The slide rule is gradually coming into its own on board ship. The growing number of technical postgraduates that are joining the fleet is one reason. Modern tendency for the average officer to dip into mathematical computations formerly left to specialists is another. Gunnery and engineering competitions provide still another.
Opposition to the slide rule has been most persistent by those resenting intrusion of fragile devices upon the sacred precincts of the bridge where the “seaman’s eye” holds sway. Justly so, too. For what assistant navigator has not thumbed a pasteboard sponge that was once an omnimeter before the shower came up?
But today the chief objection to this form of mechanical calculator really seems to lie in that common but trifling difficulty: how to fix the decimal point. This fact was forcibly brought home to the writer when three brother officers derided him for using a slide rule under circumstances that would have involved much tedious labor had the task in hand been done by outright multiplication. Such an attitude was not at once clear. Yes, they all knew how to manipulate the “slipstick.” However, so long as there was a supply of scratch pads and pencils on hand they preferred old-fashioned methods. But curiosity was not to be denied. What was the real reason behind their distaste for this indispensable labor-saving device that no officer should be without? Finally the secret was divulged. It sounded about like this: “Yes, I can use a slide rule. Yes, it is accurate and easy to use, and convenient. But it doesn’t save time.”
“It doesn’t! Why not?”
"Because every time I work anything out I have to make a rough multiplication with pencil and paper to find where to put the decimal point, which takes about as much time and invites as many mistakes as if I had done it in the old-fashioned way in the beginning!”
That sounded plausible. Also it was very enlightening. We tried other officers; common or garden naval officers who didn’t pretend to be ordnance experts or engineering sharks or radio physicists. They were unanimously divorced from the slide rule, and all named the decimal point as co-respondent.
We consulted the senior draughtsman and technician of the local establishment.
“The book tells you how to fix the decimal point,” he replied glibly to our inquiry.
“Yes, but how do you find it?”
“Oh, I just make a simple computation on paper: multiply the whole thing out in round numbers. Takes a little time. But—”
And so did his assistant; and the rest of his gang as well.
At which point we began to wonder if the whole Navy wasn’t divided into two classes: Feeble ones who, having long ago tired of chasing the decimal point, frankly admitted the mediocrity of their erudition and abandoned slide rules forever; and stubborn ones who, through a sort of perverse vanity, clung to the slide rule, but who still covertly figured with pencil and paper where to put that elusive dot which, wrongly placed, may well mean death in a turret or engine room.
Further investigation proved that such a grotesque state of affairs really does exist, at least among officers of and below the rank of commander; hence the propriety of herein calling attention to the fact that there is a short specific rule of thumb for placing the decimal point in slide rule computations.
That there may be no misunderstanding we must say here that this rule is in use at both the Westinghouse and General Electric Company laboratories, the U. S. Weather Bureau, and other places where elements of time and accuracy tolerate no “cuff computations.”
The rules given in the slide rule manual for fixing the decimal point are probably never used because they are too long, too difficult to remember, and too empirical or haphazard for intermittent and accurate application. Psychologically they are weak, in that they lack that factor of association which helps so much in memorizing.
The rule of thumb to which we refer as an easy answer to the decimal point conundrum is as follows;
Add characteristics algebraically. Correct this sum: plus one going right across the rule’s mid-point; minus one going left. The result is the characteristic of your answer.
This rule is even simpler than it looks. It requires no effort to remember. And the adding and correcting indicated are all easy mental processes during the actual multiplication.
Take the following:
+I +I +I +I
71 X 21.4 X 35 X 17
------------------------ = 21.8
8.5 X 42 X 5.8 X 20
O —I O —I
Over each figure has been placed the logarithmic characteristic with suitable sign. Ordinarily this would not be done; the addition would be made simply by inspection. The algebraic sum of them all is 2. Now as we multiply 71 and 21.4 together on the rule we cross the mid-point going right. Mentally we add 1 to our sum of the original characteristics, making 3. Dividing by 8.5 we cross to the left, making it necessary to subtract 1 from our sum, and once more leaving us 2. Multiplying by 35 and dividing by 42 we remain in home territory, so to speak. So our 2 remains untouched. Applying 17 doesn’t change the situation either. But when we intrude the divisor, 5.8, our inner limb begins to stick out over the slide like a sounding boom. So in order to give the remaining 20 a chance to get in its insidious work we resort to the 1 that still remains aboard, namely the mid-point. But, by the rule, this is neutral ground, so when we divide by the 20 we are moving from right to left and so must substract 1 from the characteristic we have been carrying along; 2 it was, and 2 minus 1 is 1. Or the characteristic of the answer is 1.
Putting it more briefly: We glance at the problem. Rapidly our eye runs along the upper figures: four i’s are 4. And, along the lower row: 1 and 1 are 2. 2 from 4 leaves 2. With that “2” in mind we take the rule and begin. We cross right: 2 and 1 (for crossing) are 3. We cross left: 3 minus x (for crossing) are 2. The 17 forces us to the mid-point. Whence we go left, and again: 2 minus 1 (for crossing) is 1. 1 is our characteristic. Whence our decimal point must free two digits as integers. Take your rule and try it.
It may be superfluous to explain why the rule works. Yet without briefly adding its derivation it may appear unsound.
The slide rule is, of course, just an adding machine, but it is calibrated in logarithmic dimensions, so when it adds it puts these dimensions together, thereby multiplying or dividing the numbers they represent on the face scale.
At risk of sounding puerile we remind the reader that a logarithm is an exponent. For logs based on 10 the logarithm of 100 is 2 because 102 equals 100. Unhappily there are many intermediate numbers. Whence we encounter logarithms that look like this: 2.2638. The 2 is the characteristic and the .2638 is the mantissa. If the logarithm of A is 2.2638 then 10 raised to the 2.2638 power equals A.
This bit of high school mathematics is put in deliberately, for to understand the decimal point rule clearly one must visualize the slide scale as divided into lengths corresponding to the mantissas of the numbers scribed upon it. In functioning the slide rule adds or subtracts only these mantissas, but in ordinary logarithmic work the characteristics must also be added or subtracted, and if the sum of the mantissas is greater or less than 1 that 1 is added to or taken away from the characteristic.
The sum of the original characteristics is taken quickly and easily by inspection as was shown above. But the change due to mantissas is not so simple. It varies every time their sum passes 1. Fortunately the sum of two mantissas can never result in a mantissa larger than 1. They wouldn’t be mantissas if it could, and fortunately, too, we find that every time their sum does exceed we cross the index or mid-point on the slide rule as a reminder.
Or, putting it another way, every time the left half of the scale is used in multiplication this means that the sum of the mantissas is less than 1; i.e., there is no change in the characteristic indicated by the original sum. In division use of the left hand scale indicates that the divisor is smaller than its dividend; another case of no change. Conversely, use of the right half of the scale means that the operation has been great enough to alter the original characteristic by plus or minus 1, depending on whether we are multiplying or dividing.
The rule is a good one. Try it. And don't be discouraged by the prolixity of its explanation, which bears no relation to the simplicity of its application.