"NOMOGRAPHY. - The methods of graphic calculation may be divided into two main groups. (a) Those in which a more or less complicated geometrical construction is performed for the solution of an isolated problem. Graphic Statics (see 17.960) may be instanced as an example of this group. (b) Those in which all the solutions of a formula which are likely to be required are embodied in a permanent diagram with figured scales, drawn once for all, and read simply by the intersection of lines or the alignment of points on it.
The methods grouped under (a) do not lend themselves readily to concise and useful generalization; they can in fact only be dealt with satisfactorily as they occur in direct connexion with a particular subject. Those of group (b), however, the application of which in scientific and engineering work generally has developed considerably in recent years, can be successfully generalized, and they form the subject of this article.
It was M. d'Ocagne who, in his Nomographie: Les calculs usuels efectues au moyen des abaques (1891), invented the word Nomographie - i.e. the graphical presentment of laws - to describe the theory, and the word Nomogramme to describe the diagrams resulting from the application of these methods.
The English forms Nomography and Nomogram have now come into general use with similar meanings.
Although the invention and introduction of some of the methods utilized date back to a remote period, there can be no dispute as to the predominatingly important position to be assigned to the work of d'Ocagne as far as the generalization and systematization of the modern treatment are concerned.
The exposition of the main principles given in this article follows the lines laid down in his works.
1. Notation. - Following d'Ocagne the different variables appearing in an equation or formula will be denoted by z 1, z 2, zs.
, and the letters f, g, h, with appropriate subscripts, will be used to denote functions of these variables. Thus fi, g1, h 1 , will denote different functions of z i; f 2, g 2, h 2 , different functions of z2; f23 a function of z 2 and z 3, and so on.
If now instead of the network in (a), we employ the network x =21D y= µ2V2 we obtain for the system (R) a system of straight lines /2 3.34 Cµ l ? radiating from the origin (fig. 6).
A ?
60
20 ?
600 1500 20 Fig: 6.
(c). Writing the formula log D - log V + log R - log 3'34 = 0 and employing the network x =µ1 log D y = 2112 log V we obtain (fig. 7) for (R) a system of parallel straight lines x_ _ y - - { - logR - l og3.34=o500 1000' 5.
mm.
(d) Fig. 3.
f123=0 described in 4, we took, corresponding to the variables z1, 22, evenly divided scales along ox, oy.
Suppose that instead of this we take the functional scales x= /-llfl y = l-12f2.
I 1
Instead of the network with evenly spaced meshes corresponding to the evenly divided scales previously employed, we shall now have = D = V 2 3 = R ' A logarithmic anamorphosis as illustrated in (c) is so frequently resorted to in practice that paper already ruled with a logarithmic network can be obtained commercially and is largely employed.
6. Graphic Representation of a Three-Variable Formula in Parallel Coordinates. - The preceding sections have dealt with Intersection Nomograms in which the answer is read from the intersection of lines in a point. For certain types of formulae, however, a representation is possible in which the three variables are arranged along three scales, and the answer is read by the alignment of points on these scales. Such an arrangement, an " Alignment Nomogram," is possible only when a diagram for the formula can be constructed, in cartesian coordinates, which consist of three systems of straight lines.
Defining the three systems of straight lines in cartesian coordinates by xfi+ygl+hl = o x { f ' 2+ yg 2-i 7h_2 = o x, J 3 +yg3 +/?3 corresponding to the three variables z 1, z 2, z 3, we arrive at an equation for the formula which is most conveniently expressed in determinant notation fi gl hl f2 g2 h2 - o (i).
f3 g3 h3 For further investigation it is necessary to introduce the idea of Parallel Coordinates referred to two parallel axes, so that a point is represented by an equation of the first degree.
These coordinates are defined as follows: - If a straight line MN (fig. 8) cuts two parallel axes Au, B y (A and B being the origins of the axes) in M and N, the coordinates of the straight line are N =AM, v=BN.
Any equation of the first degree au+bv+c = o will represent a point, and to determine this e point it is sufficient to know two solutions of the equation, and take the intersection of the straight lines resulting from these two solutions.
a _ b Along the axes Au, B y (fig. 9) take AQ = - a, BR= - The intersection of the straight lines AR, BQ in P will then give the point required, the point au+bv+c =0 This correspondence of points to straight lines and vice versa, according as to whether cartesian or parallel coordinates are employed for the geometrical interpretation, e. is an example of the Principle of Duality. As an alternative to a diagram composed of straight lines there is a correlative diagram composed of points, and if three straight lines intersect in a point in the first diagram, the three points in the second will lie on a straight line.
