Pierre-Simon De Laplace was born on 23rd March, 1749, at Beaumont-en-Auge, a village four miles west of Pont l’Évêque in Normandy. As its name would suggest, it stands on high ground : and from an old Benedictine Priory where Laplace went to school, only a stone's throw from this birthplace, there is in fine weather a superb panorama, reaching to the English Channel seven miles away.

The original documents relating to Laplace's life have almost all perished in quite recent times: most of them as a result of a fire which in 1925 destroyed the château of Saint-Julien de Mailloc, near Lisieux, the home of his great-great-grandson the Comte de Colbert-Laplace : some had been destroyed earlier, when his house at Arcueil near Paris was looted by housebreakers in 1871 : and some were lost very recently in the bombardment of Caen. This is the more regrettable, because many of the statements regarding Laplace in the histories of mathematics are false. His father, Pierre de Laplace, owned and farmed the small estate of Mérisier near Beaumont, and held the office of Syndic of the parish. When Laplace became a Marquis, it was the fashion to regard him as having risen from a very humble origin: but the fact was that he came of a country family of good social standing One of his father's brothers was a priest and one a surgeon, and his greatuncle, Maître Olivier de Laplace, had held the title of Chirurgien Royal.

The following theorems are closely connected in various ways; one common thread that runs through them is that in each a critical stage of the proof consists in filling in a missing value in the range of values for which the result is known to hold. Another common feature, of no logical importance but none the less the raison d'être for this article, is that each serves to illustrate important fallacies and omissions in current accounts of elementary algebra and analysis.

It is well known that, if.

cos t>Cn(t) or cos t<Cn(t) holds for 0< t< ∞ according as n = 1, 3, 5, … or n = 2, 4, 6, …. It will be noted that this is not obvious in itself. In fact, the absolute values of the successive terms of the alternating series do not decrease from m = 0 onward, except when t is small.