happen.
A new geometry
Khayyám also wrote a book that tackled Euclid’s fifth postulate, which had long rankled a contingent of mathematicians. The fifth postulate Euclid wrote concerned
parallel lines, and it is therefore normally referred to as the parallel postulate
.
Imagine two lines (PQ and RS) with a third (XY) crossing them. Inside PQ and RS we now have four angles, two on each side of the XY: a, b, c and d:
The parallel postulate suggests that if you add the pairs of angles on the same side of XY together (e.g. a+b and c+d) then PQ and RS will cross on the side of the line where
the sum of the angles is less than 180°. If the angles on each side add up to 180° then PQ and RS are parallel and therefore will never cross.
Mathematicians, however, have argued over the ages that this postulate is not quite as obvious as Euclid made out. Khayyám was the first to come up with a counter-example, arguing that
Euclid’s parallel postulate does not always work if the surface you are drawing on is curved. Thus Khayyám instigated the ideas of elliptical and hyperbolic geometry, a
direct challenge to the simple Euclidean geometry that had gone before. This kind of thinking would eventually help Albert Einstein to come up with his ideas of space-time and gravity.
The Middle Ages in Europe
Despite Europe’s plunge into the Dark Ages – so-called because it was thought that following the fall of the Roman Empire the continent had reverted back to a
barbaric state of tribal warfare and religious fundamentalism – there remained a coterie of individuals intent on pushing the boundaries of mathematics even during these difficult times.
B EDE (672–735)
The Venerable Bede is known more perhaps for his contribution as a historian than for the role he played in the development of mathematics. Bede was a monk living in
north-eastern England and his translation of a number of scholarly works into the English of the time helped to spread an enormous amount of knowledge.
Bede’s contribution to mathematics began when he attempted to develop a way to calculate accurately when Easter would fall. At the time it was thought to fall on the
first Sunday after the first full moon following the spring equinox. Missing Easter mass following the calculation of an incorrect date would have resulted in excommunication, and therefore
damnation, so Bede’s was no trivial task.
Dating in the Dark Ages
In order to calculate the date of Easter, it was necessary for Bede to rationalize the date of the spring equinox with the lunar calendar. This was a difficult task in itself
because the date of the equinox varied because the Julian calendar in use at the time was unreliable. Because the date of the equinox varied each year, and full moons come at alternate 29- or
30-day intervals, it meant that there was a 19-year cycle of possible dates for Easter. The procedure for calculating the date of Easter has been known as computus (meaning
‘computation’) ever since.
Once Bede had completed the computus, he decided to sort out dating the rest of history as well. Prior to Bede’s endeavours, historians had been dating things in reference to the lifetime
of the current emperor or king, for example: ‘the Vikings first attacked in the third year of Aethelred’s reign.’ This method, of course, relied on the reader knowing when
Aethelred was around in the first place. Bede decided that it would be far more sensible to date everything occurring either before or after the birth of Jesus Christ. Although not originally
Bede’s idea – that responsibility lay with Dionysius Exiguus, a south-easternEuropean monk active during the sixth century – such was his influence that we
have been using AD (
Anno Domini
, Year of the Lord) and BC (Before Christ) ever since.
Finger Talk
Bede also wrote a book called
On Counting and Speaking With the Fingers
, which allowed the reader to use hand signals for numbers into the
A new geometry
Khayyám also wrote a book that tackled Euclid’s fifth postulate, which had long rankled a contingent of mathematicians. The fifth postulate Euclid wrote concerned
parallel lines, and it is therefore normally referred to as the parallel postulate
.
Imagine two lines (PQ and RS) with a third (XY) crossing them. Inside PQ and RS we now have four angles, two on each side of the XY: a, b, c and d:
The parallel postulate suggests that if you add the pairs of angles on the same side of XY together (e.g. a+b and c+d) then PQ and RS will cross on the side of the line where
the sum of the angles is less than 180°. If the angles on each side add up to 180° then PQ and RS are parallel and therefore will never cross.
Mathematicians, however, have argued over the ages that this postulate is not quite as obvious as Euclid made out. Khayyám was the first to come up with a counter-example, arguing that
Euclid’s parallel postulate does not always work if the surface you are drawing on is curved. Thus Khayyám instigated the ideas of elliptical and hyperbolic geometry, a
direct challenge to the simple Euclidean geometry that had gone before. This kind of thinking would eventually help Albert Einstein to come up with his ideas of space-time and gravity.
The Middle Ages in Europe
Despite Europe’s plunge into the Dark Ages – so-called because it was thought that following the fall of the Roman Empire the continent had reverted back to a
barbaric state of tribal warfare and religious fundamentalism – there remained a coterie of individuals intent on pushing the boundaries of mathematics even during these difficult times.
B EDE (672–735)
The Venerable Bede is known more perhaps for his contribution as a historian than for the role he played in the development of mathematics. Bede was a monk living in
north-eastern England and his translation of a number of scholarly works into the English of the time helped to spread an enormous amount of knowledge.
Bede’s contribution to mathematics began when he attempted to develop a way to calculate accurately when Easter would fall. At the time it was thought to fall on the
first Sunday after the first full moon following the spring equinox. Missing Easter mass following the calculation of an incorrect date would have resulted in excommunication, and therefore
damnation, so Bede’s was no trivial task.
Dating in the Dark Ages
In order to calculate the date of Easter, it was necessary for Bede to rationalize the date of the spring equinox with the lunar calendar. This was a difficult task in itself
because the date of the equinox varied because the Julian calendar in use at the time was unreliable. Because the date of the equinox varied each year, and full moons come at alternate 29- or
30-day intervals, it meant that there was a 19-year cycle of possible dates for Easter. The procedure for calculating the date of Easter has been known as computus (meaning
‘computation’) ever since.
Once Bede had completed the computus, he decided to sort out dating the rest of history as well. Prior to Bede’s endeavours, historians had been dating things in reference to the lifetime
of the current emperor or king, for example: ‘the Vikings first attacked in the third year of Aethelred’s reign.’ This method, of course, relied on the reader knowing when
Aethelred was around in the first place. Bede decided that it would be far more sensible to date everything occurring either before or after the birth of Jesus Christ. Although not originally
Bede’s idea – that responsibility lay with Dionysius Exiguus, a south-easternEuropean monk active during the sixth century – such was his influence that we
have been using AD (
Anno Domini
, Year of the Lord) and BC (Before Christ) ever since.
Finger Talk
Bede also wrote a book called
On Counting and Speaking With the Fingers
, which allowed the reader to use hand signals for numbers into the
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