It has already been shown that if the number of years in the intercalary period be multiplied by seven, the number of days in the week, their product will be the number of years in the cycle. Now, in the Gregorian calendar, the intercalary period is 400 years; this number being multiplied by seven, their product would be 2,800 years, as the interval in which the coincidence is restored between the days of the year and the days of the week. This long period, however, may be reduced to 400 years; for since the dominical letter goes back five places every four years, in 400 years it will go back 500 places in the Julian and 497 in the Gregorian calendar, three intercalations being suppressed in the Gregorian every 400 years. Now 497 is exactly divisible by seven, the number of days in the week, therefore, after 400 years the cycle will be completed, and the dominical letters will return again in the same order, on the same days of the month. In answer to the question, "Why two dominical letters for leap-year?" we reply, because of the additional or intercalary day after the 28th of February. It has already been shown that every additional day causes the dominical letter to go back one place. As there are 366 days in leap-year, the letter must go back two places, one being used for January and February, and the other for the rest of the year. Did we continue one letter through the year and then go back two places, it would cause confusion in computation, unless the intercalation be made at the end of the year. Whenever the intercalation is made there must necessarily be a change in the dominical letter. Had it been so arranged that the additional day was placed after the 30th of June or September, then the first letter would be used until the intercalation is made in June or September, and the second to the end of the year. Or suppose that the end of the year had been fixed as the time and place for the intercalation, (which would have been much more convenient for computation,) then there would have been no use whatever for the second dominical letter, but at the end of the year we would go back two places; thus, in the year 1888, instead of A being dominical letter for two months merely, it would be continued through the year, and then passing back to F, no use whatever being made of G, and so on at the end of every leap-year. Hence it is evident that this arrangement would have been much more convenient, but we have this order of the months, and the number of days in the months as Augustus Caesar left them eight years before Christ. The dominical letter probably was not known until the Council of Nice, in the year of our Lord 325, where, in all probability, it had its origin. CHAPTER IV. RULE FOR FINDING THE DOMINICAL LETTER.