Let us map out the 1460 years which elapsed between two successive coincidences between the 1st of Thoth in the vague year and the heliacal rising of Sirius at the solstice, so that we can see at a glance the actual number of years from any start-point (= 0) at which the 1st of Thoth in the vague year occurred successively further and further from the heliacal rising, until at length, after a period of 1460 years, it coincided again. As the Sirius-year is longer than the vague one, the first vague year will be completed before the first Sirius-year, hence the second vague year will commence just before the end of the fixed year, and that is the reason I have reversed the order of months in the diagram. Now it is clear that, if the Egyptians really worked in this fashion, the date of the heliacal rising of Sirius, given in this way, would enable us to determine the number of years which had elapsed from the beginning of the cycle. This calendar system, it will be seen, is good only for groups of four years. Now, a system which went no further than this would be a very coarse one. We find, however, that special precautions were taken to define which year of the four was in question, and the fact that this was done goes some way to support the suggestion I have made. Brugsch,[74] indeed, shows that a special sign was employed to mark the first year of each series of four. Next, as a matter of fact it is known (I have the high authority of Dr. Krall for the statement) that each king was supposed to begin his reign on the 1st Thoth (or 1st Pachons) of the particular year in which that event took place, and the fact that this was so further supports the suggestion we are considering. During the reign its length and the smaller events might be recorded in vague years and days so long as the date of its commencement had been referred to a cycle. The diagram will show how readily the cycle year can be determined for any vague year. If, for instance, the 1st Thoth in the vague year falls on 1 Tybi of the cycle, we see that 980 years must have elapsed since the beginning of the cycle, and so on. Here, then, we have a true calendar system. If the Egyptians had not this, what had they? Dealing, then, with the matter so far as we have gone, we find that the system suggested enabled the place of the beginning of each vague year and of each king's reign to be dated in terms of the cycle of 1460 years; and further that, if they had not such a system as this, they had no means of recording any lapse of time which exceeded a year. It is not likely that any nation would put itself in such a position, least of all the ancient Egyptians. The existence of periods of 365 years and of 120 years among the Egyptians is easily explained when the existence of this great year is recognised; the 365 years' period, marking approximately the intervals from solstice to equinox and equinox to solstice, in the natural year.