewed 27 Jan 2012), p. 6. (↑)

²⁵⁷ For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

²⁵⁸ For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

²⁵⁹ http://www.islamicspain.tv/Arts-and-Science/The-Culture-of-Al-Andalus/Mathematics.htm (↑)

²⁶⁰ http://www.islamicspain.tv/Arts-and-Science/The-Culture-of-Al-Andalus/Mathematics.htm (↑)

²⁶¹ ibid (↑)

²⁶² ibid (↑)

²⁶³ ibid p. 240 (↑)

²⁶⁴ ibid pp. 244–5 (↑)

²⁶⁵ ibid p. 240 (↑)

²⁶⁶ ibid pp. 244–5 (↑)

²⁶⁷ http://www.islamicspain.tv/Arts-and-Science/The-Culture-of-Al-Andalus/Mathematics.htm (↑)

²⁶⁸ ibid p. 240 (↑)

²⁶⁹ ibid p. 240 (↑)

²⁷⁰ ibid p. 240 (↑)

²⁷¹ ibid p. 240 (↑)

²⁷² ibid pp. 244–5 (↑)

²⁷³ ibid p. 237 (↑)

²⁷⁴ ibid p. 237 (↑)

²⁷⁵ ibid p. 240 (↑)

²⁷⁶ ibid (↑)

²⁷⁷ ibid (↑)

²⁷⁸ ibid p. 237 (↑)

²⁷⁹ ibid (↑)

²⁸⁰ see for example http://en.wikipedia.org/wiki/History_of_mathematics (viewed 14 Jan 2012) and references contained therein. (↑)

²⁸¹ Carl B. Boyer and Uta C. Merzbach, History of Mathematics, Wiley, 2010, pp. 43–52. (↑)

²⁸² Marcus Asper, “The two cultures of mathematics in ancient Greece”, “, in Robson and Stedall, The Oxford handbook, p.107. (↑)

²⁸³ ibid (↑)

²⁸⁴ ibid pp.108–114 (↑)

²⁸⁵ see for example http://en.wikipedia.org/wiki/History_of_mathematics (viewed 14 Jan 2012) and references contained therein. (↑)

²⁸⁶ Carl B. Boyer and Uta C. Merzbach, History of Mathematics, Wiley, 2010, pp. 43–52. (↑)

²⁸⁷ Boyer and Merzbach, History of Mathematics, pp. 176–191 (↑)

²⁸⁸ ibid, p. 192 (↑)

²⁸⁹ Marcus Asper, “The two cultures of mathematics in ancient Greece”, “, in Robson and Stedall, The Oxford handbook, p.107. (↑)

²⁹⁰ ibid (↑)

²⁹¹ ibid pp.108–114 (↑)

²⁹² Boyer and Merzbach, History of Mathematics, p. 177 (↑)

²⁹³ Boyer and Merzbach, History of Mathematics, pp. 191–192 (↑)

²⁹⁴ Boyer and Merzbach, History of Mathematics, p. 177 (↑)

²⁹⁵ This helpfully descriptive phrase and, except where noted otherwise, all of the detail in this section is drawn from the excellent treatment in Boyer and Merzbach /A History of Mathematics//, Chapter 13, pp. 225–45. (↑)

²⁹⁶ This helpfully descriptive phrase and, except where noted otherwise, all of the detail in this section is drawn from the excellent treatment in Boyer and Merzbach A History of Mathematics, Chapter 13, pp. 225–45. (↑)

²⁹⁷ For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

²⁹⁸ For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

²⁹⁹ For a subtle discussion of the role and design of Stonehenge see for example Lionel Sims, “The ‘Solarization’ of the Moon: Manipulated Knowledge at Stonehenge”, Cambridge Archaeological Journal, vol 16, 2006, pp. 191–207 (↑)

³⁰⁰ This phrase and accepted where noted otherwise all of the detail in this section is drawn from the excellent treatment in Boyer and Merzbach /A History of Mathematics//, Chapter 13, pp. 225–45. (↑)

³⁰¹ This helpfully descriptive phrase and, except where noted otherwise, all of the detail in this section is drawn from the excellent treatment in Boyer and Merzbach /A History of Mathematics//, Chapter 13, pp. 225–45. (↑)

