Age, in my experience (I'm 59), does not so much creep up on you as – at increasingly regular intervals – leap out from behind a hedge and go "Boo!" when you least expect it. That first jolting moment when your doctor suggests a full health check ("just to make sure there's no nasties, old chap"); the day you realise that your mouth is now a knacker's yard and not a healthy colony of enamel; the first of your children's weddings; the sudden death from a heart attack in a Norfolk pub of the best man at your own wedding; and the news, given to me late last year, that you will be a grandfather.

"It's wonderful," goes the compilation of what old, experienced hands told me as we waited for the birth, "Better than having your own children... You've got real time to spend with them... All the fun and none of the worry... Power without responsibility... Best thing that ever happened to me..."

I was not entirely convinced. After all, you can kid yourself about a lot of things as the years go by – that you can still run for trains, play 18 holes of golf without a buggy, and that those grey hairs actually look more sun-bleached blond in a certain light), but the arrival of a third generation defeats any attempts at self-delusion. In a year or so's time, a little person will call you Grandad. There is no arguing with that. No amount of wearing leisure shirts outside your trousers and gelling your hair will hide it. You are now, in his mind and everyone else's, the sort of cardiganed old sort who hands out Werther's Originals. It's like getting your first mailshot from a funeral director.

He arrived on Tuesday (courtesy of a magnificent performance by my daughter-in-law, Alison, and two vigilant and constantly encouraging midwives at Mayday University Hospital in Croydon), his name is Jack, all was well, and, by the standards of babies, looked remarkably acceptable. His father – not normally noted for his sentimentality – could hardly take his wondering eyes off him. And my previous, silly, ambivalence about becoming a grandfather was forgotten in the two hours after the birth it took my wife and me to ring everyone we had ever met, and scoot up to the hospital, cameras in hand.

Before we left, I had done something which may be thought surprising, and yet within which I think lurks one of the real significances of what had just happened. I went upstairs to my home office, logged on to my Ancestry family tree, and added Jack, the first of our new generation. Nor was this spontaneous. Throughout the seemingly interminable months of pregnancy (why does it still take nine months, for heaven's sake? Isn't it about time modern medicine streamlined the process?), I found myself looking forward to the moment when I would be able to slot Jack's name into the long record of fertile (but otherwise undistinguished) Randalls. He was not just a new life, but a continuation.

This summer, prompted in part by recent family deaths and the imminent new arrival, we had journeyed down to Devon to explore my wife's roots, and to Wiltshire to fumble around in mine. Several hundred years ago, my ancestors infested North Bradley, near Trowbridge, and it was here that my great-great-great-grandfather was born, out of wedlock to a weaver's daughter. It was he who came to London as an apprentice cordwainer, and founded the line whose latest manifestation is young Jack.

We went to the ancient church of St Nicholas, and, encouraged by a woman who changed the flowers, hit upon a plan. This winter, we will bring Jack here to be blessed in the church where his great-great-great-great-great-grandfather was baptised exactly 200 years ago. We hope, when he grows up, that the knowledge of this will make Jack realise that he is part of something bigger than himself – even if it is only our family.