Newton's Birth Date and The Anni Mirabiles

It's usually said that Isaac Newton was born on Christmas Day, but
there is some ambiguity in this because England was still using the
"old" Julian calender at the time of Newton's birth, whereas the rest 
of Europe had adopted the "modern" Gregorian calender (later adopted 
by England and still in use today).  According to the modern calender, 
Newton was born on 4 January 1643, but according to the calender in 
force at the time and place of his birth, he was born on 25 December 
1642.  (It's been speculated that this fact held some significance
for the mystical side of Newton's imagination, and helps to explain
his fascination for biblical interpretation, since he can hardly
have failed to notice that he was born on Christmas Day with no
worldly father - his natural father Robert, a farmer, having died
some 3 months before Isaac's birth.)

An even trickier question is whether Newton was born in the same year
Galileo died.  Galileo died on 8 January 1642 (Gregorian) and Newton 
was born on 25 December 1642 (Julian).  But when placed on the same 
calender the two events fall in different years.  To make things even 
more confusing, many English of that time still considered March 25 
to be the first day of the calender year, so by the old English 
reading of the Italian calender, Galileo died in 1641.

Originally the Romans considered the year to begin on March 1, which 
is the reason February (the "last" month of the year) is truncated at 
28 days, and is used to sneak in an extra day every four years.  This 
also explains why the prefixes of our month names are all "off" by 
two, i.e., SEPT-ember (7), OCT-ober (8), NOV-ember (9), DEC-ember 
(10).  The Romans officially recognized January 1 as the first day 
of the year in 153 BC, but it didn't catch on everywhere right away.

By the way, it's interesting to note that the "Julian period" (not 
to be confused with the Julian Calender) well known to historians 
and astronomers, was named not after Julius Caesar, but after the 
father of Joseph Scaliger, who in 1582 devised the scheme by which 
each day is consecutively numbered beginning with January 1, 4713 
BC, which Joseph thought would cover everything of interest.  But
I digress...

Another ambiguous date associated with Newton is his famous "annus
mirabilis", or year of miracles, around 1666.  This was the period 
during which Isaac, having been sent home from college because of 
the Plague epidemic, occupied his time by inventing calculus, 
discovering the chromatic composition of light, and conceiving of 
the inverse-square law of universal gravitation... or at least this 
is how Newton later represented things.  Of course, it must be 
remembered that in later life Newton was embroiled with priority 
disputes, most notably with Robert Hooke over optics and the 
inverse-square law of gravity, and with Leibniz over the Calculus.  
Thus, it was always in Newton's self-interest to place his 
discoveries as early as possible.  The documentary evidence 
suggests that, at least with regard to mechanics and gravitation, 
his ideas hadn't actually reached a coherent state until much later, 
around 1685-1687, when he was actually composing the Principia.

In any case, it's clear that Newton devoted himself intensively
to the study of mathematics and physics during the intermission
from Cambridge, but it's often been pointed out that his burst of
activity should really be called "anni mirabiles", since it covered 
the time from 1664 to 1667, and much of that time he was actually 
at Cambridge.  The school closed in the summer of '65, and Newton 
returned in March of '66 when the school was re-opened after the
plague had gone dormant over the winter.  The plague re-appeared
so Cambridge was closed again in June of '66.  This time it
remained closed until April of '67, when Newton again returned 
to Trinity college.

Notwithstanding the caveat mentioned above about Newton's vested
interest in recalling early dates for his discoveries, it's still 
interesting to read his famous recollection of his activities during 
these years.  Following his death there was found in his papers part 
of a draft letter that the elderly Newton had written, apparently 
to the Huguenot scholar Pierre Des Maizeaux, and it contains this 
description:

  "In the beginning of the year 1665 I found the Method of 
   approximatiung series & the Rule for reducing any dignity
   of any Binomial into such a series.  The same year in May
   I found the method of Tangents of Gregory & Slusius, & in
   November had the direct method of fluxions & the next year 
   in January had the Theory of Colors & in May following I 
   had entrance into the inverse method of fluxions.  And the
   same year I began to think of gravity extending to the orb 
   of the Moon & (having found out how to estimate the force 
   with which a globe revolving within a sphere presses the 
   surface of the sphere) from Keplers rule of the periodical 
   times of the Planets being in sesquialterate proportion of 
   their distances from the centers of their Orbs, I deduced 
   that the forces which keep the Planets in their Orbs must 
   be reciprocally as the squares of their distances from the 
   centers about which they revolve: and thereby compared the 
   force requisite to keep the Moon in her Orb with the force 
   of gravity at the surface of the earth, and found them 
   answer pretty nearly.  All this was in the two plague years 
   of 1665 and 1666.  For in those days I was in the prime of 
   my age of invention & minded Mathematics & Philosophy more 
   than at any time since."

Although he says his results "answer pretty nearly", it seems that
at the time he was dissatisfied with the disagreement between his
rough calculation and his observations.  His surviving notes from 
the late 1660's show that he was using an estimate of 60 miles per
degree of latitude.  He had gotten this from Galileo's writings,
and Galileo had used this figure because that's what seamen of
that time commonly used.  Now, similar to the conflict between
the calenders of Italy and England, they also used different
definitions of a "mile".  

The Italian mile was 5000 feet, whereas the English mile was 5280 feet.
Newton was aware of this difference, so he tried his calculation both 
ways, but it still gave a result for the Moon's orbit that implied an 
object at the Earth' surface should fall only 13.2 or 13.9 feet in one 
second, whereas Newton had measured that an object actually falls about 
16.1 feet in one second at the Earth's surface.  According to William 
Whiston, "this made Sir Isaac suspect that this Powewr was partly 
gravity and partly that of Cartesius's Vortices", so he "threw aside 
the Paper of his Calculation and went on to other studies".  Similarly,
Henry Pemberton heard Newton say that "his [1666] computation did not 
answer his expectations, so he concluded that some other cause must at 
least join with the action of the power of gravity on the moon".

Not long afterwards, in 1670, the French astronomer Jean Picard
determined with very accurate measurements that one degree of latitude
equals 69.1 English miles.  In 1675 this result appeared in the Royal 
Societies "Philosophical Transactions", and Newton saw it.  He
supposedly rushed home and excitedly repeated his earlier commputation,
this time finding it "perfectly agreeable to the Theory".  This may
indeed have been what he had in mind when he later said he "found
them answer pretty near", but of course this was nine or ten years 
after the annis mirabiles.