Y. I. Nikolin.

Professor of the Tomsk Technological Institute of Emperor Nicholas II.

GRAPHICAL METHODS

OF CALCULATING

WATER SUPPLY AND SEWERAGE.

ISSUE II.

Classification and theoretical premises of nomographic methods for calculating water conduits.

With 10 tables of drawings.

Tomsk.

Typo-lithography of the Siberian Association of Printing, ъ. Dela, corner of Dvoryanskaya St. and Yamskoy Lane, own building.

1913.

The broad development, in the last decades of the nineteenth century and at the beginning of the twentieth century, of the construction of water-supply and sewerage works has raised, in the most urgent manner, the question of simplifying methods of hydraulic calculation, with a view to saving labor and time. Under the pressure of the practical necessity of calculating pipelines, a new tendency has appeared, expressed in the striving, as far as possible without detriment to practical accuracy, to simplify calculation formulae, to develop auxiliary tables extensively, and, finally, to apply methods of graphical calculation. All these methods, proceeding by different paths toward one goal—the simplification of the process of hydraulic calculation—deserve the attention of engineers and specialists, both for study and for further development. Especially interesting, from both the practical and the theoretical point of view, are graphical methods of calculation, which reduce to various methods of transforming hydraulic formulae into graphical form, namely to representing them in the form of diagrams of curves or straight lines, or of special logarithmic scales. And indeed, in recent times these applications have begun to attract more noticeable attention. However, the technical literature devoted to this question, both in Russian and in foreign languages, is still confined almost exclusively to more or less brief communications scattered through various journals concerning individual methods of graphical calculation; moreover, the insufficiency of theoretical foundations and generalizations is conspicuous. The materials that might serve to elucidate the question, both theoretical and practical (tables, albums), are still even more scattered and fragmentary. These circumstances, and in particular the theoretical interest and practical importance of the new tendency, compel one to regard as timely the study of the question of applying graphical methods of every kind to the calculation of water supply and sewerage, with a view, on the one hand, to systematizing and elucidating these methods together with their theoretical, mathematical, and hydraulic premises, and with the generalizations and conclusions following from them, and, on the other hand, to popularizing these methods among specialists, which may facilitate their practical application and further development.

Working in this direction according to a definite plan on individual methods of graphical calculation of pipelines, I con—

I therefore consider it not superfluous to set forth in the present article, in as brief a form as possible, certain information and considerations relating to the methods of such calculation in general. I mean to systematize the existing methods of grapho-hydraulic calculation, to acquaint the reader with their essence and form, and to indicate the mathematical basis to which they reduce. My aim in the present case is to draw attention to this question in all its fullness and to give persons who might become interested in developing individual methods general information on the theoretical foundations from which these methods proceed, and in which one must seek the paths for further development. With the same purpose I give at the end of the article as complete as possible a list of the literature of the question.

The graphical methods of calculating pipelines that have been proposed up to the present time may be classified as follows.

1) Diagrams for determining various coefficients of hydraulic formulae.

2) Graphical tables for representing relations between quantities in the form of a system of straight and curved lines (diagrams of isopleth curves). Such are the diagrams of Baumeister, Gobrecht, Gerhard, Gürten, Collignon, D’Aubuisson and Villeroy, Coffin, E. and G. Taylor, P. F. Gorbachev.

3) Logarithmographic tables for representing relations between logarithms of quantities in the form of a system of straight lines (diagrams of isopleth straight lines). Such are the diagrams of Thiem, Frank, Fromm, von Meyden, V. A. Satkevich, M. S. Yasyukovich.

4) Representation of relations between logarithms of quantities in the form of diagrams of conjugate scales, constructed by the method of scales of functions. Such are the diagrams according to Lampe’s formula for metric measures by Vener, and for Russian measures by Prof. N. K. Chizhov.

5) Application of logarithmic slide rules for hydraulic calculation (chiefly in England), which represent one of the forms of conjugate scales.

6) Diagrams of conjugate scales, constructed according to the method of points of rectilinear intersection (diagrams or “abaci” of isopleth points). Such are the diagrams: according to the Lévy–Vallot formula for metric measures by Daries; according to the Flamant formula for metric measures by Bertrand and (abridged) Daries; for English (and Russian) measures, placed in the course of Prof. M. M. Cherepashinsky.

7) Applications of graphical methods to the calculation of a closed water-supply network, for example, the method proposed by M. S. Yasyukovich.

The enumerated graphical methods of calculating pipelines must be divided into two types, essentially differing from one another.

another, namely into methods that are simply graphical and methods that are nomographic. The latter term applies to those methods which are based on the principles of the mathematical science known as Nomography. In order to establish the distinction between the two just-mentioned types of grapho-hydraulic calculation, it is sufficient to explain the general character of Nomography.

By Nomography is meant the theory of the graphical representation of mathematical laws expressed by equations with any number of variables.

If, for all the variables that are connected by a given equation, one constructs a system of geometrical elements (points or lines), graduated in accordance with the values of these variables, and if the relation between the variables established by the equation is expressed geometrically by an easily determined relative position of the corresponding geometrical elements, then the totality of the latter constitutes the diagram of the given equation. By Nomography is meant the theory of such diagrams.

In its application to practice, Nomography sets itself the aim of reducing computations which are necessary in various branches of engineering to simple reading from graphical tables compiled once and for all.

This permanent character of the diagrams gives grounds for distinguishing between Nomography and graphical calculation in the proper sense of the word.

Nomographic diagrams depict the results of a known relation for all possible values of the given elements within definite limits. One may say that a nomographic diagram represents a synthesis of the geometrical constructions corresponding to an infinite number of different values of the elements entering into the calculation. As for graphical calculation in the proper sense of the word, here, in application to the data of each particular case, the numerical calculation is replaced by the drawing of a diagram. Each time, for a new set of data, it is necessary to construct a new diagram.

Precisely such a calculation constitutes the subject of Graphical Statics.

Of the seven methods of graphical calculation of pipelines listed above, the first six belong to the category of nomographic calculation, while the seventh (network calculation) represents a type of calculation that is simply graphical, which, in distinction from the nomographic, might be called idiographic.

In the present article we confine our exposition to nomographic methods of calculation and shall arrange the information relating to the separate methods of this category in the order of sequence—

…of the remarkable development of those nomographic methods which are founded on them.

