1. One is often struck by the fact that nomograms or alignment charts are coming more and more into general use in engineering subjects. In England they are frequently used in gunnery work. In France they are particularly popular. In the U. S. Naval Service we are beginning to see them. Usually, since they are so simple to work and since they are presented without explanation of their theory, those who use them do not learn the theory of their construction. One may not know where to turn to to study up the matter and may hesitate to show apparent ignorance by inquiring. Since occasionally one might want to make an alignment chart of his own for some special use, it is thought that an elementary discussion of the simplest good all-around form of alignment chart may not be out of place. It is hoped that they will in time be more commonly used in our naval service.
2. After looking over various books in which the subject is treated, we have here put it briefly in the way which seems clearest to us. "Graphical and Mechanical Computation" by Lipka, John Wiley & Sons, is suggested as very suitable for reading by naval officers who may be interested in going beyond this elementary discussion.
3. Suppose we want to multiply two variables x and y to get a product z. The slide rule uses logarithmic scales and affords a ready solution. But consider also the following alignment chart construction (Fig. 1):
At equidistant points a, c and b on the horizontal base line, erect perpendiculars and lay off on the perpendiculars from a and b scales running simply as logarithm x and as logarithm y. Then if using a straight-edge we connect any point on the x scale as h with a point on the y scale as d, where will the straight-edge cross the z scale? Obviously at a point f, which is the mean height of ah and bd. In other words, cf would be
It we graduate the scale of z on cb as half of the ordinary logarithms, z will be read off as the product of x and y, for log z= log x
+log y, and, by marking off the z scale as one-half of the logarithms of z, we get around the fact that the mean is only half of the sum.
4. Now suppose we had wanted to get not z=xy, but z=x2y.
Then log z=2 log x+ log y. We want then, in summing up the x
and y logarithms, to give double weight to log x. We can do this by making the point c twice as close to a at the foot of the log x scale as it is to b at the foot of the log y scale (Fig. 2):
Then 
so this time we must graduate the scale for z as 1/3 of the logarithms of z.
5. So with simple logarithmic scales for x and y, i. e., with both drawn natural size, we must have the ratio ac/cb the same as the desired ratio of the exponents of x and y:
6. But in going from simple products to products involving powers of x and y, there is an alternative method. Instead of sliding c over twice as close to the x scale to give logarithm x twice the weight in the sum, we may get this same effect by leaving c still in the middle of ab, but making the x scale twice as large as formerly, or as 2 log x. In this case (Fig. 3) | 
and we have then only to graduate the scale as half the logarithms of z to make this sort of alignment chart work.
7. The foregoing is not a proof but is just a preliminary glance over the matter to illustrate the nature of the nomograms so as to have an understanding of them in mind while proving the constructions for the more general forms like kxlymzn, etc.
8. Such alignment charts as these are really only logarithmic scales summed up graphically with the weights of the exponents of the variables or, what is the same thing, with the values of the coefficients of the logarithms in the sum of the logarithms, taken care of by either of the following methods:
(1) A shifting of the position of the intermediate scale z, or
(2) An increase in the size of units used in plotting the logarithmic scales of one or both of the variables x and y.
Or, we may use both (1) and (2) at the same time, especially if it is desirable to make the scales of more convenient lengths. If, for instance, x has only half the range of y, it would be better to use the method of Fig. 3 and plot the logarithm x scale double size rather than to leave its scale of inconveniently short length while merely shifting the scale closer to it (as in Fig. 2). Convenience of scale length is a governing factor and one can suit himself.
9. Now about any constants in the formula. It seems that since we are dealing with logarithmic scales, everything we have said still holds even if constants come into the formula. For a bodily vertical displacement of the logarithmic z scale will take care of all the constants connected with z. We will find that we can lay off the x and y scales so that the extreme values are at the top and at the bottom (with each scale 10 inches long); next carefully locate where the z scale comes between the other scales. Then we need construct only one point on the z scale to get the proper origin for the scale of z. With this and with a knowledge of how far apart the graduations on the logarithmic scale should be spaced, the graduations of the z scale can be put in place.
10. If one wishes to make an alignment chart for three factors he can first make one for two factors (scales a, c, b in Fig. 4):
The point P on the c scale (often called the "support") gives the product of the first two. Then this is considered as the start of another similar nomogram, the scale at d carrying the next factor z. The product of xyz is read off of scale e. And one can extend this process to any desired number of factors. Support scales need not be graduated as the partial products shown on them are not needed in the work.
1. By adjusting the horizontal intervals between the scales and by adjusting the size of the logarithmic graduations on the scales, x, y and z can be given any desired exponents and the scales themselves can be made any desired length (10 inches for all is a good value). We can have division alignment charts as well as multiplication alignment charts, for if in Fig. 1, where z=xy, we reletter interchanging y and z, we have z=y/x. Or we can lay a logarithmic scale off the other way, i. e., from top to bottom, to take care of a negative exponent. In fact one can handle any expression like Kxlymzn, etc., where the exponents are fractional or negative, or anything he likes to make them, and where the number of factors like xyz is as great as one pleases.
12. There are literally hundreds of different forms of alignment charts. But when studied, this particular form proves to be of extreme generality. It has therefore been selected as the illustration with which one can best introduce the subject to those unfamiliar with alignment charts and their construction.
