158 6. CALCULATION
Year 0 1 2 3 4 5 6
Full Moon Apr. 14 Apr. 3 Mar. 23 Apr. 11 Mar. 31 Apr. 18 Apr. 8
Year 7 8 9 10 11 12 13
Full Moon Mar. 28 Apr. 16 Apr. 5 Mar. 25 Apr. 13 Apr. 2 Mar. 22
Year 14 15 16 17 18
Full Moon Apr. 10 Mar. 30 Apr. 17 Apr. 7 Mar. 27
Using this table, calculate the date of Easter for the years from 2040 through
- You can easily compute the day of the week for each of these dates in a given
year, starting from the fact that March 21 in the year 2000 was a Tuesday. [Note:
If the first full Moon of spring falls on a Sunday, Easter is the following Sunday.]
6.27. Prosthaphaeresis can be carried out using only a table of cosines by making
use of the formula
. cosfa + â) + cos(o! — â)
cos a cos â —.
2
Multiply 3562 by 4713 using this formula and a table of cosines. (It is fair to use
your calculator as a table of cosines; just don't use its arithmetical capabilities.)
6.28. Do the multiplication 742518 · 635942 with pencil and paper without using a
hand calculator, and time yourself. Also count the number of simple multiplications
you do. Then get a calculator that will display 12 digits and do the same problem on
it to see what errors you made, if any. (The author carried out the 36 multiplications
and 63 additions in just under 5 minutes, but had two digits wrong in the answer
as a result of incorrect carrying.)
Next, do the same problem using prosthaphaeresis. (Again, you may use your
hand calculator as a trigonometric table.) How much accuracy can you obtain this
way? With a five-place table of cosines, using interpolation, the author found the
two angles to be 50.52° and 42.05°. The initial digits of the answer would thus
be those of (cos(8.47°) + cos(92.57°))/2, yielding 47213 as the initial digits of the
12-digit number. On the other hand, using a calculator that displays 14 digits,
one finds the angles to be 50.510114088363° and 42.053645425939°. That same
calculator then returns all 12 digits of the correct answer as the numerical value of
(cos(8.45646866242°) +cos(92.563759514302°))/2. Compared with the time to do
the problem in full the time saved was not significant.
Finally, do the problem using logarithms. Again, you may use your calculator
to look up the logarithms, since a table is probably not readily available.