³⁰² The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁰³ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁰⁴ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁰⁵ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁰⁶ This phrase and accepted where noted otherwise all of the detail in this section is drawn from the excellent treatment in Boyer and Merzbach /A History of Mathematics//, Chapter 13, pp. 225–45. (↑)

³⁰⁷ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁰⁸ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁰⁹ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³¹⁰ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³¹¹ See for example, http://www.astrocal.co.uk/metonic-cycle.htm (viewed 25 Jan 2012) (↑)

³¹² See, for example, “Archimede’s Planetarium”, Museo Galilleo, http://brunelleschi.imss.fi.it/vitrum/evtr.asp?c=8253 (viewed 25 Jan 2012). (↑)

³¹³ See for example, http://www.astrocal.co.uk/metonic-cycle.htm (viewed 25 Jan 2012) (↑)

³¹⁴ See, for example, “Archimede’s Planetarium”, Museo Galilleo, http://brunelleschi.imss.fi.it/vitrum/evtr.asp?c=8253 (viewed 25 Jan 2012). (↑)

³¹⁵ See for example, http://www.astrocal.co.uk/metonic-cycle.htm (viewed 25 Jan 2012) (↑)

³¹⁶ See, for example, “Archimede’s Planetarium”, Museo Galilleo, http://brunelleschi.imss.fi.it/vitrum/evtr.asp?c=8253 (viewed 25 Jan 2012). (↑)

³¹⁷ See for example, http://www.astrocal.co.uk/metonic-cycle.htm (viewed 25 Jan 2012) (↑)

³¹⁸ See, for example, “Archimede’s Planetarium”, Museo Galilleo, http://brunelleschi.imss.fi.it/vitrum/evtr.asp?c=8253 (viewed 25 Jan 2012). (↑)

³¹⁹ See for example, http://www.astrocal.co.uk/metonic-cycle.htm (viewed 25 Jan 2012) (↑)

³²⁰ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³²¹ See for example, http://www.astrocal.co.uk/metonic-cycle.htm (viewed 25 Jan 2012) (↑)

³²² See, for example, “Archimede’s Planetarium”, Museo Galilleo, http://brunelleschi.imss.fi.it/vitrum/evtr.asp?c=8253 (viewed 25 Jan 2012). (↑)

³²³ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³²⁴ For a description of the Roman understanding of the celestial movements, the implications for the hours in a day, and the construction of sundials and water clocks to measure the passage of time in relation to these movements, see Marcus Vitruvius Pollo (~ 80 BCE–15 BCE), de architectura, translated by Morris Hicky Morgan as The Ten Books of Architecture, Book IX, http://en.wikisource.org/wiki/Ten_Books_on_Architecture/Book_IX (viewed 25 Jan 2012). (↑)

³²⁵ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³²⁶ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³²⁷ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³²⁸ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³²⁹ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³³⁰ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³³¹ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³³² see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³³³ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³³⁴ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³³⁵ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³³⁶ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³³⁷ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³³⁸ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³³⁹ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³⁴⁰ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³⁴¹ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³⁴² ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³⁴³ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³⁴⁴ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁴⁵ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁴⁶ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³⁴⁷ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³⁴⁸ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁴⁹ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁵⁰ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁵¹ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁵² The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁵³ ibid, Translator’s Preface, http://www.history.rochester.edu/steam/hero/translators.html (viewed 25 Jan 2012) (↑)

³⁵⁴ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁵⁵ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁵⁶ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁵⁷ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁵⁸ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁵⁹ The Pneumatics of Hero of Alexandria, Translated for and Edited by Bennet Woodcraft, Taylor Walton and Maberly, London, 1851. Reproduced at http://www.history.rochester.edu/steam/hero/section50.html (viewed 25 Jan 2012) (↑)

³⁶⁰ see index, ibid, http://www.history.rochester.edu/steam/hero/index.html (viewed 25 Jan 2012) (↑)

³⁶¹ Boyer and Merzbach, History of Mathematics, p. 177 (↑)

³⁶² Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁶³ See for example, http://www.astrocal.co.uk/metonic-cycle.htm (viewed 25 Jan 2012) (↑)

³⁶⁴ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁶⁵ Philip Ball, “Complex clock combines calendars”, Nature, vol 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁶⁶ T. Freeth, Y. Bitsakis, X. Moussas, J.H. Seiradakis, A.Tselikas, E. Magkou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M.G. Edmunds, “Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism”, Nature vol 444, 2006, pp. 587–591 (↑)

