CLONMELSH and MATHEMATICS
Fr Ingram Memorial Lecture, Carlow, 24th November 2000
Maurice OReilly, St Patrick’s College, Drumcondra, Dublin 9 *
Where is Clonmelsh?
Clonmelsh is a parish about 6 or 7 km south of Carlow town, on the way to Muine Bheag.
According to Thomas Fanning, Department of Archaeology, NUIG, there is little
evidence, on any but the first edition of the Ordinance Survey map for County Carlow
(sheet 12), of earthworks in the townland of Garryhundon identified as Killogan.
Quarrying and dumping had badly damaged the site as evidenced by an aerial photograph
of 1967. However, in 1892, an early granite cross was found there dating from between
the late 8th and early 10th century. "The nature of the enclosure at Killogan and
particularly the presence of the cross point to the existence of an early monastic site or
cell, clearly one of a number in the immediate area." [1]
I visited Clonmelsh on the chilly yet dry afternoon of 24th November 2000. It was indeed
as Fanning had described it: a quarry with little hint of its impressive past of which more
below. At the centre of Killogan burial ground which was quite overgrown, the cross
which had been repaired in the mid 1980s, was again in two pieces, its moss-covered
head leaning, forlorn, against the socket stone. The following morning, in a misty
drizzle, I returned to the site with Elizabeth Oldham and Olivia Bree – the first IMTA
fieldtrip to Clonmelsh!
In this paper, I shall try to treat some of the disparate ideas about the interaction of place
and history, of culture and mathematics, which have distracted me from time to time over
the past four years.
Clonmelsh & Rath Melsigi
Today, Clonmelsh is unlikely to attract visitors in the same numbers as does, say, St
Mullins or Glendalough; yet there is strong evidence that, in the 7th century, if not a
popular destination, it was at least a significant centre of learning, under the name of Rath
Melsigi. Indeed, Dáibhí Ó Cróinín has argued that manuscripts associated with the
monastery of Echternach in Luxembourg may well have been written at Rath Melsigi [2].
Moreover, his evidence contributes to the debate, still unresolved, about the provenance
of the great 'Insular' illuminated manuscripts including the Book of Durrow, the
Lindisfarne Gospels and the Book of Kells. Another great manuscript of this family is
the Echternach or Willibrord Gospels, now in the Bibliotèque Nationale, Paris. Even if
the Echternach manuscripts were not actually written in Carlow, Ó Cróinín argues that
Rath Melsigi was their "most natural historical setting". How could this have been?
References in Bede
Bede, the Venerable, lived from about 673 to 735 apparently without leaving
Northumbria. In book III, chapter 27 of his Ecclesiastical History of the English People
[3], he wrote:
*
e-mail: maurice.oreilly@spd.ie
In this year of our Lord 664 there was an eclipse of the sun on 3 May about 4
o'clock in the afternoon. In the same year a sudden pestilence first depopulated
the southern parts of Britain and afterwards attacked the kingdom of Northumbria,
raging far and wide with cruel devastation and laying low a vast number of
people. Bishop Tuda was carried off by it and honourably buried in the
monastery called Pægnalæch. The plague did equal destruction in Ireland.
At this time there were many in England, both nobles and commons, who, in the
days of Bishops Finan and Colman, had left their own country and retired to
Ireland either for the sake of religious studies or to live a more ascetic life. In
course of time some of these devoted themselves faithfully to the monastic life,
while others preferred to travel round to the cells of various teachers and apply
themselves to study. The Irish welcomed them all gladly, gave them their daily
food, and also provided them with books to read and with instruction, without
asking any payment.
Among these were two young Englishmen of great ability, named Æthelhun and
Egbert, both of noble birth. The former was a brother of Æthelwine, a man
equally beloved of God, who, later on, also went to Ireland to study; when he had
been well grounded he returned to his native land and was made bishop in the
kingdom of Lindsey, over which he ruled for a long time with great distinction.
Æthelhun and Egbert were in a monastery which the Irish call Rath Melsigi, and
all of their companions were carried off by the plague or scattered about in
various places, while they themselves were all stricken by the same disease and
were dangerously ill.
The account continues to explain that Æthelhun perished but Egbert was spared. Egbert
intended to follow the footsteps of Wilfred (of York) in the mission to the Frisians who
lived in the flat marshy land between the Rhine and the North Sea [4]. He was
unsuccessful in this mission, as was his successor Uuichtberct. After Uuichtberct,
Willibrord left for Fresia in 690, founded Echternach (in 699) and, under the patronage of
Pippin II, lead the conversion of the Frisians, the Danes, the Thuringians and more
besides [5]. The key point Ó Cróinín makes is that although the Frisian mission began
from Northumbria, Egbert, Uuichtberct and Willibrord set out on their mission from Rath
Melsigi.
