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Okay, let’s have a bit more of these calculations:
Form AAm we learn that, the earliest time for the awakening of the father of the Dwarves is Valian Year 1050 (that is when the elves awoke). The Eldar enter Beleriand in Valian Year 1125. So the maximum time we have is 75 Valian Years (VY).
In LotR, Appendix A we are told that the Dwarves get their first children about the 100th year of the Sun (SY) of their life. That would mean that the average time between one generation and the next is a bit longer, but for the easier calculation let us stick with 100 SY.
In addition we learn: ‘The number of dwarf-men that marry is actually less than one-third.’ In the ‘Line of the Dwarves of Erebor’ the greatest number of sons recorded for one Dwarf is 3. If that would be a general fact the race would not grow but dwindle. So we have to assume that earlier the Dwarves that married got more children, let us assume 4 sons of which 1.3 in average would marry and beget children on their part. (This is already a stretch, since it would mean 5.3 children in average for each couple including the daughters.)
For easy reference I will now call the 7 Fathers generation 0 (G0), their sons and daughters G1 and so on.
Now we are told that Durin had no wife and that his people were gathered from other houses. That means that Durin did take part in the propagation of his race only starting from generation 1 (G1) and that we have to split the total number of Dwarves starting from generation 1 by 7 to get the number of Dwarves in each House.
The split between the Petty-Dwarves and the Dwarves of the Ered Luin will not be taken into account by the calculation. It must have occurred early in the time period described above, but probably later then generation 2 and the gathering Durin’s people.
The mathematic:
A) The Second Generation:
We do not want the total number of Dwarves in that generation, we are only interested in the couples for producing the further generations: The prolific of spring of the 6 fathers plus Durin and his wife: 6 x 1.3 + 1 = 8.8
We will round that to 9 and start from that number after a time of 100 SY.
B) Assuming 1 VY = 9.582 SY:
75 VY x 9.582 SY/VY = 718.65 SY.
That means we have G7 as children, G6 as parents and G5 as grandparents. To calculate the total number of Dwarves in G5 we need the number of prolific couples in G4:
9 x 1.3 x 1.3 x 1.3 = 9 x 1.3**3 = 19.773
Total number of Dwarves in G5 is: 19.773 x 5.3 = 105
Number of prolific couple in G5: 9 x 1.3**4 = 25.7049
Total number of Dwarves in G6 is: 25.7049 x 5.3 = 136
Number of prolific couple in G6: 9 x 1.3**5 = 33.41637
Total number of Dwarves in G7: 33.41637 x 5.3 = 177
Total number of Dwarves alive: 105 +136 + 177 = 405
Total number of Dwarves in Beleriand and Ered Lindon: 405 x 2 / 7 = 116
I don’t think that is enough.
C) Assuming 1 VY = 144 SY:
75 VY x 144 SY/VY = 10800 SY.
That means we have G108 as children, G107 as parents and G106 as grandparents. Doing the same mathematic as above would yield 38.5 x 10^12 Dwarves in Beleriand and Ered Lindon, which is far too much. That means in this case we could go with fare more realistic procreation rates and would none the less come to realistic numbers of Dwarves.
Conclusion:
If we assume 1 VY = 9.582 SY we need the additional Dwarves laid to sleep near the fathers. If we assume 1VY = 144 SY we do not need them.
Respectfully
Findegil
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