He Has His Father's Eyes!
We've all heard stories of a red-headed baby being born to parents who are decidedly not red-headed and immediately thought of infidelity. Well, that's not necessarily true as anyone who knows a bit about genetics will tell you. It's a complex science but a simple treatment of the subject might help explain it.
In 1866 a monk named Gregor Mendel published a paper explaining how he had noticed a strange pattern in cross-pollination of plants and put forward a theory as to the mechanisms to explain it. His experiments were with peas but the same principles apply for all heredity. He noticed that if he took two different pure species of pea and cross-pollinated them, all the offspring were pure copies of one of the species. However, if he cross-pollinated those offspring he always generated three-quarters of one species and one-quarter of the other. The simplified explanation is as follows.
Each plant has two genes, which in a pure species are the same. In cross-pollination, both species contributes, at random, one of the genes. In that first generation, given that both genes are the same in any species, all will have one gene from one and one gene from the other e.g.
AA bred with BB produces AB, AB, AB, AB, first with first, second with first, first with second and second with second. Now one of these genes will dominate the other and will alway be the one to show so if A dominates B they will all show as type A.
The second generation is more interesting though. There are four possibilities of cross-combination. Taking one gene from the first one and one gene from the second one, AB bred with AB can produce AA, AB, BA and BB in equal proportions. If A dominates B then three combinations will show as type A and one will show as type B, the proportions he always observed. Therefore two parents which show as type A have produce offspring which shows as type B, your red-headed baby!
Human genetics are complex but let's say there is one gene which controls eye colour. The father may be carrying a gene-pair AB and the mother CD. The child will inherit one gene from either parent but one will dominate the other so if the father has eyes showing as type A, there is only a quarter chance that the child will as well. On the next generation the same game is played out and there is only a one-in-eight chance that A will show as there is only a half-chance that the particular gene will be passed on.
Of course, in certain populations there are limited numbers of genes which is why the Chinese all have dark hair and eyes but in our mongrel race there seems to be much greater variations in appearance due to the cross-fertilisation from various gene pools. In the example above, it could be that the parents have the gene pairs AB and AA so the chances of "daddy's eyes" is increased threefold. The child would also have mummy's eyes though.
What this is leading to is that when we refer back to someone a few generations back, if they only contribute half of their genes and only half of them show and that happens at each generation then the influence over the years is rather diluted. You are much more likely to resemble your mother than your great-great-great granny. However, g-g-g-g's red hair might surprisingly reappear in your own grand-daughter. I mention red hair because that is renowned as being a recessive gene, one which allows others to dominate. Therefore it has more chance of being passed on than showing.
When considering your family history, most people put too much emphasis on the line which provided the surname. Genetically speaking, all lines are just as important in making you who you are.
A final thought - the maternal line is certain, given that the pregnancy is a clear indication of parenthood but how certain is the paternal line? Just one wee white lie somewhere in the line and you're looking at the wrong ancestry. That is why some societies, including our own Picts, employed the maternal line to decide on succession.