Learning in early medieval Ireland
In the Late Roman Empire, the imperial curriculum comprised seven disciplines:
grammar, rhetoric, logic, arithmetic, geometry, astronomy and music. The first three of
these known as the trivium, the last four, the quadrivium. During this period, many
textbooks were compiled such as Martianus Capella's The Marriage of Philology and
Mercury (410-429), and the works of Donatus and Priscian [6]. In 410, the Visigothic
leader, Alaric, sacked Rome. Ireland, having been outside the Empire, was about to
undergo momentous, yet peaceful, change.
We tend to think of learning in Ireland as beginning with the coming of Christianity and
Latin probably in the late 4th century. This is, of course, a simplification: these two
foreign elements arrived in a society enjoying a rich and structured cultural framework
which was propagated orally since writing had not yet been introduced. This civilisation,
by all accounts, embraced the new written learning, secular and, especially, religious
from the ailing Roman Empire.
After about two centuries of taking root (from, say, 400 to 600), a new scholarship
flourished in Ireland throughout the next two centuries and gradually waned in the 9th
century [7]. Evidence of the Irish 'Golden Age' is plain to see in metalwork, stone
carving and illuminated manuscripts [8]. But there is more besides: the enormous legacy
of this period found in manuscripts scattered throughout Europe bears testimony to Irish
zeal for travel or peregrinatio, and learning or ubera sapientiae [9].
The Irish curriculum in this Golden Age was dominated by three areas of study:
grammar, which concerned all aspects of the study of Latin, exegesis, which included
biblical, liturgical and patristic studies as well as hagiography, and, finally the computus,
the study of the calendar. In all of this there are two areas especially where it is fruitful to
look for mathematics: the geometry of the illuminated manuscripts and the computus.
What geometry did the illuminators know?
Let us return to Clonmelsh, or at least to one of its leading alumni, Willibrord, who
studied there for twelve years from 678 to 690. Ó Cróinín [2] suggests that Rath Melsigi
is at least as good a candidate as anywhere in Northumbria for the scriptorium where the
Echternach Gospels were written. He offers various historical evidence to support his
claim, rather than the arguments from archaeology and art-history which had been used to
suggest a Northumbrian provenance. He quotes Lowe that the manuscript belonged to
Saint Willibrord himself.
As mathematicians, our interest is in how the illuminators conceived geometry. Whether
or not they knew, for example, Pythagoras' theorem we may never know; that they had at
least a practical flair for geometry is certain.
Several authors have investigated the geometrical forms in the Insular manuscripts, but
none so comprehensively as Robert D. Stevick, Professor of English at the University of
Washington, Seattle [10]. According to Stevick, his interest arose 'as an attempt to
understand the Anglo-Saxons' divisions of their longer vernacular poems'. He continues,
'this study evolved into a dual investigation of form in art and poetry of the same locale
and time'. He emphasises the possibility of practical reconstructions of geometrical
forms which underlie, for example, illuminated pages from Insular manuscripts, while
acknowledging that any such proposed form was not necessarily used by the artist.
Unfortunately, evidence of compass marks in the vellum is unusual, an interesting
exception being folio 26v. of the Lindisfarne Gospels. He also points out that individual
artists were not necessarily familiar with the use of mathematics in any sophisticated
way, nor with mathematical proof: 'Geometers' techniques were being used for creating
forms, not for mathematical inquiry or problem solving'. His project is 'to recover the
repertory of construction techniques that were used to create the forms'.
I have used Maple to examine Stevick's proposal for the Imago Leonis page (75v.) of the
Echternach Gospels. Here is the final result:
The page itself, as its title suggests, has a wonderful rampant lion positioned diagonally
across it. On this page, Stevick identifies occurrences of ratios involving 2 and the
golden ratio, . Indeed, he claims that the entire geometrical form of the page can be
constructed using these two 'true measures'. Without more attention to the details of
draughtsmanship, it is difficult to take into account the thickness of the plan outline,
which, on the original, is probably about 2.5mm. Possible discrepancies arising from
ignoring this thickness can give rise, in the most extreme cases, to errors in estimating
linear measurements of about 15%. With such a possible margin of error, it is difficult to
distinguish between the measurements 2+2 = 3.414… and 1+22 = 3.828…, for
example.
