I have read over Mr. Keiths paper carefully, and endeavoured to make out his meaning. He has indeed expressed himself so very loosely that it is not easy to say what he intended. One thing however is clear, that he meant to depreciate the Rod-pendulum; and this he has done in a manner that does no credit to his Candour or Abilities. We have only his assertion, unsupported by any thing that can be called reasoning, “That it is impossible to fix an invariable or nearly equal measure from the Rod of a Cylindrical Pendulum”; and he barely hints at an Objection which you may remember I made to that Pendulum, and which I take to be far the most important, I mean the difficulty of making it precisely of the same thickness throughout. As to the other difficulty, which he likewise slightly mentions, the placing one extremity of the rod exactly in the center of motion, I conceive every workman will pronounce it too trifling to merit attention.

Mr. Keith must certainly have wrote down his very first thoughts on the subject, which he never afterwards considered or corrected. For I think I never saw so much no meaning, so ill expressed and in so few lines as his paper contains. He observes that “the length of Mr. J—s Standard is not the length of the seconds pendulum.” I believe no one ever said it was, in the sense in which Mr. Keith sometimes uses the word pendulum, and sometimes does not. He adds, “the length of the pendulum has nothing to do with the whole length of the rod.” This I conceive is not strictly true in any sense. The length of the rod will not in all cases determine the length of the pendulum, but all other circumstances being given the whole length of the rod does determine the length of the pendulum. He says, “it is the distance between the point of suspension and the center of Oscillation which the Philosopher calls a pendulum.” A Philosopher might indeed call that distance the length of his pendulum, but I conceive he would require something more than pure space to give the name of pendulum to. All this is, however, perhaps nothing more than cavilling at some expression the Secretary had used. For instance, he may have called his cylindrical Rod simply a pendulum, instead of a Rod-pendulum.1

Mr. Keith says, “if a Rod-pendulum could be made of equal thickness, and density of matter, and vibrated from the very point of suspension in a Cycloid and in Vacuo, the true length of the pendulum would be exactly two thirds of the distance between the two extremities of the Rod.” This I think compleats Mr. Keiths period, but it is surely very imperfect. For, not to mention the Burlesque Idea of making a rod vibrate in a Cycloid, or the inaccuracy of not stating one extremity of the rod to be at the point of suspension, it is egregiously false unless the thickness of the rod be limited, which he has not done. The next paragraph far exceeds all that this gentlemen has said besides in absurdity; it is as follows. “If this rod were not very fine, and if the center of motion were not at the top of the rod, or point of suspension, the length of the pendulum would be two thirds of the distance between the center of motion and that of Oscillation.” That is, the length of the pendulum would be two thirds of what Mr. Keith had, a little before, defined to be the length of the pendulum. But this wonderfull passage may be more easily reconciled to truth and common sense than Mr. Keith himself would perhaps imagine. It is but striking out the little word, not, twice, and writing the lower extremity of the rod instead of the Center of Oscillation; and with these alterations the sentence is unexceptionable.

Mr. Keith now seems to think he has demolished the rod-pendulum; having shewn it to be useless for the purpose intended; but before he quits the subject adds, “For, measured from its extremities, a second pendulum is of all possible lengths, from 39 to 58 ½ inches.” How is it possible for us to guess what kind of pendulum he means in this place? Is it the Ball-pendulum, or is it the cylindrical Rod-pendulum? The assertion however is not true; or at least is very inaccurate, of any kind of pendulum whatever. According to his own definition of the word pendulum, a seconds pendulum can be neither more nor less than 39 inches and a fraction. We must therefore suppose that he uses the word in a sense similar to that for which he blames Mr. J. for using it in. There are I think but three kinds of pendulums which he could have had in view. If he meant any other he ought certainly to have described it. The first is the Ball-pendulum in which the weight of its slender rod is not considered. The limits of its length, measured from one extremity to the other, I find to be 39.1 and 55.9 inches. The next is the Simple cylindrical rod-pendulum; which, its thickness not being given, may be of any length whatever under 58.7 inches. But if the thickness does not exceed half an inch its limits will be nearly comprized between 58.7, and 58.7005, the difference being about 1/2000 of an inch. The third is the pendulum compounded of both Ball and rod, and by varying the proportions between them it may be made of any length whatever. As to what Mr. Keith says of the Monstrous Error in the Cubic foot derived from the Rod-pendulum, he certainly meant nothing more than that the difference between it and the English cubic foot is very considerable, and I believe nobody ever thought otherwise, or asserted the contrary. I am, Dr. Sir, your most obedt.