The earliest of such methods is the principle of Descartes’ rectangular coordinates, which also forms the basis of analytic geometry. The graphical representation of a function of one variable,

\[ y=f(x) \tag{1} \]

or, what is the same thing, of an equation between two variables,

\[ \varphi(x,y)=0. \tag{2} \]

Along the abscissae of a rectangular system of coordinates (Fig. 1) the values \(x\) are laid off, and along the ordinates the values \(y\); a curve is thereby obtained which represents the dependence of these two variables.

In practice this nomographic device is constantly used for representing the dependence between quantities obtained as the result of physical experiments.

In its application to the calculation of pipelines it is used in those cases when the quantity sought is a function of one variable. Precisely such a case is presented by diagrams showing the variation of the value of the coefficient of velocity \(k\) as a function of the hydraulic radius of sections for a specified degree of roughness, or by diagrams of discharges for a specified diameter at different hydraulic gradients, which one often has to construct specially, for convenience of calculation, as applied to the working conditions of the pipelines being designed. In this case the diagrams are usually drawn on a grid of mutually intersecting coordinates with uniformly increasing marks. As an example of nomographic diagrams of this kind, a diagram (Fig. 2) is given for the value of \(k\), for hydraulic radii from 0.010 to 0.500, determined by the abbreviated Ganguillet–Kutter formula

\[ k=\frac{100\sqrt{\rho}}{b+\sqrt{\rho}}, \tag{3} \]

with the coefficient of roughness \(b\) equal to 0.35, i.e. for the case of ordinary cast-iron pipes. The use of such diagrams is self-evident.

The method of nomographic representation of equations with three variables was first applied by Pouchet in his Arithmétique linéaire in 1795, then in the works of Obenheim, Piombert, Bellencentre, Allix, and was theoretically elucidated by Terquem and Lalanne (these works belong to the first half of the nineteenth century). Pouchet’s method serves as the basis for representing the dependence between the elements of hydraulic calculation in the form of systems

curved and straight lines, which we call diagrams of conjugate curves.

Before explaining this method, we must give a notion of one of the fundamental principles of nomography, which will be applied both in the present case and in the further exposition.

Let there be some function \(f(x)\) of an independent variable \(x\), within such limits that for each value of the variable \(x\) there is only one definite value of the function. Let us lay off on the axis \(OX\), from the origin of coordinates \(O\) (Fig. 3), the lengths

\[ \begin{aligned} l_1&=\lambda f(x_1),\\ l_2&=\lambda f(x_2),\\ l_3&=\lambda f(x_3),\\ &\ldots \end{aligned} \tag{4} \]

where \(\lambda\) is an arbitrarily chosen length, and let us write above the points marking the ends of the segments \(l_1, l_2, l_3,\ldots\) the corresponding values \(x_1, x_2, x_3,\ldots\) of the variable.

The set of points thus obtained, with numerical marks, constitutes the scale of the function \(f(x)\). The length \(\lambda\) is called the modulus of this scale.

If the scale of the function must be bounded by two particular values of the variable, for example \(x_0\) and \(x_n\), then it may be constructed starting from the lower limit \(x_0\), without the origin of coordinates \(O\) taking part.

To obtain the points \(x_1, x_2, x_3,\ldots\), one must lay off on the axis, starting from an arbitrarily chosen point with the mark \(x_0\), the segments

\[ \begin{aligned} l_1&=\lambda [f(x_1)-f(x_0)],\\ l_2&=\lambda [f(x_2)-f(x_0)],\\ l_3&=\lambda [f(x_3)-f(x_0)],\\ &\ldots\\ L&=\lambda [f(x_n)-f(x_0)], \end{aligned} \tag{5} \]

where \(L\) is the length of the scale.

Taking

\[ f(x)=x \tag{6} \]

and changing it by an equal and round number of units of one or another decimal order, we obtain, by means of the indicated construction, a normal scale. Depending on the problems subject to graphical solution, it is sometimes necessary to apply the construction to other functions, and then scales of functions of another character are obtained, for example logarithmic, segmental, isograded¹.

Thus, let us now suppose that we have an equation with 3 variables of the form

\[ F(x, y, z)=0, \tag{7} \]

which we wish to represent in graphical form. The method used for this purpose by Pouchet reduces to the following. Let us give one of the variables—preferably the one that is most often expressed as a function of the other two, for example \(z\)—a definite value. Then we obtain one equation with two variables. Such an equation is easily represented in the form of a curve drawn on a rectangular coordinate grid, determined by the equalities

\[ \begin{aligned} x'&=\lambda_1 x,\\ y'&=\lambda_2 y, \end{aligned} \tag{8} \]

where \(\lambda_1\) and \(\lambda_2\) are, respectively, the chosen scale moduli for plotting the quantities \(x\) and \(y\) along the coordinate axes (cf. Fig. 4).

The equation of this curve will be

\[ F\left(\frac{x'}{\lambda_1},\frac{y'}{\lambda_2},z\right)=0. \tag{7′} \]

Such a curve, along which the element \(z\) retains one and the same value, was called by Lalanne a curve of equal element (courbe d’égale élément), and then by the German author Vogler an isopleth curve (\(\ἴσος\)—equal, \(\pi\lambdaῆθος\)—magnitude); this latter term was subsequently adopted by Lalanne as well.

Constructing in the same manner curves corresponding to an entire series of values of \(z\), increasing by definite intervals, and writing beside each curve its corresponding value of \(z\), it is, of course, sufficient to draw only that part of each curve within the rectangle formed by two pairs of perpendiculars drawn to the axes \(OX\) and \(OY\) through the points corresponding to the limiting values of \(x\) and \(y\).

In this way we have obtained a system of curves within a rectangle (Fig. 5), divided by rows of coordinates into cells in the form of a grid. This system represents our variables graphically within the assigned limits. Diagrams of this kind, and by analogy with them other diagrams with graduation and numerical markings, were given by Lalanne and his French pupils the name abacs (les abaques des lignes isoplèthes). We shall call them simply diagrams, in the present case diagrams of isopleth curves.

If we agree to designate by the terms horizontal and vertical lines parallel respectively to \(OX\) and \(OY\), then the method of using such a diagram for the purpose of determining the value of \(z\) from given \(x\) and \(y\) may be formulated as follows: read...

the mark of the curve passing through the point of intersection of the vertical with mark \(x\) and the horizontal with mark \(y\).

It goes without saying that, when reading, interpolation is applied if necessary. The same diagram makes it possible to determine \(x\) or \(y\), if one of these variables is given, as well as \(z\). For example, if \(y\) and \(z\) are given, then \(x\) can be obtained by reading the mark of the vertical passing through the point of intersection of the horizontal with mark \(y\) and the isopleth curve with mark \(z\).