A computer algebra package with good graphics such as Maple or, The Geometer's
Sketchpad makes investigation of the detail of such analysis more controllable, and may
well lead to new insights into geometry in Early Medieval Ireland.
Learning in Europe in the 'Dark Ages'
Mathematics is considered by many as a cumulative discipline, one in which theories are
built up in painstakingly upon solid foundations involving axioms and theorems bound
together by the super glue of logic. Fashions may come and go - conic sections will
probably never again enjoy the popularity of the days of George Salmon's textbooks [11].
Some ideas such as the differential calculus are here to stay in spite of having severe
doubts cast on their philosophical basis by George Berkeley 70 years after its discovery
[12].
Euclid laid solid foundations for geometry about 300BC, and, although the parallel
postulate was questioned repeatedly leading to non-Euclidean geometries in the 19th
century, the rigour of the Greeks is still used as an exemplar. The Greeks knew how to
state axioms and formulate theorems - all who engage in the true art of mathematics
should do likewise! It is interesting that the real line was not axiomatized until 1872
independently by Dedekind and Cantor.
In his wonderful book, Journey through Genius [13], William Dunham discusses twelve
great theorems from the 5th century BC to the late 19th century. In this celebration of the
great moments of mathematics, there is a gap of almost 1500 years between Heron's
formula for triangular area and Cardano's solution of the cubic. This glaring time-gap is
known in the history of mathematics and elsewhere as the Dark Ages when classical
knowledge was lost to the Western world.
Yet we know that mathematics is more than the rigour of axioms and theorems, however
important they may be. Beyond the south-eastern extremities of Europe, in Baghdad,
algebra arose in the 9th century [14]. It was to have a major impact on the future
development of European mathematics in spite of Europe's obsessions with its Hellenic
legitimacy. (See Cifoletti, Høyrup and others in [15].) I believe that the mathematics
practised in the north-western extremities of Europe, in Ireland in the 7th and 8th centuries
deserves more attention, if not among the global mathematical community, at least
among those interested in mathematics in Ireland and its neighbours.
The calendar & the computus
Now let us turn to the question of the calendar. Today we take the calendar for granted,
accepting details like months of differing lengths and leap years without a second
thought. Because of millennium fever, quite a few popular books have been published
recently on the calendar. EG Richards’ book, Mapping Time, contains a good deal of
interesting material including several algorithms for calendar calculations [16]. The final
part of this book is on Easter, the issue which, above all, motivated the study of the
computus in the early Christian era.
The two units of time which are the main concern of the computus are the solar year and
the synodical or lunar month. Nowadays, the solar month is defined as the period of time
during which the earth makes one revolution around the sun, measured between two
successive vernal equinoxes, equal to 365.242199 days. On the other hand, the synodical
month is the period of time taken by the moon to make one complete revolution around
the earth, measured between two successive new moons, equal to 29.53059 days [17].
Thus the solar year is 12 synodical months and 10.875 days
The Easter question arose from the Jewish calendar in which the date of the Passover (or
Pesach) depended on the lunar rather than the solar calendar. In the early church, the
question of universal agreement on the date of celebration of Easter caused much
controversy. There were theological issues such as deciding on which day of the week
Easter should be celebrated, or, having decided on Sunday, what the permitted range of
dates for Easter should be. These were not the only questions to be resolved; there was
also the technical puzzle of how to predict the relationship between the solar and lunar
calendars in advance [18].
Returning to Clonmelsh, or Rath Melsigi, Ó Cróinín [2] argues a strong case that a
particular Echternach manuscript (again in the Bibliotèque Nationale, Paris) containing
much computistical material, was written there. We can be confident that the computus
was a significant element in the education of students at Rath Melsigi.
The technical computistical challenge involved the establishment of a periodic cycle to
model the solar and lunar calendars. The relative position of the sun and moon would be
approximately the same at the end of each cycle as at its beginning. Moreover, a cycle,
just like the familiar sine function’s cycle of period 2 radians, represented the
characteristic pattern of the intertwined motions of sun and moon. By the middle of the
7th century there were two main contenders: a 19-year cycle favoured by both Alexandria
and Rome and an 84-year cycle which still enjoyed support in parts of the British Isles.
