SEA-COAST MORTAR FIRE.

66

By J. T. HONEYCUTT, FIRST LIEUTENANT, FIRST ARTILLERY, In the May-June 1897 Number of the Artillery Journal is printed by permission of the Major-General, Commanding the Army," the Report of a Board "Convened by the Commanding Officer, U. S. Artillery School, to consider and report upon a paper by 1st Lieutenant John T. Honeycutt, 1st Artillery."

The subject of the paper referred to was "Parallel and independent fire, with special reference to mortar batteries."

The paper referred to stated: "The demonstration given below was prepared more than four years ago, being intended to supplement a different one of the same general principle, which latter demonstration, though never disproved, had been officially discredited by high authority.

"The demonstration here given I sent at the time to Professor Merriman of Lehigh University, author of a standard work on probabilities, and one of the foremost mathematicians of the country, requesting him to make a careful examinations of it, and inform me of his opinion as to its soundness. This he did and pronounced it to be correct.

The adverse opinion which I then held, and expressed, in reference to the proposed method of firing, and to the form of mortar battery which seemed to have been founded on this method of firing as its corner-stone, I have never seen any sufficient reason for changing; although I fully recognize that the defects of the system were greatly exaggerated in the paper referred to."

Further on the paper stated: "The "probable number of hits" and "the probability of at least one hit" are two distinct mathematical questions, and it is only the first of these two questions that my demonstration, the one referred to above and the one given below, deal with."

The conclusion drawn from my demonstration was expressed in italics as follows: "The probable number of hits from a battery of mortars will be greater if the mortars are aimed independently at the target than would be the case were the mortars aimed by the parallel method."

My proposition was stated mathematically by the inequalily:

in which ' &c., represented the probabilities of hitting by the individual mortars when aimed by the independent method; and P1 &c., the same by the parallel method.

Conclusion 10 of the Report of the Board reads:

10. The application of these principles to Lieutenant Honeycutt's proposition shows that the expression C+ C+... >P1 + P2 + . . . is not generally true, and that the relation. between Q and P depends upon circumstances."

And referring to a table in which the results of the investigations of the Board are summarized, page 344, the report reads: "The table is based on the assumption that the designated position of the target and the center of impact coincide,—the most favorable case for the independent method. Column 5 shows that even under this most favorable supposition the independent method will be inferior in probable number of hits to the typical parallel method, in 6.72 cases out of 7.72."

After learning what the Report was, by seeing it in the Journal, and thoroughly believing my proposition to be absolutely true, I sent copies of the Journal to a number of gentlemen of high authority on the subject of probabilities, and asked for their views on the Report. Among others I sent copies to Professor Merriman of Lehigh University, and Prof. Johnson of the Naval Academy, who are authors of standard works on probabilities, and who were recommended to me by the Secretary of the American Mathematical Society as being of the highest authority, the other two recommended being Prof. Simon Newcomb, and Prof. Woodward of Columbia University. The latter two have informed me that they could not give the necessary time to the subject.

Professor Merriman writes: "I find no occasion to change my opinion regarding the correctness of your paper," and, after a critical survey of the Report: "It is thus seen that the Board has deduced nothing which reflects upon your proposition proving that the probable number of hits by a volley from the independent battery is greater than that by a volley from the parallel battery."

Professor Johnson writes: "I have not been able to give such a study to the subject as would enable me to criticise the Report's conclusions, except as to your actual proposition as stated, in which you are clearly right. Q is the maximum of P, because is the maximum of P. * * * Thus the Board by

adopting your notation put themselves in the position of main taining a proposition 'simply absurd' as you say." The quotation "simply absurd" was from a short type-written paper I had prepared in reply to the Report.

The eminent astronomer and mathematician, Professor Holden of the Lick Observatory writes: "As I understand the case your method is entirely correct. It is also entirely general and your conclusion (in italics at the foot of page 289) is established mathematically."

Lieutenant Harlow, 1st Artillery writes: "I have been over the article on mortar firing in the Journal and believe fully that your proposition as I understand it, viz. : that the independent method will in the long run give the greater number of hits, is mathematically true."

In view of the weight of authority given above, I offer no further argument in support of my proposition as stated.

The Report would seem to indicate that the Board had in mind the "probability of at least one hit in a volley," though the Report nowhere says so.

