328 Mr Watts" Remarks on Captain Kater'^s 
But I ought to state, that Captain Kater has been very for- 
tunate in committing the error which relates to the correction 
due to the amplitude of the arc of vibration, as it is nearly com- 
pensated by a contrary error which he has fallen into, by not 
employing the appropriate formula for the correction of the arcs 
of vibration ; and that while I take upon me to expose the er- 
rors of others, 1 ought in justice to acknowledge my own. I 
allude here to a correspondence which I have held with Mr 
Gilbert, relatively to the pendulum experiments of Captain Ka- 
ter ; and in which I also employed a formula for the correction 
of the arcs of vibration, that does not give correct results. But 
in doing this I was misled by the authority of M. Biot, imagin- 
ing that his theorem was the right one, because he had, at least 
apparently, deduced from it, by way of corollary, the formula 
of the Chevalier Borda, which is known to be correct. But 
there are fallacies in some of the steps of Biot" s investigation ; 
and it will be found, by a comparison of the following demon- 
stration Avith that of M. Biot, given in his Astronomie Physique^ 
tom. hi. pp. 169,-172, 2d edition, that this apparent agreement 
is owing to a compensation of errors, and that his theorem Avili 
not give correct results. But by treading AAnrily in his steps, 
and carefully aA^oiding his mistakes, 1 have deduced a formula 
similar to that delivered by Borda, Avithout demonstration, in 
his fine memoir on the measure of the meridian, as the diffe- 
rence between his theorem and mine Avill be found to be no- 
thing in its practical applications. 
I am not aware that a correct demonstration of this formula 
is to be found any Avhere else. I lately, indeed, saAV a the- 
orem Avith Captain Kater, Avithout the demonstration, which 
he gave to a friend of mine, Mr George Harvey of Ply- 
mouth, AA'hen Ave Avaited on him in London, but I think there 
is some difference between them. As I state this, hoAvever, 
merely from memory, I cannot speak Avith any degree of cer- 
tainty on this head. 
Before I proceed to give the demonstration of my formula, it 
may not be amiss to premise the following extracts from Captain 
Kater"s paper, (see the Beport ordered to be printed by the House 
of Commons, p. 10.), as they Avill serve, not only to introduce 
the subject in question, but also to throAv additional light upon it. 