It is not difficult to see that in diagrams of isopleth curves the graphical representation of an equation is created by means of horizontal sections of the surface defined by equation (7), in which \(x\), \(y\), and \(z\) are taken as Cartesian coordinates of space.

The method of isopleth curves, as applied to graphical calculation, has become widely used in all branches of engineering. The number of diagrams of isopleth curves for the calculation of pipelines is also very considerable. They are applied to formulae of both non-logarithmic and logarithmic form. Without intending to present an exhaustive list of diagrams of this type—which would hardly be possible—we may point to the diagrams of Hobrecht (used for calculating the Berlin sewerage system—Hobrecht, Die Kanalisation von Berlin), Gerhard (Gerhard’s Diagramm für Abzugskanäle, Gesundheits-Ingenieur, 1883), Baumeister (Bäaumeister, Städtische Strassenwesen und Städtereinigung), Hürten (Hürten, Kurventafeln zur Bestimmung der Leistungsfähigkeit unter Druck liegender Bauwerke in Entwässerungs und Bewässerungsgräbens), Colignon (Colignon, Cours de mécanique, II, Hydraulique), d’Aubrive and Villerupt (d’Aubrive et Villerupt, L’album des abaques pour le calcul des conduites d’eau), Coffin (Coffin, The graphical solution of hydraulic problems), E. B. and G. M. Taylor (E. B. and G. M. Taylor’s Diagrams of the discharge of pipes in accordance with Kutter’s formula), and P. F. Gorbachev (P. F. Gorbachev, On the Calculation of Flow Velocities and Carrying Capacity in Water Pipes and Watercourses).

As an example of diagrams of isopleth curves may serve the diagrams for calculating water-supply pipes by d’Aubrive and Villerupt. These diagrams, five in number, for discharges from 0 to 20,000 liters per second and for pipe diameters up to 3.00 meters, are constructed on the basis of the logarithmic formula of Lévy-Vallot¹

\[ D = 0.324 \left(\frac{Q}{\sqrt{i}}\right)^{3/8}, \tag{9} \]

where \(D\) is the pipe diameter,
\(Q\) is the discharge,
\(i\) is the hydraulic gradient.

Putting

\[ j=\sqrt{i}, \tag{10} \]

we obtain

\[ D=0{,}324\left(\frac{Q}{j}\right)^{3/8}, \tag{9'} \]

\[ \frac{Q}{j}=\left(\frac{D}{0{,}324}\right)^{8/3}. \tag{9''} \]

To find an expression for the velocities \(v\), we note that

\[ v=\frac{4Q}{\pi D^2}, \tag{11} \]

and, introducing instead of \(D\) its expression in terms of \(Q\) and \(j\), we obtain

\[ v=\frac{4Q^{1/4}j^{3/4}}{(0{,}324)^2\pi}, \tag{12} \]

whence the equation

\[ Qj^3=\frac{(0{,}324)^8\pi^4}{256}\,v^4. \tag{13} \]

Let us take the axes of rectangular coordinates \(OQ\) and \(OJ\) (Fig. 6) and lay off the discharges in the form of a normal scale along the axis \(OQ\), and the slopes in the form of a scale of the function

\[ j=\sqrt{i}. \tag{10} \]

Then the isopleths of discharges will be represented by straight lines parallel to \(OJ\), and the isopleths of slopes by straight lines parallel to \(OQ\). The isopleths of velocities will be curves, which can be constructed point by point on the basis of the preceding equation. The isopleths of diameters will be represented in the form of radial straight lines issuing from the origin of coordinates.

To obtain, for given \(Q\) and \(i\), for example, the velocity, one must find the intersection of the corresponding isopleths of discharge and slope and take the velocity from the isopleth of velocities passing through the point of intersection found. In doing so, of course, interpolation is applied when necessary.

In Fig. 7 there is shown a diagram of isopleth curves from the album of d’Ocagne and Williot, for the calculation of water-supply pipes. It includes diameters from 0.45 to 1.25 meters and discharges from 100 to 2000 liters per second.

Another example of diagrams of isopleth curves for the calculation of pipelines is the well-known diagrams of Baumeister. They are constructed according to one and the same principle for circular and oval cross-sections, with different coefficients of roughness, whereby

coefficient of velocity may be taken on the basis of the shortened Ganguillet–Kutter formula

\[ k=\frac{100\sqrt{D}}{0.7+\sqrt{D}}. \tag{14} \]

In Fig. 8 such a diagram is presented for circular sections at degrees of roughness, according to Ganguillet–Kutter, IV (good cast-iron pipes; good brick masonry) and VI (fouled cast-iron pipes; rubble masonry).

For its construction the velocities \(v\) are laid off as abscissas, the discharges \(Q\) as ordinates, the diameters \(D\) are given by oblique rays, and the falls \(i\) by curves, of which the smaller graduation is valid for degree of roughness IV, and the larger for degree of roughness VI. The diagram thus contains four quantities \(Q\), \(v\), \(i\), and \(D\). By means of it, two of the latter can be determined when the other two are known.

In diagrams for ovoid sections the diameters \(D\) are replaced by the heights \(H\) of the profile (Fig. 9). Consequently, in this diagram the discharges \(Q\) are given by ordinates, the velocities \(v\) by abscissas, the heights \(H\) by rays, and the falls \(i\) by curves.

For expressing more complex functions encountered in hydraulic calculation, more or less complicated combinations of diagrams of isopleth curves are sometimes used. Such a combination is represented, for example, by the diagram for the graphical determination of the velocity coefficient \(k\) according to the complete Ganguillet–Kutter formula, given, among others, by Prof. M. M. Cherepashinsky.^1

In constructing diagrams of isopleth curves there may be special cases in which the curves \(z\) turn into straight lines. Considering equations (7), (7′), and (8), we see that this occurs when equation (7) can be reduced to the form

\[ \frac{x'}{\lambda_1} f(z)+\frac{y'}{\lambda_2}\varphi(z)+\psi(z)=0 \tag{15} \]

and, by virtue of (8),

\[ x f(z)+y\varphi(z)+\psi(z)=0. \tag{15′} \]

In the more general case, when for constructing equation (7) it proves more convenient to lay off along the coordinate axes not \(x\) and \(y\), but \(f_1(x)\) and \(f_2(y)\), in such a way that

\[ \begin{aligned} x'&=\lambda_1 f_1(y),\\ x'&=\lambda_2 f_2(y), \end{aligned} \tag{8′} \]

then, in order that the curves \((z)\) of the diagram should turn into straight lines, equation (7) must have the form

\[ \frac{x'}{\lambda_1} f(z)+\frac{y'}{\lambda_2}\varphi(z)+\psi(z)=0 \tag{16} \]

^1 M. M. Cherepashinsky, Water Supply, pp. 6–7.

or

\[ f_1(x) f(z) + f_2(y)\,\varphi(z) + \psi(z) = 0. \tag{16′} \]

Nomographic diagrams distinguished by this property are called diagrams of isopleth straight lines.

