TO FIND THE DAY OF THE WEEK. 



r.y JAMES ASHKR. 



Four new methods of finding the day of the week in any 

 year, when the corresponding date in some other year is given, 

 have been recently devised by the writer. These will 

 presently be described. 



Before these methods can be used it is desirable to know 

 how to tell whether a year is leap year, also to tell how many 

 leap days are between a date in a given year and the corres- 

 ponding date in any other given year. The writer has also 

 devised new methods for these. 



A year is leap year if the number expressed by the last two 

 figures is a multiple of 4. Thus 1908 was leap year for 08 is 

 a multiple of 4. The year 1927 will not be leap year, for 27 is 

 not a multiple of 4. On dividing 27 by 4, the remainder 

 shows that 1927 will be the third after leap year. 



Since 1752, centennial years not divisible by 400 are not 

 leap years. Thus 1800 and 1900 were not. but 2,000 and 

 2,400 will be leap years. The centennial years before 1752 

 were all leap years from the time of Julius Caesar. 



It was decreed that, in England and its colonies, the day 

 after September 2nd. 1752. should be called the fourteenth. 

 It was also decreed that, thereafter, of the centennial years, 

 only those which were multiples of 400 should be leap years. 

 Further, it was ordered that the year should begin on 

 January 1st, instead of March 25th. 



To find the number of leap days between a given date in 

 one year and the corresponding date in another : — 



Find the number of years from first February 29th, after 

 the date in first year to the first February 29th, after the date 

 in the last year, and divide by 4. If any centennial j-ears, after 

 1752, not multiples of 400, intervene between the first Feb- 

 ruary 29th, after the date in the first year and the first 

 February 29th after the date in the last year, subtract one 

 for each, after making the simple calculation. 



But, when the first and the last year are both leap years, it 

 is better to subtract the first from the last, and divide by 4. 

 If any centennial years since 1752, not multiples of 400, 

 intervene between the first and the last year, subtract one 

 for each, after performing the division. 



If the day of the month in the first year of the period is on 

 or before September 2nd, 1752, and the last year is after 

 1752, do not use the same day of the month in the last year, 

 but use a date 11 days later, in making calculations according 

 to the methods to be presently described. 



First problem, first method : — 



Magna Charta was signed by King John and the barons, 

 June 15th, 1215. 



Find the day of the week, knowing that June 26th, 1912, 

 was Wednesday. 



Solution : — 



The period is 697 years. From February 29th, 1216, to 

 February 29th, 1916, are 700 years. The fourth of 700 is 

 175. We subtract 2 from this because 1800 and 1900 were 

 not leap years. Thus we find that there were 173 leap days. 



In the 697 tropical years from June 15th, 1215, to June 26th, 

 1912, there are 

 697X365 + 173 = 254,578 days, or 36,368 weeks and 2 days. 



When the number of weeks is a whole number, in other 

 words when no days remain, the day of the week required is, 

 evidently, the same as the given day of the week. In solving 

 the problem we have found 2 for remainder. This shows that 

 the required day of the week is two days farther back than 

 our given day of the week. Now the given day is Wednesday. 

 Count backward 2 days to Monday, the answer. 



In solving problems by any of the methods described in this 

 article, if the given day of the week is in the first year, in 

 using the remainder, count forward. 



Second problem, by a second method: — • 



Christopher Columbus discovered America, October 12th, 

 1492. Find the day of the week, knowing that October 23rd, 

 1912, will be Wednesday. 



Solution : — 



The period is 420 years. The first and last year are leap 

 years. The fourth of 420 is 105. We subtract 2 from this, 

 because 1800 and 1900 were not leap years. We thus find 

 that there were 103 leap days in the period. 



In the 420 years there were 420 X 365 + 103 days. 



It will be seen that 420 is a multiple of 7, conseejuently 

 420 X 365 days is a whole number of weeks, and we need do 

 nothing with 420 and 365. Divide 103 by 7 and find 5, the 

 remainder. Count backward 5 days to Friday, the answer. 



This method can be used only when the number of years is 

 a multiple of 7. 



Third problem, by a third method : — 



Queen Victoria was married February 10th, 1.S40. Find 

 the day of the week, knowing that February 10th, 1891, was 

 Tuesday. 



Solution : — 



The period is 51 years. The first February 29th after 

 February 10th, 1840, was February 29th, 1840. The first 

 February 29th after February 10th, 1891, was February 29th, 

 1892. From February 29th', 1840, to February 29th; 1892, 

 were 52 years. The number of leap days is found by dividing 

 52 by 4. We thus find that there were 13 leap days between 

 February 10th, 1840, and February 10th, 1891. 



In the 51 years there were 51 X 364 + 51 + 13 days. 



The number 7 is a factor of 364, therefore 51 X 364 days is 

 a whole number of weeks, and nothing need be done with it. 

 Divide the sum of 51 and 13 by 7, and find that 1 is the 

 remainder. Count backward one day from Tuesday to 

 Monday, the answer. 



Our proceeding may be set down as a rule thus : Call the 

 number of years in the period, days ; to this add the number 

 of leap days in the period, then divide by 7. Count backward 

 from the given day of the week as many days as there are in 

 the remainder. But if the first date is on the given day of the 

 week, count forward. 



Fourth problem, by the third method : 



Standard time was adopted, in the greater part of North 

 .America, November 18th, 1883. Find the day of the week, 

 knowing that November ISth, 1912, will be Monday. 



Solution : 



The period is 29 years. The number of years from 

 February 29th, 1884 to February 29th, 1916, is 32. The 

 fourth part is 8. Subtract 1 because 1900 was not leap year. 

 We thus find that there were 7 leap days. 



We may omit some of the figures that were shown in 

 solving a problem by the third method. We add together a 

 number of days equal to the number of years in our period, 

 and the number of leap days, then divide by 7. Now the sum 

 of 29 and 7, divided by 7, gives 1 for remainder. Count 

 backward 1, from Monday to Sunday, the answer. 



Fifth problem, by a fourth method : — 



Charles Robert Darwin and Abraham Lincoln were born 

 on the same day February. 12th. 1809. Find on what day of 

 the week was the hundreth anniversary of their births, 

 knowing that February 12th, 1912, was Monday. 



Solution : — 



The days of the week seem to step ahead one day each 

 year as you go forward, and to step backward one day if you 

 go backward in reading. In leap year, after leap day, the 

 day of the week will be found two days ahead of the day on 

 the corresponding date in the previous year. 



In our problem the period is 3 years. There are no leap 

 days, because our period ends before leap day, or February 



367 



