121 MEMOIR EXPLANATORY OF A NEW PERPETUAL CALENDAR. 



SUPPLEMENT TO MR. McILVAINE'S MEMOIR. 

 Read December I8th, 1846. 



When constructing the " New Perpetual Calendar," which I had the honour of present- 

 ing to the Society, last year, I purposely avoided furnishing any rule for the Era before 

 Christ, lest the requisite explanations in regard to leap years, (which, for that period, are 

 so expressed by chronologists, as never to be multiples of 4,) might unduly extend or 

 complicate the Tablet. Since its publication, however, in the Quarterly Bulletin, where 

 it appears in a reduced form, with the Examples conveniently separated from the Rules, I 

 have been led to believe that, without exceeding the limits of a single page, in the next 

 volume of the Transactions, at large, the 1st of the annexed Supplemental Rules might 

 readily be subjoined to it, and thus render the plan applicable to all Time. 



The Rule is a mere corollary from the general principles of the Tablet; for after a.d. 1, 

 the numbers of the two scries 1, 29, 57, &c, and 1, 20, 39, &c, (see pages 109 and 113) 

 never again coincide until a. d. 533, 1065, 1597, &c, each of which years, like a.d. 1, is the 

 first after leap year, begins and ends on Sat., and has the same Epact, 11. Reversing the 

 order of time, the chronol. year 1, B. C, or astron. year 0, is the 28th of the Solar, and 19th 

 of the Lunar, cycle, next preceding a.d. 1. It is a leap year, begins on Th., ends on F., 

 has the Epact 29, and corresponds with a.d. 532. The chronol. year 2, B.C., or the astron. 

 year 1, is the 27th of the Solar, and 18th of the Lunar Cycle, next preceding a. d. 1. It. is 

 the third after leap year, begins and ends on W., has the Epact 18, and corresponds with 

 a. d. 531 ; and so backwards without, limit, through all the combinations of Old Style cycles 

 concurring at regular intervals of 532 entire years. 



I have added a convenient Rule, the 2d, for years of the Julian Period; a 3d Rule, with 

 formulas, to serve as proofs; also, a simple method, Rule 4th, of solving converse problems, 

 which, though more curious, perhaps, than useful, I beg leave to append to the memoir. 



I was recently much gratified by learning from Mr. Galloway and Mr. De Morgan, who 

 have, at my request, had the goodness to consult for me the fifth volume of Clavius's mathe- 

 matical works, and to furnish me with a few particulars from the explication there given 

 of the Reformed Roman Calendar, that my conjecture respecting the discrepancy, in a 

 single case, between my results and those set down in the great Table of Clavius, as well 

 as in the text of Delambre, turns out to be correct. On tracing the difference to its source, 

 it appears that the Table and the text are alike erroneous, but accidentally so. 



The substance of flic communications with which I have been honoured on the point 

 in question, is this. In the Table there is an "obvious misprint,''' at the angle where the 

 line belonging to the year 3860 meets the column of Easters; in which column the months 

 of March arid April arc not otherwise distinguished than by their initial letters M and A. 

 The Easter fur that year " stands 22 M, but ought to be 22 A;" for, in the same line, and 

 in nearly adjoining columns on the left, the Paschal full moon is made to fall "18 d 18 h A," 

 the Paschal 14th is made " 18 A," and all the moveable feasts accord with those of an 

 Easter that occurs on the 22d day of April. Moreover, in the column immediately on the 

 right stands "Pentecost 10 June," which is absurd, when Easter is any where in March, 

 and renders the fact more strange that Delambre should have "missed seeing the error.'''' 



