It is very common to say that Lee took the average life, or expectation, as it is wrongly called, for his term: and this I have done myself, taking the common story. Having exposed the absurdity of this second supposition, taking it for Lee's, in my Formal Logic,[[345]] I will now do the same with the first.
A mathematical truth is true in its extreme cases. Lee's principle is that an annuity on a life is the annuity made certain for the term within which it is an even chance the life drops. If, then, of a thousand persons, 500 be sure to die within a year, and the other 500 be immortal, Lee's price of an annuity to any one of these persons is the present value of one payment: for one year is the term which each one has an even chance of surviving and not surviving. But the true value is obviously half that of a perpetual annuity: so that at 5 percent Lee's rule would give less than the tenth of the true value. It must be said for the poor circle-squarers, that they never err so much as this.
Lee would have said, if alive, that I have put an extreme case: but any universal truth is true in its extreme cases. It is not fair to bring forward an extreme case against a person who is speaking as of usual occurrences: but it is quite fair when, as frequently happens, the proposer insists upon a perfectly general acceptance of his assertion. And yet many who go the whole hog protest against being tickled with the tail. Counsel in court are good instances: they are paradoxers by trade. June 13, 1849, at Hertford, there was an action about a ship, insured against a total loss: some planks were saved, and the underwriters refused to pay. Mr. Z. (for deft.) "There can be no degrees of totality; and some timbers were saved."—L. C. B. "Then if the vessel were burned to the water's edge, and some rope saved in the boat, there would be no total loss."—Mr. Z. "This is putting a very extreme case."—L. C. B. "The argument
would go that length." What would Judge Z.—as he now is—say to the extreme case beginning somewhere between six planks and a bit of rope?
MONTUCLA'S WORK ON THE QUADRATURE.
Histoire des recherches sur la quadrature du cercle ... avec une addition concernant les problèmes de la duplication du cube et de la trisection de l'angle. Paris, 1754, 12mo. [By Montucla.]
This is the history of the subject.[[346]] It was a little episode to the great history of mathematics by Montucla, of which the first edition appeared in 1758. There was much addition at the end of the fourth volume of the second edition; this is clearly by Montucla, though the bulk of the volume is put together, with help from Montucla's papers, by Lalande.[[347]] There is also a second edition of the history of the quadrature, Paris, 1831, 8vo, edited, I think, by Lacroix; of which it is the great fault that it makes hardly any use of the additional matter just mentioned.
Montucla is an admirable historian when he is writing from his own direct knowledge: it is a sad pity that he did not tell us when he was depending on others. We are not to trust a quarter of his book, and we must read many other books to know which quarter. The fault is common enough, but Montucla's good three-quarters is so good that the fault is greater in him than in most others: I mean the fault of not acknowledging; for an historian cannot read everything. But it must be said that mankind give little encouragement to candor on this point. Hallam, in his
History of Literature, states with his own usual instinct of honesty every case in which he depends upon others: Montucla does not. And what is the consequence?—Montucla is trusted, and believed in, and cried up in the bulk; while the smallest talker can lament that Hallam should be so unequal and apt to depend on others, without remembering to mention that Hallam himself gives the information. As to a universal history of any great subject being written entirely upon primary knowledge, it is a thing of which the possibility is not yet proved by an example. Delambre attempted it with astronomy, and was removed by death before it was finished,[[348]] to say nothing of the gaps he left.