13. If we have three parallel axes as ah, bd and cf (Fig. 5) carrying scales for x, y and z, it is of great interest to know how the lengths of the intercepts are related in terms of the spacing ratio m1/m2 between these axes. For with this knowledge we can give what relative weight we desire to the other scales in reading off a weighted sum of the x and y logarithms on the z scale between them. We can get the idea from rereading paragraph 4, but, to show it more generally, we have, in Fig. 5, made acb a transversal instead of a horizontal base line. We will be interested to find how this proof shows the different scales can without complications be moved up or down to get them opposite each other for convenience.
So if we plot logarithms their normal size in both scales in this way, then in getting a sort of weighted mean on the z scale we find we have given a relative weight to each logarithm proportional to the interval between the z scale and the scale of the other variable. (In Fig. 2, for instance, a weight of 2 was given to log x by making the interval between the z and y scales twice the interval between the z and x scales.) We will have weighted the logarithms inversely as the spacing ratio.
16. In general, however, one would not plot log x and log y normal size, but would expand or contract to make them fill up the desired lengths. Then one can select the spacing ratio m1/m2 for locating the z scale in such a way as to take care not only of the scale enlargements used, but to take care, at the same time, of the exponents desired for x and y. To get at this sort of construction we may first place equation (3) in proper form for direct addition of scales by dividing through by m1m2 to get the following:
Then the intercept on fc (the z scale) will represent the sum of the intercepts ah (on the x scale) and db (on the y scale), provided we select, on the appropriate scales, moduli proportional to these denominators in (4). If this is done, everything is kept clear and we will be virtually adding the logarithms of whatever powers of x and y we select and getting direct answers in the product.
17. So, therefore, after plotting log x on the x scale to some convenient size which will make the scale the desired length, we examine to see how many times we have enlarged the logarithm of the function of x to increase it so as to give the scale actually laid off. We call this modulus m1. For instance, if we wanted to have (x)1/3 appear in the product and have multiplied log x by 5 to get a convenient 10-inch length for the x scale, then m1 is 15. Similarly one obtains m2. Then m3 the modulus for the z scale has to be m1m2/m1+m2. This defines the amount of enlargement of log z on the z scale. The position of the origin of the z scale depends upon the constants. So if we construct one point and calculate the origin of the z scale graduations, the positions of the divisions on the z scale are completely defined. Also ac is abm1/m1+m2. and cb is abm2/m1+m2, so the position of the z scale, as well as the size and location of the divisions on it, is now determined.
18. This is all there is to the construction of such alignment charts but, as in navigation, it is best to develop the ability to apply the theory by including after the explanation an illustrative practical example.
19. Example.—
and let us say x varies from 1 to 8, y from 30 to 40 and z from 100 to 3500. First it will be determined what multipliers are needed for the logarithms to give 10-inch scales:
This can be done with sufficient accuracy on the omnimeter or on the slide rule. It will be seen that if we had used the same multiplier for log x, log y and log z, we would not have efficiently utilized the full available length of all the scales. It is not necessary to insist upon all scales being precisely 10 inches long (or any other exact length for that matter). 11, 80 and 6 are sufficiently accurate and are the more convenient multipliers which will be used.
20. Sometimes specially printed logarithmic paper scales are used for putting the graduations in place. These, by folding, can be made to show logarithms in any degree of enlargement, but they are not quite so practicable as they sound. Also they are not necessary and it will be assumed that the reader has none but will use computation instead.
21. There should be, say, 20 divisions calculated on each scale. For each variable the logarithms of about that number of equidistant values are picked out and the work carried through in the manner shown by the following skeletonized tables. In the third table it is desirable to insert a certain number of extra values in the part of the scale where the spacing is inconveniently large. The fourth column is obtained by subtracting the initial value in the third column from each of the other figures in that column.
23. The next step is to arrange the horizontal spacing of the scales. We will place the x scale on the left, the y scale 7 inches and the z scale 8 inches over to the right. This sort of wide spacing of the y scale gives more room between scales in working the alignment chart and makes the straight-edge cut intermediate scales less obliquely. The support scale should be 7 X m1/m1+m2 = 2.03 inches to the right of the x scale. The remaining space over to the z scale is 5.97 inches. So the w scale is located 5.97 x m3/m3+m4 =4.08 inches to the right of the support scale, or 6.11 inches from the extreme left.
24. We can therefore lay these out as in Fig. 6. If we calculate the point for the bottom of the w scale (x=1, y=30, z=100) the value of 1, 048, 000 is obtained. Its logarithm, times 5.59 or 33.65370, must therefore be subtracted from m5 log w in laying off the z scale. If we work out the top of the z scale from the maximum values of all the variables (x=8, y=40, z=3500) w=51, 742, 500 is obtained. It should be 9.48 inches up the scale from construction, and this checks with the value obtained by computing it like the other points on the w scale. A tabular form, similar to those of paragraph 21, is used in computing the w scale.
26. In using alignment charts the full lines need not be drawn across in the construction, as only short pencil marks where the scales are crossed are necessary. Or else we may use a pin pushed into the paper to mark accurately where the straight-edge crosses a scale. Fig. 6 is necessarily shown reduced in size. Practically any requirements in the way of alignment charts can be readily met by studying Lipka's book or other similar works, but it is thought desirable to confine this article to a description of and an introduction to the subject and to give, though in considerable detail, but a single illustrative example.