Effecting such a dualistic transformation the three systems of straight lines will now be represented by three systems of points ufi+vgi-I-hl =o ufz+vg2 +h2 =o uf3+vg3 +h3=o forming three scales arranged along a straight line or a curve, according as to whether the straight lines of the correlative system meet in a point or not,' and when three points are taken on these scales whose graduations correspond to three values of z l, z 2, z3 satisfying (r), the three points will lie on a straight line, since the correlative straight lines meet in a point.
Hence to use such a diagram (shown schematically in fig. io) we join any two values of two of the variables on their respective scales by a straight line, and the point of intersection of this straight line with the third scale gives the corresponding value of the third variable.
It will not be necessary actually to draw I / the straight line on the diagram; a piece of lz,? (22) thread stretched across it will give the alignment, or a strip of transparent celluloid, having a straight line engraved down the centre, may be employed for the same purpose.
Given a diagram consisting of straight lines only, representing a three-variable formula in cartesian coordinates, the correlative diagram representing the same formula in parallel coordinates can be constructed geometrically without knowing the analytical expression of the formula represented.
Let D (fig. i I) be a straight line of the left-hand diagram. Take a point M on this straight line whose cartesian coordinates are OH, OK.
The correlative straight line H'K' will be one whose parallel coordinates are AH' = OH, BK' = OK.
Taking in this way the correlative straight lines to any two points on the straight line D, we get by their intersec tion the point P, correlative to the straight line D.
Thus we might take BX', AY', the correlatives of X and Y, the points where the straight line D cuts the axes Ox, Oy, making AX' = OX, BY' = OY.
Proceeding in this way we can replace all the straight lines of the intersection diagram by points.
As an example we have taken the Intersection Nomogram, fig. 6, and constructed from it an Alignment Nomogram, fig. 12.
Suppose, for instance, we want to know the value of R for V =70 m./h., D = r, ioo ft. All that it is necessary to do in fig. 12 is to join 70 on the (V) scale to 1,100 on the (D) scale. This straight line will be found to cut the (R) scale at 1 5% (see transverse line in fig. 12), the required value of R.
Comparing the two figures the advantages of the Alignment Nomogram will be evident. The disadvantages re ferred to in §q. have disappeared, for Fig: 12. there is no tracing back along a line to read its graduation, and any interpolation by eye is only necessary on simple graduated scales.
Proceeding to the direct construction of Alignment Nomograms, without the preliminary construction of an Intersection Nomogram, certain types will now be considered which are particular cases of the general equation (i).
Writing the formula I+0 00011,) 7)3-0 z1 =L, f1 = L =V, f2= 1+000010 - W)3 l construct the scales u= /2lf =/21 L = and taking u The support of the scale (W) is the straight line joining the zero of the (L) scale to the zero of the f2 scale. This latter zero is at an inconveniently great distance from the top graduation on the (v) scale, but the support can readily be obtained without the actual use of the zeros of the f i and f 2 scales by the use of the formula al referred to above, or by a cross alignment in the following way: - Take W = loo and work out L for v =60 and 80. The straight lines joining these two values of L and v will intersect at the point 200 on the W scale. Joining this point to the zero of the (L) scale gives the support, and the remainder of the scale can be graduated by taking v = 50 say, and working out L for W = 20, 30, 40, 50.... Joining 50 on the (v) scale to fLI these values of the (L) scale in 10 tons turn, will give an intersection on pi the support for the corresponding graduations of the (W) scale.
The completed diagram is shown in fig. 16. To use it sup pose, for instance, we require the / sovalue of W for L =7 tons, v = 70 m./hour. The straight line join ing 7 on the (L) scale to 70 on s / o sothe (v) scale (see dotted line, fig.
16) cuts the (W) scale at about 82 lb./yd., the required value.r 4 w-rr, 7 (L Type C. - Nomograms with two parallel rectilinear scales and one 130 '30- ' curvilinear scale. If the formula to be represented can be put Ito loin the form (7) it can be represented by two systems of points (z1), (22), arranged along two parallel straight lines, and the third system (z 3) arranged along a curve.