The construction of such diagrams is, of course, considerably easier than that of diagrams with curved lines.

Hence it is clear that in practice one more readily resorts to the use of diagrams of this type, provided only that the equation determining the relation among the elements of the problem can be reduced to the form (15′) or (16′).

An example of a diagram of isopleth straight lines could be the diagram of d’Ocagne and Willereau shown in Drawing 7, if one confines oneself to constructing the relation between the discharge \(Q\), the slope \(i\), and the diameter \(D\). The character of the diagram in this case is explained by the fact that the equation expressing the dependence among \(Q\), \(i\), and \(D\) (9) represents a special case of equation (15′).

It must be noted that every individual curve constructed in a coordinate system (rectangular or otherwise) can be transformed into an equivalent straight line, in terms of its values, by means of a simple graphical process. Suppose, for example, that we have the curve OMP (Drawing 10), obtained as the result of construction by the method of Cartesian coordinates, with a normal scale for plotting the variables. Let points with the marks \(x\) and \(y\) on the coordinate axes correspond to the point M on the curve OMP. Extend the line \(yM\) to its intersection with the line OP at the point \(M\), and from this point drop a perpendicular to the axis OX until it intersects at the point A. If we now assign to the point A the mark \(x\) and perform the same operation with the other points of the curve OMP, then it is clear that the straight line OP will completely replace the curve OMP. In this, evidently, the scale of plotting along the axis OY remains normal, while the scale of plotting along the axis OX changes its character. Such a replacement, in homography, of curved lines by straight ones bears the name anamorphosis. This most primitive method of anamorphosis, in distinction from others, may be called graphical.

It is obvious, however, that such an anamorphosis, applied to an individual curve, cannot bring any substantial simplification in the use of diagrams. Far more important are methods that make it possible to replace all the curved lines of an entire diagram by straight lines.

Lalanne, Lallemand, and later investigators showed in various ways that diagrams containing curved lines (diagrams of isopleth curves), in certain cases, by means of certain preliminary operations, can be artificially replaced by diagrams consisting exclusively of straight lines,

parallel or nonparallel among themselves, i.e. diagrams of interlaced straight lines.

The matter is thereby reduced to transforming the equations that are to be graphically represented, by one method or another, into the form (15′) or (16′), which is usually accompanied by a change in the scales of the functions along the coordinate axes; sometimes it is necessary to resort to the introduction of a new coordinate system or of a special third scale of functions (Lallemand’s hexagonal diagrams). Lalan called such methods of transformation geometric anamorphosis. In those cases where, in transforming the equations, one must resort to taking logarithms (which is definitely reflected in the character of the resulting diagrams of interlaced straight lines, which are then called logarithmic-graphical tables), the anamorphosis is called logarithmic.

Among the equations that represent special cases of type (16′) are the equations

\[ f_1(x) + f_2(y) + f_3(z) = 0, \tag{17} \]

which are very often encountered in practice.

In this case, applying the previous method of plotting on the coordinate axes, we have

\[ \begin{aligned} x' &= \lambda_1 f_2(x),\\ y' &= \lambda_1 f_2(y), \end{aligned} \tag{18} \]

whence the equation of the curves \((z)\) is obtained:

\[ \frac{x'}{\lambda_1} + \frac{y'}{\lambda_2} + f_3(z) = 0, \tag{19} \]

Equations of this kind give a diagram of interlaced straight lines parallel to one another.

Another form of equations, very useful in practice and representing a special case of type (16′), is the following:

\[ f_1(x) \cdot f_2(y) = f_3(z). \tag{20} \]

Equations of this type, when constructed directly, give diagrams of interlaced straight lines in the form of a series of radii issuing from the origin of coordinates, as we have seen in a particular example.

But, in addition, equations of type (20) can be reduced to the preceding form by taking logarithms of both sides:

\[ \lg f_1(x) + \lg f_2(y) = \lg f_3(z). \tag{21} \]

In this form the equation can be represented graphically as a diagram of parallel straight lines (on a logarithmic grid). The diagram of interlaced straight lines thus obtained, by virtue of the logarithmic character of the transformation and plotting, has in practice been assigned, as we have already mentioned, the name logarithmic-graphical tables.

This simple and convenient method of transformation has very great significance. It belongs to the well-known French engineer Lalanne, who is rightly considered the founder of nomography as a science.

The point is that for a long time the principle of interlaced lines and the graphical method of calculation based on it found no wide application. The development, in the 1840s, of the network of railways in France raised the problem of the rapid computation of large quantities of earthwork. To solve this problem, the engineers Lalanne (Léon-Louis-Chrétien Lalanne, 1811–1892) and Devaine had the idea of applying graphical methods. To this circumstance we owe the fact that Lalanne systematized and developed methods of graphical calculation and, among other things, discovered (in 1843) the remarkable principle of logarithmic anamorphosis. Lalanne’s discovery was published in the form of a memoir in the Annales des Ponts et Chaussées for the year 1846 under the title Mémoire sur les tables graphiques et sur la géométrie anamorphose appliquée à diverses questions qui se rattachent à l’art de l’ingénieur. This principle underlies the method of calculation by means of logarithmographic tables. The method itself is also often called Lalanne’s method, and logarithmo-graphical tables—Lalanne diagrams.

The work mentioned laid the foundation for the further development of nomography and by itself provided a basis for a whole series of applications to technical calculation. Lalanne himself was extremely interested in the practical application of his theory and prepared several manuals on graphical calculation (Tables nouvelles pour abréger divers calculs, Tables graphiques à l’usage des chemins de fer, Description et usage de l’abaque ou compteur universel, Instruction sur les règles de calcul, etc.; he also invented a special device for the mechanical performance of computations—balance à calcul, balance algébrique).