The figures of the Report do not give in any case the relative "probability of at least one hit" but the ralative "probable number of hits" if, the ship is in a certain position with respect to the center of impact. This if as one of my high authorities wrote me, does not belong to a case of a priori probability. We do not know before firing what the position of the ship will be with respect to the center of impact. In firing a single shot the probability that the ship in any one position will be hit is, as the same authority writes me, the product of the probability that the shot will strike in that location multiplied by the probability that the ship will be there. This latter probability is indicated in the Report by the ordinates of the range-finder curves. The Board nowhere multiplied by this factor. Consequently the Report gives just as much weight to a probable hit within a rectangle equal to the deck of the ship at a long distance from the center of impact as to a probable hit within such rectangle at or near center of impact. This could be justified only under the supposition that the ship is no more likely to occupy the position. of the rectangle at or near center of impact than it is to occupy a rectangle remote from this center. This is equivalent to supposing that there is no accuracy at all, but that the firing is entirely at random; in which latter case the range-finder (or accuracy) curve becomes a straight line; and the ordinates, our accuracy factors, become all equal.

In all the tables accompanying the plates the numbers should have been multiplied by the corresponding ordinates of the rangefinder curve; since the probability of hitting the ship in any position is equal to the probability of the shot's falling in that position multiplied by the probability of the ship's being in that position.

Also, since the aim of the Board was, as it appears, to show the relative probabilities of "at least one hit in a volley" for the two batteries, there should have been made further computations to show in each case the probability of "at least one hit."

Failing to do this the Board had no figures from which to draw conclusions either as to the "probable number of hits" or the "probability of at least one hit."

Any two corresponding columns of the tables in the Report if carried out to infinity, will when added up give equal sums, since the areas of the curves are equal, each containing all the errors. But after we have multiplied through by the ordinates of the range-finder curve, the Q columns give the greater sums because the larger numbers at the heads of the columns would be multiplied by the larger range-finder ordinates, thus proving at once my proposition of the greater "probable number of hits" to be in favor of the independent battery.

We will now give some special consideration to the subject of the probability of "at least one hit in a volley." We should not sacrifice every other consideration to this one. To do that would be equivalent to acting under the supposition that one hit is as good as two or more, or that a single hit will always disable the ship. Even though this "at least one hit" should be our main object, the Board has failed to bear in mind an evident truth; namely that while to get the best effect, some scattering of the projectiles is necessary, there is a limit beyond which the effect. ceases to increase and begins to decrease. For example if there are left at and near the center of impact vacant spaces larger than the deck of the ship, it is evident that the scattering has become too great. We would then be leaving places of safety for the ship near the center of impact for the sake of a possible hit in a position more remote from the center, where there is a smaller probability of the ship's being than there is of its being at the center, however small the probability in the latter case may be.

This is analogous to the case of the shot-gun. Up to a certain Journal 19.

degree the effect of scattering increases with its extent, but it may become too great even for the very poor marksman.

By substituting in the probability integral successive values for the probable error in range, I have learned that if this probable error be as great as 110 yards there will be for the range zone extending 122 yards on each side of the center of impact a probable number of hits less than one. This shows that with the scattering corresponding to this error in range we are leaving vacant spaces at and near the center greater than the width of the target, and therefore that for the lateral position of ship we have passed beyond the limit of maximum advantage of scattering, however great the inaccuracy of aim may be. I have selected this case of extreme inaccuracy merely for convenience to make sure of getting a case where the scattering is too great, which case I wish to use for a purpose made evident further on. With anything like a reasonable degree of accuracy this limit would of course have been far smaller. With perfect accuracy we would require no scattering, and as we pass from perfect accuracy to the greatest inaccuracy, so long as the firing is not at random, we should for a target 25 yards wide increase our scattering from that determined by the probable error o in range to that determined by the probable error 110 yards.

We at least know then that for all cases of probable error as great as 110 yards the independent method would for the lateral position of ship give a greater probability of "at least one hit " than the parallel method, since the latter method gives the greater scattering. Yet in all these cases the probability curves would, as in the cases of the groups of shot treated in the Report, intersect at some distance from the vertical axis. From the center of impact to the foot of the ordinate of intersection of the curves there would be for all positions of the ship a superiority on the side of the independent method, and for all positions of the ship beyond this point, extending indefinitely, or at least as far as there is any possibility of a shot's going, the superiority would be on the side of the parallel method. Therefore the method of reasoning employed in the Report would here, as well as in the cases dealt with, show a superiority for the parallel method, the probability of the ship's being between center of impact and the point determined by the intersection of the curves being very small, and the inaccuracy being great at the long range necessary to produce so great a shot dispersion.

We see then that the method employed by the Board would