As in the preceding types we take functional scales =µi fs u =µ2f2 along the parallel axes Au, B (fig. 17). We then have for the system (23) g which with our usual axes defines the system of points x - 1 _ h + y _ /Llh3 8 and we ca determine any number of points on the system (23) by means of these equations, or by a series of cross alignments. This latter method is especially indicated in cases in which a double-entry table of corresponding values of z l, 22, 23, is already available.
Proceeding by whichever of these ways is most convenient, we can obtain the complete scale (z3), tracing the curvilinear support through the points determined.
As before it is advantageous, where the variable z 3 is generally the unknown, for the scale (z 3) to lie between the scales (z1), (22). This will be the case if g3 is positive, and this can always be arranged, if necessary, changing the signs of both f 2 and h3.
Having constructed the scales (z l), (22), (z 3) as described above, any straight line drawn across them will cut the three scales at corresponding values of zl, z2, 2 3 as defined by (7).
As an example of Type C take the formula used for the thickness of cast-iron pipes in waterworks, t =o000 125 P d -F- 0 15 'VT where t = Thickness of metal in inches.
P = Pressure of water in pounds/inch.
d =Internal diameter of pipe in inches.
Writing this, 0 000125 P d - t -}- 0.15 o and putting z1 f1=P z3 =d, f 3 =0 15 1/71, g3=d, h8= we see that it is of type C.
We construct the scales u =µ1P y=/22t along two convenient parallel axes (fig. 18).
We then determine sufficient points on (d), by cross alignments, to draw the curve and graduate the scale.
When P is zero, t =0.15 ¦d giving us an easily calculated series of alignments for d = 5, to, 15... For the cross alignments it will be convenient to take P = loo, and calculate t for d = 5, 10, 15, ... as before.
producing the straight line to cut the (t) scale (see dotted line, fig. 18), we get, at the point of intersection, the required value t = P3 in.
7 Graphic Representation of Formulae with more than Three Variables. (i.) Double Alignment Nomograms. - Certain types of formulae containing four variables can be dealt with by breaking them up into two or three variable formulae with a common auxiliary variable.
Consider for instance a formula which can be written in the form fl (8).
Introducing an auxiliary variable z 5 we can construct two partial nomograms f1-1112 (9) f3 +f (10) 7t':: of Type A, having the scale (25) in common.
Such an arrangement is shown schematically in fig. 19, the central line representing the auxiliary scale.
If we take values of 21, 2 2, 2 3, 2 4 satisfy-, ing equation (6), the alignment of z 1 with 2 2, and of z 3 with 2 4 will intersect on the scale (25). The central line need not be c--" d graduated as it is only required as a refer- ence line, the nomogram being read in the following way: Suppose we require the value of z 4 for z1= a, 2 2 =b, z 3 = c. Join a on (z l) to b on (22) cutting the reference line at e. Join c on (23) to e and produce to cut (24) in d, which will give the corresponding value of 24.
Such a nomogram from the way in which it is read is termed a Double Alignment Nomogram.
It will be noticed that as the unit of (25) is the same in both (9) and (to) we must have the relationship I I I I /14 while the distances of the scales from the reference line (fig. 19) will be given by 63 62 112' 64 Hence for the practical construction we graduate any of the three scales, (21), (z 2), (23), say, from three conveniently chosen origins on the supports of their scales. We then determine a point on the scale (24) by means of four values of z i, 2 2, 23, 2 4, (a, b, c, d, say) which satisfy equation (8).
The alignment of c and the intersection of the alignment ab with the reference line then determine the point d on the scale (z4)
and as we know /14 we can construct the scale (24) completely. As an example take the formula for the discharge of gas in pipes, Q - 1050 D"- N 1 H 0.45L where L =Length of pipe in yards.
D = Diameter of pipe in inches.
H =Head of water in inches equivalent to the pressure.
Q =Quantity of gas discharged in cub. ft. per hour. Writing it log log L = 2 5 log D +1 log H -{- const. We put z1=Q, z2 L, 23=H, z4=D.
0-
-`
_ look
0
c?J
?,:
- so 80
1.6-
d
along two convenient parallel axes. The scale (L) is an evenly divided scale, and to graduate the scale (v) a series of values of v and f2 are calculated v = 20 30 40 50 60 708012-0961 - 09171-0862 - 0 800 - 0.735 - 0 671 - 0.610 f Ig34-f2h3+f =0 Suppose now we wish to know the thickness of a pipe of so-in. bore to stand a pressure of 130 lb./in. Joining 130 on the (P) scale to 30 on the (d) scale, and 40- u _0 Z Z (Z4) 0. 62 6, 63 19.
