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In recent years there has been much discussion on the subject of the accuracy of our naval guns at long ranges, brought about not only by the natural increase in practicable ranges, due to the development of guns of larger caliber and greater velocity, but also to the laudable desire to push the effective battle range to the extreme limit possible.

Undoubtedly greater accuracy of weapons and superior skill of personnel have their most telling effects at long ranges, for at these ranges superior accuracy and skill affect most decidedly the relative number of hits obtained.

Some 17 years ago, I undertook a mathematical investigation to ascertain the value of the advantage accruing to a naval force engaging an adversary numerically equal but so placed that a portion of his force was masked or beyond effective range. The result of this investigation showed that the advantage was measured, not by the ratio of the number of the opposing ships actually engaged, but by the square of this ratio. Here was the first, statement of the so-called " n square law," a law discovered independently some years later by an English writer.

The application of this law to long range fighting is this: Suppose our greater accuracy of weapons and superior skill makes us feel confident that at a certain long range we can

make 4 hits to our adversary's 3. Then if we engage a numerically equal force at that range our superior hitting ability makes this numerically equal force equivalent to one only as large possessing, ship for ship, a hitting power equal to our own. Therefore, according to the n square law our chances of victory are not as 4 is to 3, but as 16 is to 9. Moreover this law shows that in this case the superior force after annihilating the inferior would have left the equivalent of V of its original force.

Then square law is based upon the material damage inflicted and sustained and does not take into account the moral effect due to the knowledge of the inequality between the damage inflicted and that sustained, a knowledge giving rise to a depression on one side and to a corresponding elation on the other. Courage and grim determination may serve to minimize this moral effect, but in the inanimate vessels themselves is found no counterbalancing force to offset the physical advantages conferred by superior accuracy of weapons coupled with greater skill of personnel.

The foregoing brief discussion suffices to indicate the importance of attaining the maximum effective battle range. It may be well, here, to determine generally what may logically be called the maximum effective battle range. At short ranges the target presented by an adversary is wholly a vertical one and, in estimating the accuracy of our weapons, we are wholly concerned with vertical and lateral errors. As we increase the range we reach a point where the tangent of the angle of fall is approximately equal to the enemy's freeboard divided by his beam. At this point the target may be regarded either as a vertical or horizontal one. Beyond this point, the target is largely a horizontal one and we are concerned with range and lateral errors only in estimating the accuracy of our weapons. These errors increase as some function of the range, whereas the dimensions of the target remain fixed. It is obvious that with a given accuracy of weapons and skill of personnel a range may be established beyond which the probable number of hits obtainable by the expenditure of a ship's allowance of ammunition is insufficient to inflict serious damage upon an adversary.

The range so determined may be regarded as the extreme effective battle range. The accepted maximum effective battle range should be as little inside this extreme range as our judg

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