³⁶⁷ Philip Ball, “Complex clock combines calendars”, Nature, vol 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁶⁸ T. Freeth, Y. Bitsakis, X. Moussas, J.H. Seiradakis, A.Tselikas, E. Magkou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M.G. Edmunds, “Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism”, Nature vol 444, 2006, pp. 587–591 (↑)

³⁶⁹ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁷⁰ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁷¹ Philip Ball, “Complex clock combines calendars”, Nature, vol 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁷² T. Freeth, Y. Bitsakis, X. Moussas, J.H. Seiradakis, A.Tselikas, E. Magkou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M.G. Edmunds, “Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism”, Nature vol 444, 2006, pp. 587–591 (↑)

³⁷³ Philip Ball, “Complex clock combines calendars”, Nature, vol 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁷⁴ T. Freeth, Y. Bitsakis, X. Moussas, J.H. Seiradakis, A.Tselikas, E. Magkou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M.G. Edmunds, “Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism”, Nature vol 444, 2006, pp. 587–591 (↑)

³⁷⁵ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁷⁶ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁷⁷ Philip Ball, “Complex clock combines calendars”, Nature, vol 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁷⁸ Philip Ball, “Complex clock combines calendars”, Nature, vol 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁷⁹ T. Freeth, Y. Bitsakis, X. Moussas, J.H. Seiradakis, A.Tselikas, E. Magkou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M.G. Edmunds, “Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism”, Nature vol 444, 2006, pp. 587–591 (↑)

³⁸⁰ Philip Ball, “Complex clock combines calendars”, Nature, issue 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁸¹ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, issue 456, 31 July 2008, pp. 614–617. (↑)

³⁸² Philip Ball, “Complex clock combines calendars”, Nature, vol 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁸³ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁸⁴ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁸⁵ Philip Ball, “Complex clock combines calendars”, Nature, issue 454, published online 30 July 2008, p. 561, http://www.nature.com.ezp.lib.unimelb.edu.au/news/2008/080730/full/ (viewed 14 October 2011) (↑)

³⁸⁶ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, issue 456, 31 July 2008, pp. 614–617. (↑)

³⁸⁷ Tony Freeth, Alexander Jones, John M. Steele and Yanis Bitsakis, “Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism”, Nature, vol 456, 31 July 2008, pp. 614–617. (↑)

³⁸⁸ Menninger, A Cultural History of Numbers, Fig 183, p. 306. (↑)

³⁸⁹ Menninger, A Cultural History of Numbers, Fig 183, p. 306. (↑)

³⁹⁰ Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹¹ Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹² Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹³ Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹⁴ Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹⁵ Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹⁶ Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹⁷ Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹⁸ original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

³⁹⁹ original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

⁴⁰⁰ Original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

⁴⁰¹ original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

⁴⁰² Image is from Museo Nazionale Ramano at Piazzi delle Terme, Rome. Made of brass plate it is approximately 115 x 99 mm with nine long slots (equipped with four sliders for 1–4) and eight short slots each with one slider (for 5). The eight left most columns run in powers of ten from units on the right to millions on the left. The ninth slot is for a fraction of ounces. [reproduced from http://www.ee.ryerson.ca/~elf/abacus/roman-hand-abacus.html. Further details in Cesare Rossi, Flavio Russo and Ferrucio Russo, ”Ancient Engineers and Inventions, History of Mechanism and Machine Science, Springer, Vol. 8, 2009, p. 42 (↑)

⁴⁰³ original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

⁴⁰⁴ original images Rien van de Weygaert, Sept. 2002, National Archaeological Museum Athens, http://www.astro.rug.nl/~weygaert/antikytheraoriginal.html (viewed 24 Jan 2012 (↑)

⁴⁰⁵ Image is from Museo Nazionale Ramano at Piazzi delle Terme, Rome. Made of brass plate it is approximately 115 x 99 mm with nine long slots (equipped with four sliders for 1–4) and eight short slots each with one slider (for 5). The eight left most columns run in powers of ten from units on the right to millions on the left. The ninth slot is for a fraction of ounces. [reproduced from http://www.ee.ryerson.ca/~elf/abacus/roman-hand-abacus.html. Further details in Cesare Rossi, Flavio Russo and Ferrucio Russo, ”Ancient Engineers and Inventions, History of Mechanism and Machine Science, Springer, Vol. 8, 2009, p. 42 (↑)