Fig. shows the resulting nomogram constructed with µl=2/12=2µ3=µ4 ioo-yd. pipe of 1-in. bore, with a head of water of Suppose now that we want to know the rate of discharge from a I in.
26
1
0.5
(Q)
_500
o -
_100
(0)
Fig: 20.0' 6 Join 1 on the (D) scale to 1 on the (H) scale. Join the point where this straight line cuts the reference line to ioo on the (L) scale, and produce the straight line backwards to cut the (Q) scale at 201 ft./hour (see dotted lines, fig. 20), the required rate of discharge.
These Double Alignment Nomograms can be constructed by combining any two of the types, A, B, or C where the four-variable formula can be written in the appropriate form.
Take for instance the formula H = 18,400 (log B 1 - log B 2) (I + 0.003670) giving the difference in level (H metres) between two stations at which the barometric readings are B 1 and B2 mm. respectively, the mean temperature being 0°C. = t 1 2 t2) Writing it B and A 600 Fig. 21 shows the resulting nomogram, and, to illustrate its use, suppose B 1 = 750 mm., B2 = 670 mm. and it is required to find H.
Join 670 on (B 2) to 750 on (B 1), and produce to cut the reference line. Join the point thus obtained on the reference line to on (0). This straight line produced will cut (H) at 935 mm. (see dotted line, fig. 21), the required difference in height.
(ii). Combination of an Alignment Nomogram with a Network. Suppose we have a network (z 1, z 2), composed of two systems of figured curves (z5), (72) crossing each other (fig. 22).
If we take any point on this network, a curve of both systems will pass through this point, and we may assign to the point a value of both z i and of z 2, taking the values from the curves of the systems (z 1), (z 2) which intersect in the point. The point has thus in a sense two values and is termed a binary point. The general equation in parallel coordinates of such a binary point will be of the form 112+g12u±h12v =o and its coordinates in cartesians with the usual axes will be = h12 - g12 = µf12 x - Xh g12 y h12 + We can obtain the equations of the systems (z 1), (72) forming the network (z 1, 7 2) by eliminating in turn z 2 and z 1 between the above expressions for x and y. Consider now a formula that can be put in the form fig34+f2h34+f 34 =0 This can be represented by the rectilinear parallel scales u=Alf' v=µ2f2 and the network µ2g34u?µ1 +µ1µ2 34 0 or µ1h34 µ2g34' y µ1h34+ µ2g34 As an example take the Compound Interest Formula M =PR" where P is the principal, M the amount, R the amount of I for one year at r % per annum (i.e. R = 1 -} n the number of years. Writing it and taking z l =P, f 1 =log P z 2 =n, 2=n z3=r, f = - log M z 4 = 111, g34 = I, h34 = log R the nomogram will consist of the parallel scales u = / 1 1 log P v =µ2n and the network (r, M) defined by µ2u +µ1 log Rv - µ1µ2 log M =o or in cartesians µl log R-112, _ µ1 µ2 log M Al log R+ /12 y /11 log The expression for x is independent of M, so that we have for (r) a system of straight lines parallel to (P) and (n).
For the system (M) we have, eliminating R between the above expressions for x and y, 2Xy = / L i log M (5 - x) hence (M) consists of straight lines radiating from the point x=X, y =o (i.e. the zero of the n scale), and cutting the straight line x= - X (i.e. the P scale) at the points y= - A i log M so that the lines of the system (M) are easily drawn from the graduations of (P).
Fig. 23 shows the completed diagram. Suppose, for instance, we want to know the amount of £300 in 10 years at 5% compound interest. Joining 3 on the (P) scale to 10 on the (n) scale, this straight line cuts the 5% line (see dotted line, fig. 23) at a point corresponding to the line £490 of the system (M).
Bibliography. - M. D'Ocagne, Traite de Nomographie (1899); Calcul Graphique et Nomographie (1908); Principes usuels de Nomographie avec Application a divers Problbmes concernant L'Artillerie et L'Aviation (1920); Lt.-Col. R. K. Hezlet, Nomography (1913); J. Lipka, Graphical and Mechanical Computation (1918); C. Runge, Graphical Methods (1912). (R. K. H.) -8 6, P 2 5 ?
18 Fir. 23.
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