Hydraulic (as well as every other kind of) calculation formulas having a logarithmic form are amenable to graphical representation in the form of logarithmographic tables, i.e. those in which the variables or their functions enter exclusively in the form of products or powers. In other words, this property is possessed by equations that can be reduced to the form

\[ f_1(x)\cdot f_2(y)=f_3(z). \tag{20} \]

It is not difficult to see that this equation is a special form of equation (16′)\(^1\) and therefore, without any further transformation, can be represented in the form of a diagram of interlaced straight lines.

\[ \begin{aligned} &{}^1\ \text{In fact, equation (20) can be rewritten in the form}\\ &\qquad f_1(x)\cdot f_2(y)-y^0\cdot f_3(z)+0\cdot y=0,\\ &\text{which fully corresponds to (16′).} \end{aligned} \]

Indeed, let us plot \(f_1(x)\) and \(f_3(z)\) as scales of functions along the axes of rectangular coordinates \(OX\) and \(OZ\), with scale moduli \(\lambda_1\) and \(\lambda_3\), on the basis of the equalities

\[ x'=\lambda_1 f_1(y), \qquad z'=\lambda_3 f_3(z). \tag{22} \]

Substituting the values of \(f_1(x)\) and \(f_3(z)\) into equation (20), we obtain the equation

\[ z'=\frac{\lambda_3}{\lambda_1}\, f_{x2}(y). \tag{23} \]

Such an equation defines a family of straight lines that change their position depending on the change in the value of \(y\).

It is easy to see that all these straight lines pass through the origin. Such lines, let us note, are called radiants in nomography.

Thus we see that equations of the form (20) can be represented graphically directly in the form of diagrams of interlaced straight lines. In these diagrams there will be three systems of straight lines: the lines of one system will be parallel to the axis of abscissas, the lines of the second system will be parallel to the axis of ordinates, and the lines of the third system will issue from the origin in the form of radiants.

Equations of the type

\[ f_1(x)\cdot f_2(y)=f_3(z) \tag{20} \]

may also be represented in the form of a diagram of interlaced straight lines parallel to one another. This is achieved by reducing equation (20) to the form (17) by taking the logarithms of both sides. Then one obtains

\[ \lg f_1(x)+\lg f_2(y)=\lg f_3(z). \tag{21} \]

Such an equation can be represented graphically in the form of a diagram of parallel straight lines placed on a rectangular coordinate grid. The difference between such a diagram and the diagram of interlaced straight lines for equations of the form

\[ f_1(x)+f_2(y)+f_3(z)=0 \tag{17} \]

will consist in the fact that the graduation points of the coordinate axes, having uniformly increasing marks, will be located at distances that are not equal, but vary according to a logarithmic law. Owing to this, the coordinates passing through the graduation points will form an irregular grid of a quite characteristic kind, which is called a logarithmic grid.¹ Such a diagram bears the name of a logarithmographic table.

¹ In France there is on sale a special paper with a printed grid of logarithmic divisions. When such paper is used, the construction of diagrams of the logarithmic type is carried out just as conveniently as the drawing of ordinary rectilinear diagrams on ordinary squared paper.

To represent equation (21) graphically in the form of such a table, let us lay off along the coordinate axes \(OX\) and \(OY\) (Fig. 11) \(\lg f_1(x)\) and \(\lg f_2(y)\) in the form of function scales, with scale moduli \(\lambda_1\) and \(\lambda_2\). Drawing through the graduation points of the coordinate axes a series of lines perpendicular to these axes, we see that the lines perpendicular to \(OX\) will represent the geometric loci of points corresponding to various values of \(\lg f_1(x)\), while the lines perpendicular to \(OY\) will in like manner represent various values of \(\lg f_2(y)\). These lines will be rectilinear isopleths of \(x\) and \(y\). In this case the equalities will hold

\[ \begin{aligned} x' &= \lambda_1 \lg f_1(x),\\ y' &= \lambda_2 \lg f_2(y). \end{aligned} \tag{24} \]

Determining from them the expressions \(\lg f_1(x)\) and \(\lg f_2(y)\) and substituting into equation (21), we obtain a new equation

\[ \frac{x'}{\lambda_1}+\frac{y'}{\lambda_2}=\lg f_3(z). \tag{25} \]

Assigning to \(z\) a series of constant values, we can, on the basis of this equation, construct a series of lines which will be isopleths of the quantity \(z\). It is not difficult to see that this will be a series of straight lines parallel to one another. The angular coefficient of these straight lines is constant and equal to \(-\dfrac{\lambda_1}{\lambda_2}\).

For constructing the isopleths of the quantity \(z\), it is first necessary to determine their direction. This direction is obtained if one connects the point of the axis \(OX\) marked \(\lambda_1\) with the point of the axis \(OY\) having the mark \(\lambda_2\). After this it is sufficient to determine for each isopleth one point. For this purpose one may take the points of their intersection with the axis \(OX\), determined by the equality

\[ x'=-\lambda_1 \lg x_3(z). \tag{26} \]

It may also be found, which is still more convenient, the points of intersection of these isopleths with the line perpendicular to their direction and passing through the origin of coordinates. The equation of such a line is

\[ \lambda_1 x'=\lambda_2 y'. \tag{27} \]

If we denote by \(z'\) the segments measured along this straight line, beginning from the origin of coordinates \(O\), it is not difficult to see that such a segment for the isopleth \(z_0\), representing nothing other than its distance from the origin of coordinates, is determined by the equality

\[ z'_0=\frac{\lambda_1\lambda_2\lg z(x_0)}{\sqrt{\lambda_1^2+\lambda_2^2}}. \tag{28} \]

This means that if, along the line \(OZ\), determined as shown above, we lay off the values \(\lg f_3(z)\) in the form of a scale of the function, with a modulus obtained from the moduli \(\lambda_1\) and \(\lambda_2\) taken by us earlier, on the basis of the equality

\[ \lambda_3=\frac{\lambda_1\lambda_2}{\sqrt{\lambda_1^2+\lambda_2^2}}, \tag{29} \]

and through the points obtained draw straight lines perpendicular to \(OZ\), then we shall also obtain the corresponding isopleths of the quantity \(z\).

Thus, in the end, the construction of a logarithmo-graphic table for equation (20) reduces to laying off \(\lg f_1(x)\), \(\lg f_2(y)\), \(\lg f_3(z)\) in the form of scales of functions along the axes \(OX\), \(OY\), and \(OZ\), with moduli \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\), satisfying equality (29), and to drawing through the graduation points perpendiculars, which will be rectilinear isopleths of the variables \(x\), \(y\), and \(z\).

As the simplest example of a logarithmo-graphic table, we may point to the application of this method to the multiplication of numbers, given by the inventor of the method himself, Lalanne.

