[[p. 305]] Sir,--As the experiments made by me at the Old Bedford River are elaborately criticised by Mr Carpenter's report, may I be permitted to point out the fallacies and misstatements with which it abounds, and which may perhaps confuse and mislead some of your readers who are not very conversant with practical geodesy.

    1. Mr Carpenter defines a "straight line" in a manner totally new, as being absolutely identical with a "level line," thus introducing at the outset confusion of terms, and rendering all clear reasoning impossible. It is an abuse of the English language to confine the general term "straight line" to the one special meaning of a "level or horizontal straight line."

    2. Mr Carpenter objects to the value of the view in the large telescope, "because it showed but two points, when a comparison had to be instituted between three;" but he omits to state that the telescope itself was placed accurately at the third point, just as was the spirit-level telescope--to the view shown by which he makes no such objection.

    3. He objects that the telescope was not levelled, and goes into a long argument, in which the words "straight line" occur four times, and which, whatever meaning is given to that term, is utterly confused and misleading. He says that I intended to prove that the central signal was five feet above a "straight line" joining the two extreme points. This I both intended to prove and did prove, using the word "straight line" in its proper sense; but Mr Carpenter should not impute to me the absurd mistake he makes himself of thinking that there is, or is supposed to be, a rise above the level at the centre point.

    4. Mr Carpenter's "argument" exhibits a total ignorance of the use of the spirit level, and of the simplest principles of optics and geometry. We have three points taken, at equal distances, above what Mr C. maintains to be a true horizontal straight line--the surface of standing water. The eye is placed at one of the extreme points, and, looking at the other two points, they do not coincide, as they must do if in a straight line with the eye. Again, the cross hair in the telescope of the spirit level marks the direction of "the straight line at right angles to the plumb line at the point of observation" (as Mr C. very accurately defines the true level); and as the middle signal appeared considerably below this line, that alone proved that the water surface was not truly level. The distant signal being apparently as much below the middle signal as that was below the cross hair, is absolutely inconsistent with the three being in any straight line, still less with their being in the Carpenterian "straight line," but is perfectly consistent with the three being points in a circle of about the assumed radius of the earth. This is a question of elementary geometry about which there can be no dispute. I may add that the fact of the apparent "equality " of these distances (so dwelt upon by Mr C. in his "argument"), and the views from both extremities of the six miles agreeing so closely, both prove the very great accuracy of the level used, and that it may be depended on to show that the surface of water does really sink below the true level line in a continually increasing degree as the distance is greater; but the proof of convexity in no way depends on this accuracy, as it was shown still better by the large telescope without a spirit level.

    5. Mr Carpenter's objection No. 3 is answered above. No. 4 is an entire delusion. No. 5 is an assertion destitute of proof. Nos. 6 and 7 are verbal quibbles with which I have nothing to do. Nos. 8, 9, and 10 rest on the fallacy of there being a "rise" shown by the observations, which is a pure figment of Mr C.'s brain. I have never used the word "rise" in connection with these experiments, and all the observations go to show, not a rise, but different degrees of depression below the true level line. Nos. 11, 12, and 13 are misconceptions. The curvature shown by the large telescope, according to the diagram, is about 5 1/2 ft. at the middle signal, three miles distant--equal to 11 ft. if measured at the distant signal; and the depression below the cross hair or true level line, being, according to Mr Carpenter, an equal amount, makes 22 ft. in all, leaving less than 2 ft. for refraction to bring it to the full theoretical amount, which is something less than 24 ft.

    In conclusion, I beg to state that I rely on the three views as shown by the diagrams, which substantially agree, and which demonstrate that the three points equidistant from the surface of water were not in a straight line, but deviated in a vertical direction very nearly as much as is required by the assumed dimensions of the earth; and I challenge Mr Carpenter to place three objects at equal distances apart in a true straight line (three oranges on the parapet of Waterloo-bridge, for instance), and then with a telescope at either end, in the place of one object, make the centre object appear considerably raised above the distant one. Till he can do that, all his wordy argumentation is utterly valueless.

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The Convexity of Water (S163: 1870)

    [[p. 317]] Sir,--I should hardly have thought it worth while to answer Mr Carpenter's letter in your last had you not invited me to do so, as the question of my verbal accuracy is one quite beside the main Issue.

    In Mr Carpenter's "Objections," 8, 9, and 10 (see Field, March 26), he speaks of there being "a rise" shown from the point of observation to the central signal, and argues that, if so, the point of observation must be in a depression or "circular concavity." Here then "a rise" is used in the surveyor's sense of "rise above the level of the point of observation," and I replied that I had not used the word "rise" (of course meaning in the same sense) in connection with these experiments. It is, therefore, quite beside the question for Mr Carpenter to quote me as saying that the middle signal would be seen "rising" above the others. His own diagrams show that it did so; but at the same time it "fell" below the point of observation (as every surveyor will tell him) by its being seen below the cross hair in the level telescope, allowing of course for the inverted image.

    The fallacies in the remainder of Mr Carpenter's letter have been so ably refuted (by anticipation) by your correspondent Mr J. Tanner, that I need say no more about them. I would ask Mr Carpenter, however, to state, for the information of your readers, whether the universally-accepted and only known method of deciding whether three distant points are in a straight line is true or false. That method is to place the eye (whether aided by a telescope or not) at or behind one of the extreme points, and see whether the other two or all three coincide, the nearer hiding or covering the more distant. If so, they are in a straight line. Every carpenter who looks along the edge of a floor board, every surveyor who runs his base lines across the country, every builder who sets out a long wall, uses this method. Does Mr Carpenter say they are all wrong, and that every line thus set out is a crooked or curved line? If so, let him prove this elementary point by experiment and diagrams, and thus found a totally new and hitherto unimagined geometry. If, on the contrary he admits that lines so set out are straight, then the middle and end signal which did not so coincide when seen from the other end signal could not be in a straight line, or there would be two diverging straight lines terminating in the same points, and inclosing a space!

    Mr C. has confounded actual with apparent equi-distance in the field of view of a telescope, between which there is no connection, as Mr Tanner's diagrams show. If Mr Carpenter will not try any such simple experiment as I proposed in my last, I must decline to spend any more time in refuting arguments founded on total ignorance alike of facts and of geometrical principles.

    The "men of common sense" to whom Mr C. so confidently appeals are very slow in coming forward. The solitary individual he so triumphantly quotes against me (Mr Westlake) now confesses to an oversight, and cruelly deserts him. Mr Hampden, in his letter to me, continually appeals to "public opinion" as being against the fairness of your verdict. It has, however, now clearly spoken through your widely-circulated columns, and, unless he can prove that letters on the other side have been refused insertion, he would do well, as a man of honour and of sense, to bow to its decision.