In the 19-year cycle there were 19365 = 6935 days (excluding the bissextiles or leap
days, one every four years). These 19 years comprised 12 common years, each of 12
lunar months, together with 7 embolistic years, each of 13 months. In each year, there
were 6 ‘hollow’ months of 29 days together with 6 ‘full’ months of 30 days, while the
extra embolistic month was also full. Adding all this up yields 12354 + 7384 = 6936
days, or one day too many. This error was rectified by the saltus lunae which shortened
one of the full months by one day, thus making it hollow. In a leap year, the month
containing the 24th February (or the 6th kalends of March) was given an extra day, the
‘second sixth’ or bissextus in both the solar and lunar calendars without causing any
disruption to the cycle. Likewise, the 84-year cycle of 84365=30660 days comprised 53
common years and 31 embolistic years, with 6 salti lunae to correct the discrepancy.
The Synod of Whitby of 664 – the same year as the pestilence – brought victory to the
camp of the 19-year cycle. The so-called Padua latercus (or Paschal table) which was
rediscovered in 1985 by Ó Cróinín [19], having been lost for about 1300 years, brought
to light the details of the 84-year cycle. Following this discovery and noting its complete
dependency on a text called De ratione paschali, attributed to the third century Anatolius
of Laodicea, McCarthy [20] has suggested that the corruption of tables in the latter may
have been deliberate and politically motivated.
Conclusions
It is a difficult task for those enjoying riding the Celtic Tiger, or indeed those hanging on
by his tail, to begin to appreciate what mathematical activity might have been conducted
by the Irish and their circle over 1300 years ago. For one thing the notation was
different: it was to take over three more centuries before Arabic numerals were
introduced to Western Europe. Nonetheless, I believe that the material available to us
contains evidence of mathematics at least of an algorithmic nature. By taking an interest
in this evidence, we gain important insights to pass on to our students, not least a feeling
for how mathematics and culture were inter-related in our past, and continue to be today.
The spirit of the Irish ‘Golden Age’ offers much to inspire us: an appreciation of learning,
including the geometric intricacies of the illuminated manuscripts, the arithmetic – almost
algebraic – complications of the computus, and generous hospitality at Clonmelsh which
fostered it all.
References
1. Fanning, T, Some Field Monuments in the Townlands of Clonmelsh and
Garryhundon, Co Carlow, Peritia 3 (1984) 43-49.
2. Ó Cróinín, D, Rath Melsigi, Willibrord, and the Earliest Echternach Manuscripts,
Peritia 3 (1984) 17-42.
3. Bede, The Ecclesiastical History of the English People, J McClure & R Collins (eds),
Oxford UP, 1999.
4. Collins, R, Early Medieval Europe 300-1000, Macmillan, 1991.
5. Fletcher, R, The Conversion of Europe from Paganism to Christianity 371-1386,
Fontana Press, 1998.
6. Fossier, R (ed), The Cambridge Illustrated History of the Middle Ages 350-950, CUP,
1997.
7. Ó Cróinín, D, Early Medieval Ireland 400-1200, Longman, 1997.
8. Harbison, P, The Golden Age of Irish Art, Thames & Hudson 1999.
9. Richter, M, Ireland and her Neighbours in the Seventh Century, Four Courts Press,
1999.
10. Stevick, RD, The Earliest Irish and English Bookarts - Visual and Poetic Forms
before AD 1000, Univ. Penn. Press, 1994.
11. Houston, K (ed), Creators of Mathematics - The Irish Connection, UCD Press, 2000.
12. Fauvel, J & Gray J, The History of Mathematics - A Reader, Macmillan, 1987.
13. Dunham, W, Journey through Genius - The Great Theorems of Mathematics, Wiley,
1990.
14. Rashid, R (ed), Histoire des sciences arabes, 2 - Mathématiques et physique, Seuil,
Paris 1997.
15. Goldstein, C, Gray & Ritter (eds), L’Europe mathématique/ Mathematical
Europe, Editions de la Maison des sciences de l’homme, Paris 1996.
16. Richards, EG, Mapping Time – The Calendar and its History, OUP 1998.
17. Encyclopædia Britannica, 15th edn (1978), Micropædia, vol. 3, pp595-599.
18. McCarthy, DP, Easter principles and a fifth century lunar cycle used in the British
Isles, J for the History of Astronomy, 24 (1993), pp 204-24.
19. McCarthy, D & Ó Cróinín, D, The Lost Irish 84-year Easter Table Rediscovered,
Peritia, 6-7 (1987-8), pp227-42.
20. McCarthy, DP, The Lunar and Paschal Tables of De ratione paschali Attributed to
Anatolius of Laodicea, Archive for History of Exact Sciences, 49 (1996), pp 285-320.
Acknowledgement
The author expresses his gratitude to Dr Dan McCarthy, Trinity College, Dublin, for
drawing attention to certain misleading passages in an earlier draft of this paper. Any
such passages that remain are, of course, entirely the responsibility of the author!