Suppose we wish to represent graphically the equation

\[ xy=z. \tag{30} \]

Taking logarithms of both sides, we obtain

\[ \lg x+\lg y=\lg z. \tag{31} \]

Put

\[ \begin{aligned} x'&=\lambda \lg x,\\ y'&=\lambda \lg y. \end{aligned} \tag{32} \]

Then for the isopleths \(z\) the equation obtained is

\[ \frac{x'}{\lambda}+\frac{y'}{\lambda}=\lg z, \]

\[ x'+y'=\lambda \lg z. \tag{33} \]

The logarithmo-graphic table representing this equation is shown in Drawing 12.

Logarithmo-graphic tables (or Lalanne diagrams) for the calculation of pipelines are used chiefly and with the greatest convenience for the graphical representation of formulas of hydraulic calculation having a logarithmic form, but there are examples of their application also to formulas of non-logarithmic form. To this type of graphical tables belong the diagrams of Thiem (Thiem, Ueber graphische Durchmesserbestimmung bei Wasserleitung, Journ. für Gasbeleucht. u. Wasserversorg., 1885), Frank (Frank, Die Formeln über die Bewegung des Wassers in Röhren, Civiling., 1881; Frank, Berechnung der Kanäle und Rohrleitungen), Fromm (Fromm, Diagramm

for “Träger, Schützen, städtische Entwässerungskanäle,” van Muyden (van-Muyden, “Abaque logarithmique pour le calcul des conduites d’eau sous pression”), V. A. Satkevich (“Calculation of a water-supply pipe network by means of a logarithmographic table,” Stroitel, 1898), M. S. Yasyukovich (“Calculation of gutters by means of logarithmographic tables”).

As an example, in Fig. 13 the logarithmographic table of von Meyden for the calculation of water-supply pipes, published in Switzerland, is reproduced. This table, apparently, enjoys some circulation, since it has already gone through five editions. In the first two editions the author based the development on Darcy’s formula. But later he abandoned this formula and constructed a new table on the basis of M. Lévy’s formula, which he considers more suitable, especially for large diameters. The table is compiled for pipes that have been in service and are covered inside with deposits.

M. Lévy’s formula in its basic form is

\[ \left(\frac{v}{n}\right)^2 = Ri \left(a + b \sqrt{R}\right), \tag{34} \]

where \(v\) is the velocity of motion of the water,
\(R\)—the radius of the pipe,
\(n, a, b\)—coefficients.

The latter have different values for new and old pipes.

For the present case

\[ \begin{aligned} n &= 20.5,\\ a &= 1,\\ b &= 3. \end{aligned} \]

For convenience of construction the author subjected M. Lévy’s formula to an artificial transformation and brought it to the logarithmic form

\[ Q = k \sqrt{D^5 i}, \tag{35} \]

where \(Q\) is the discharge.

Taking it in the latter form and taking logarithms, we obtain

\[ \lg Q = \lg k + 2.5 \lg D + 0.5 \lg i. \tag{36} \]

Putting

\[ \begin{aligned} x &= \lg k + 2.5 \lg D,\\ y &= 0.5 \lg i,\\ z &= \lg Q, \end{aligned} \]

we obtain the equation

\[ x + y = z, \tag{37} \]

which is easily constructed in the form of a logarithmic table, as indicated above.

Nomographic Methods of Calculation

This table contains rectilinear isopleths for diameters, slopes, and discharges. In a similar way one obtains the equation and the isopleths for velocities.

On the logarithmic table of Falk-Meiden (Fig. 13), a graduation of diameters is plotted along the axis of abscissas (from 0.05 to 3.00 meters), and a graduation of hydraulic slopes along the axis of ordinates (from 1 to 1000 mm per 1 running meter). Lines drawn through the graduation points perpendicular to the coordinate axes represent isoplethic straight lines of diameters and slopes. The isopleths of discharges (from 1 to 100 liters per second) and of velocities (from 0.25 to 5.00 meters) are represented by two systems of straight lines intersecting the coordinate axes and parallel to one another.

Thus the table represents isoplethic straight lines of four quantities connected with one another by the equations of water motion. If any two of these quantities are known, the other two can be determined by reading the table. For this it is sufficient to find the point of intersection of two isoplethic straight lines corresponding to the given quantities. Then the isopleths of the desired quantities passing through the point found determine, by their marks, the values of the latter corresponding to the given quantities. If the isopleths of the desired quantities do not pass directly through the point found, then interpolation must be performed, or the values of the nearest isopleths must be taken, depending on the nature of the problem.

Among other logarithmographic tables, those compiled by Russian specialists V. A. Satkevich and M. S. Yasjukovich, on the basis of Lampe’s formula, deserve special attention for us.¹

With the transition from isoplethic curves to isoplethic straight lines, the method of using diagrams changes little. This method, however, is not free from certain inconveniences, which consist chiefly in the fact that, in order to obtain readings, one has to follow the lines with the eyes and interpolate between them, which is tiring and does not sufficiently guard against errors.

Nomography makes possible a graphical representation of the equations of calculation by means of diagrams of another kind, in which, as elements with numerical marks, exclusively points are used. Such are the special diagrams which we call diagrams of conjugate scales.

There are two types of such diagrams. Being to some extent similar in their external appearance, they differ noticeably from one another ...

from one another both in the principles laid at their foundation and in the method of construction and use. The first of these types, to the explanation of which we now turn, represents only an application of the nomographic principle of function scales; the second, of which we shall speak later, has as its basis, in addition to the above-mentioned one, another principle as well, namely the principle of points of rectilinear intersection.

The nomographic principle of function scales was explained by us above. Of the various scales obtained as a result of constructing functions, the logarithmic scale is of the greatest importance. If in equations (4) we take

\[ f(x)=\lg x \tag{38} \]

and apply to this case the methods of construction indicated above, then the logarithmic scale of a function is obtained.

The divisions of a logarithmic slide rule may serve as its model. Such a scale is used for the construction of all logarithmic diagrams, including diagrams of conjugate scales.

Concerning this logarithmic scale it must be noted that, if it is extended beyond 10, then in the interval from 10 to 100 it will have the same length and the same divisions as from 0 to 10, the divisions corresponding to quantities 10 times greater. The same would hold from 100 to 1000, only with markings still 10 times greater, and so on. From this it follows, first, that on a logarithmic scale constructed within the limits from 0 to 10, the units may refer to any decimal order whatever; secondly, that the degree of relative accuracy of reading, when applied to any order, remains constant.

