It was thought, at the time of the General Council of Nice, which was holden in the Year of our Lord 325, that 19 Julian Solar Years were exactly equal to such a Cycle of 19 Lunar Years, or to 235 Synodical Months; and therefore, that, at the End of 19 Years, the New Moons or Conjunctions would happen exactly at the same Times, as they did 19 Years before: And upon this Supposition it was, that, some time afterwards, the several Numbers of that Cycle, commonly called the Golden Numbers, were prefixed to all those Days in the Calendar, on which the New Moons then happened in the respective Years corresponding to those Numbers; it being imagined, that whensoever any of those Numbers should for the future be the Golden Number of the Year, the New Moons would invariably happen on those Days in the several Months, to which that Number was prefixed.
But this was a Mistake:
| For 19 Julian Solar Years contain | 6939d, | 18h, | 0´, | 0´´, | 0´´´ |
| Whereas 235 Synodical Months contain only | 6939, | 16, | 31, | 56, | 30 |
| And are therefore less than 19 Julian Solar Years by | 0, | 1, | 28, | 3, | 30. |
This Difference amounts to a whole Day very nearly in 310.7 Years, the New Moons anticipating, or falling earlier, by 24 Hours in that Space of Time, than they did before: And therefore now in the Year 1750, the New Moons happen above four Days and a half sooner, than the Times pointed out by the Golden Numbers in the Calendar.
In order therefore to preserve a sort of regular Correspondence betwixt the Solar and the Lunar Years, and to make the Golden Numbers, prefixed to the Days of the Month, useful for determining the Times of the New Moons, it would be necessary, when once those Golden Numbers should have been prefixed to the proper Days, to make them anticipate a Day at the End of every 310.7 Years, as the Moons will actually have done; that is to set them back one Day, by prefixing each of them to the Day preceding that, against which they before stood.
But as such a Rule would neither be so easily comprehended or retained in Memory, as if the Alteration was to be made at the End or at the Beginning of complete Centuries of Years; the Rule would be much more fit for Practice, and keep sufficiently near to the Truth, if those Numbers should be set back nine Days in the Space of 2800 Years; by setting them back one Day, first at the End of 400 Years, and then at the End of every 300 Years for eight times successively: whereby they would be set back, in the whole, nine Days in 2800 Years. After which they must again be set one Day back at the End of 400 Years, and so on, as in the preceding 2800 Years. By which means the Golden Numbers would always point out the mean Times of the New Moons, within a Day of the Truth.
It is plain however that the Lunar Year will have lost one Day more than ordinary, with respect to the Solar Year, whenever the New Moons shall have anticipated a whole Day; as they will have done at those times, when it is necessary that the Golden Numbers should, by the Rule just now given, be set back one Day: and consequently the Epact, for that and the succeeding Years, must exceed by an Unit the several corresponding Epacts of the preceding 19 Years.
For the Epact is the Difference, in whole Days, betwixt the common Julian Solar and the Lunar Year; the former being reckoned to consist of 365, and the latter of only 354 Days. If therefore the Solar and the Lunar Year at any time should commence on the same Day, the Solar would, at the End of the Year, have exceeded the Lunar by 11 Days; which Number 11 would be the Epact of the next Year: 22 would be the Epact of the Year following, and 33 the Epact of the Year after that, the Epacts increasing yearly by 11. But as often as this yearly Addition makes the Epact exceed 30, those 30 are rejected as making an intercalary Month, and only the Excess of the Epact above 30 is accounted the true Epact for that Year. Thus when the Epact would amount to 31, 32, 33, 34, &c. the 30 is rejected, and the Epact becomes 1, 2, 3, 4, &c.
Since therefore the Lunar Year will have lost a Day more than ordinary, in respect of the Solar Year, whenever it is necessary to set the Golden Numbers one Day back, as was before observed; it follows, that the Epact must at the same time be increased by an Unit more than usual: the Difference betwixt the Solar and the Lunar Year having been just so much greater than usual. That is, 12 must be added, instead of 11, to the Epact of the preceding, in order to form what will be the Epact of the then present Year. Which Addition of an Unit extraordinary to one Epact will occasion all the subsequent Epacts (which will follow each other in the usual manner, each exceeding the foregoing by 11) to be greater by an Unit than their respectively corresponding Epacts of the preceding 19 Years.
If therefore, instead of the Golden Numbers, the Epacts of the several Years were prefixed, in the manner the Gregorians have done, to the Days of the Calendar, in order to denote the Days on which the New Moons fall in those Years whereof those Numbers are the Epacts; there would never be Occasion to shift the Places of those Epacts in the Calendar; since the Augmentation by an Unit extraordinary of the Epacts themselves would answer the Purpose, and keep all tolerably right.