The first is self-evident. And that the relative accuracy (or percentage accuracy) of calculation when using a logarithmic scale is everywhere the same is clear from the following reasoning. At any place on the scale the greatest magnitude of the error in reading by eye is, generally speaking, the same in length. This magnitude represents the difference of two logarithms of certain values, between which the true value lies. Thus, denoting the extreme values by \(x_n\) and \(x_{n+1}\), we may say that

\[ \lg x_{n+1}-\lg x_n=\text{const}. \tag{39} \]

But

\[ \lg x_{n+1}-\lg x_n=\lg \frac{x_{n+1}}{x_n}, \]

whence

\[ \lg \frac{x_{n+1}}{x_n}=\text{const}. \]

$$ \frac{x_{n+1}}{x_n}=\mathrm{const}. $$

$$ \frac{x_{n+1}-x_n}{x_n}=\mathrm{const}. \tag{40} $$

Thus, if, in reading on a logarithmic scale, we assign to its numerical marks values 10 times greater, then the numerical values of the limits of error will also increase 10 times, while the relative magnitude of the error and the percentage accuracy of the calculation remain the same.

The theory of constructing diagrams of conjugate scales by the method of scales of functions is very simple. Suppose we have a calculation formula in the form of an equation of logarithmic type

$$ f_1(x) f_2(y)=f_3(z). \tag{41} $$

Taking logarithms of both sides of this equation, we obtain

$$ \lg f_1(x)+\lg f_2(y)=\lg f_3(z). \tag{41'} $$

Let us lay off the functions \(\lg f_1(x)\) and \(\lg f_2(y)\) in the form of two separate scales of functions (Fig. 14), with one and the same scale modulus \(\lambda\), on the basis of the equalities

$$ \begin{aligned} x'&=\lambda \lg f_1(x),\\ y'&=\lambda \lg f_2(y). \end{aligned} $$

Expressing the functions \(\lg f_1(x)\) and \(\lg f_2(y)\) on the basis of these equalities and substituting into equation \((41')\), we obtain a new equation

$$ \frac{x'}{\lambda}+\frac{y'}{\lambda}=\lg f_3(z) $$

or

$$ x'+y'=\lambda \lg f_3(z). \tag{42} $$

The last equation shows that, if we construct the function \(\lg f_3(z)\) in the form of a third scale of functions with the same modulus \(\lambda\), then the determination of each particular value of the quantity \(z_0\), from given \(x_0\) and \(y_0\), can be made as follows. One must take, on the scale of the function \(x\), the segment from the origin of the scale to the mark \(x_0\), and add to it the length corresponding to \(y_0\) on the scale \(y\). Laying off the sum of these lengths on the scale \(z\) from its origin, we obtain the mark determining the corresponding value \(z_0\).

From the preceding it is clear that the representation of calculation formulas by means of diagrams of conjugate scales, which for brevity we shall call Veiner diagrams, is applicable only to such formulas as represent equations of logarithmic type either with respect to the variables themselves, or, at the very least, with respect to certain functions of them, from which a simple and convenient transition to the variables themselves is possible.

The method described for the graphical representation of formulas for hydraulic calculation was proposed by G. Vener in the article Ein Beitrag zur Berechnung des Rohrwiderstandes in der Praxis (Gesundheits-Ingenieur, 1897), the diagram being drawn up for Lampe’s formula in metric measures. In the same year, Prof. N. K. Chizhov applied this method to the same Lampe formula, but, in accordance with the conditions of Russian practice, compiled his diagram for measurements in feet. The latter diagram appeared in an article entitled “A Mechanical Method for Calculating Loss of Head” (Stroitel, 1897).

As a specimen of diagrams of this kind we append (Fig. 15) the diagram of N. K. Chizhov.

The method of constructing the diagram is as follows. Lampe’s formula¹ is used in practice in the form

\[ i=\frac{n^{1.8}}{\rho^{1.25}}, \tag{43} \]

where

\[ \begin{aligned} i&\text{—hydraulic gradient,}\\ \rho&\text{—hydraulic radius of the section,}\\ v&\text{—velocity,}\\ n&\text{—coefficient of roughness.} \end{aligned} \]

For the coefficient of roughness \(n\), Prof. N. K. Chizhov adopts the following values:

\[ n_1=0.0000675 \]

—for entirely new asphalted cast-iron pipes conveying clean water;

\[ n_2=0.00009\text{—}0.00001 \]

—for cast-iron pipes covered with deposits (this value is especially suitable for calculating municipal water mains);

\[ n_3=0.000095 \]

—for drainpipes.

Taking logarithms of this formula, we obtain

\[ \lg i-\lg n=1.8\lg v-1.25\lg\rho. \tag{44} \]

On an arbitrary straight line (see the lower scale of the diagram \(i\)—\(n\), Fig. 15) we mark \(\lg i\) and \(\lg n\) as functional scales, i.e., we lay off, on an arbitrary scale, divisions proportional to \(\lg i\) and \(\lg n\), and above the divisions write the corresponding values of \(i\) and \(n\). On another arbitrary straight line we similarly mark (see the upper scale of the diagram, \(\rho\) and \(v\)) on one side \(1.8\lg v\), and on the other side \(1.25\lg\rho\). In doing so, the scale modulus for these functions must be the same as for the scale of \(\lg i\) and \(\lg n\). The absolute lengths, however, of the new two scales between equal divisions will be different, namely 1.8 and 1.25 times greater than the corresponding lengths on the scale of \(\lg i\) and \(\lg n\).

Nomographic Methods of Calculation

The laying off must be begun from the common point, marked with the numerical notation 1.0 both for \(v\) and for \(\rho\). This follows from the following consideration. For the case when

\[ \lg i_0 - \lg n_0 = 0, \tag{45} \]

equation (44) becomes

\[ 1.8 \lg v_0 - 1.25 \lg \rho_0 = 0. \tag{46} \]

If at the same time

\[ \rho = 1.0, \]

then

\[ 1.8 \lg v_0 = 0, \]

\[ v_0 = 1.0. \]

Having inscribed at the divisions the corresponding values of \(\rho\) and \(v\), we obtain a double logarithmic scale of hydraulic radii and velocities.

The two constructed scales are sufficient for determining any of the quantities \(i\), \(v\), and \(\rho\) entering into Lampe’s formula (43), if two of them are given. But, in order also to introduce into the calculation the discharge \(Q\), logarithmic scales of discharges (in cubic feet) have also been constructed on the diagram for 20 different diameters, beginning with \(2''\) and up to \(30''\). These scales are constructed on lines parallel to the scale of velocities, and moreover in such a way that the divisions of the discharge scale corresponding to one and the same velocity for different diameters lie on one vertical line with the division denoting this velocity. The modulus of the scale satisfying this condition is determined on the basis of the relation

\[ \lambda_0 \left(\lg \frac{\pi D_0^2 v_0}{4} - \lg \frac{\pi D_0^2}{4}\cdot 1\right) = 1.8\,\lambda \lg v_0, \tag{47} \]

where \(\lambda\) is the modulus of the scale of \(\lg i\) and \(n\),

\(\lambda_0\) is the modulus of the scale of the corresponding discharge \(Q\).

Whence

\[ \lambda_0 = 1.8\,\lambda. \tag{47'} \]

This means that the intervals between identical divisions on the discharge scale will be the same as on the velocity scale.

It should be noted that the discharge scales for different diameters end on the left at points (marked on the diagram by small circles) that lie on one vertical line with the divisions indicating on the scale of \(\rho\) the value of the hydraulic radii for the given diameters. The significance of this circumstance will be clear from the description of the use of the diagram.

In order to determine, for a given coefficient of roughness \(n\), from given, for example, values of the hydraulic slope and hydraulic radius (or diameter), the velocity, it is necessary

take with the dividers on the \(i\)—\(n\) scale the distance between the divisions corresponding to the given \(i\) and \(n\). In this way we obtain a length corresponding to \(\lg i-\lg n\). Then lay off the length thus obtained, on the upper connected scale, from the division denoting the given \(\rho\). Then the other end of the dividers will indicate to us the division corresponding to the sought velocity \(v\).

In exactly the same way, from the given slope and velocity, the hydraulic radius of the section may be determined. The determination of the necessary slope from the given hydraulic radius (or diameter) and velocity is carried out in an analogous manner, only in the reverse order.

When, from a known diameter and hydraulic slope (for a specified coefficient of roughness), it is required to find the discharge, one could first determine the velocity and then pass to the discharge. But the operation is simplified by the fact that the origins of the discharge scales for different diameters lie on one vertical line with the marks of the corresponding hydraulic radii. Therefore it is more convenient, taking with the dividers on the \(i\)—\(n\) scale the distance between the corresponding divisions, to lay it off from the origin of the discharge scale for the given diameter. Then the other end of the dividers will directly indicate to us the required discharge.

The determination of the slope from a known diameter and discharge is made just as simply in the reverse order. The determination of the diameter from a given discharge and hydraulic slope is carried out by means of successive trials.

The great convenience of N. K. Chizhov’s diagram consists in the fact that it permits the use of any coefficient of roughness.

On examining the method of using diagrams of the Venera type or of N. K. Chizhov, it is not difficult to see that the matter reduces to operations on logarithmic scales drawn on paper, similar to those used when employing the well-known calculating logarithmic rules. It should be said that the application of the principle of the logarithmic rule to hydraulic calculation, in its most concrete form, appeared considerably earlier. Similar logarithmic rules, with divisions marked on two fixed scales and on a movable middle part, are manufactured in England.

Another type of diagram of conjugate scales, a type based on the principle of points of rectilinear intersection, belongs to d’Ocagne (by whose name, for brevity, we shall call such diagrams). He was the one who drew attention to certain inconveniences of diagrams of isopleth curves and straight lines and found that it is more desirable to use, as elements with numerical marks for the graphical representation of equa-

…equations, as far as possible, only points. Ocagne proposed a method which makes it possible to attain this goal with respect to certain equations: namely, the method of points of rectilinear intersection.

It should be said in passing that d’Ocagne (Maurice d’Ocagne, professor at the Paris École des Ponts et Chaussées) deserves special credit for continuing Lalanne’s conceptual work and for developing nomography. He devoted a whole series of works to the question of the graphical representation of equations, and in 1884 published a new general principle he had found in this field, to which he then applied the transformation of Lalanne diagrams and which forms the basis of the method of points of rectilinear intersection (méthode des points alignés).

The foundations of this method were first set forth in d’Ocagne’s memoir Procédé nouveau de calcul graphique (Annales des Ponts et Chaussées, 1884). Subsequently the same question was treated more extensively in a later article by d’Ocagne, which appeared in 1890, Méthode de calcul graphique fondée sur l’emploi des coordonnées parallèles (Génie civil, 1890), and finally received full development in his extensive monographs Nomographie (1891) and Traité de Nomographie (1899).

The method of points of rectilinear intersection proposed by d’Ocagne is based on the principle of geometrical dualism. This principle consists in the fact that, for every system of straight lines, there exists such a system of points that three mutually intersecting straight lines of the first system correspond, in the second, to three points lying on one straight line. Every transformation based on this property is called dualistic.

Suppose that we have applied such a transformation to a diagram consisting of three systems of certain straight lines, preserving, of course, in passing from one figure to the other, the numerical mark of each element. Then we shall obtain a new diagram (Fig. 16), in which to each of the variables \(x\), \(y\), and \(z\) there will correspond a system of points with numerical marks, arranged along a line—in the general case, a curve. These three systems of points with numerical marks will constitute curvilinear scales.

Just as in the first diagram three straight lines with numerical marks of the values \(x\), \(y\), and \(z\), satisfying the equation, met one another, so here the three corresponding points will lie on one straight line intersecting all three scales. This new diagram represents the most general form of the diagram of points of rectilinear intersection. Hence the manner of using a diagram of this kind: the straight line joining the points marked \(x\) and \(y\) on two curvilinear scales meets the third scale at the point marked \(z\).

In practice, under the name of d’Ocagne’s method of points of rectilinear intersection, there is applied a special case of the above-mentioned dia…

diagrams, when the curvilinear scales of functions, owing to the special features of the equations subject to graphical representation, turn into straight lines parallel to one another.

Such a case occurs when the equations have the form (17) or (20)—(21). In this case the diagram has the form of three (or several) scales of functions, arranged parallel to one another and standing in such a mutual relation that any straight line crossing these scales meets them at points with such numerical values of the variables that, when substituted simultaneously, they satisfy the equation represented by the diagram.

The method in question was originally given by its inventor, d’Ocagne, the name method of isoplethic points (méthode des points isoplèthes), by analogy with Lalanne’s method of isoplethic straight lines. But this name was later recognized by the author as unsuccessful (in fact, a point cannot be isoplethic, since in essence it represents a single value), and for some time he used a new term—méthode des points cotés (method of points with numerical marks), wishing to emphasize that in his diagrams it is precisely points furnished with numerical marks that serve for reading, and not lines. However, here the consideration arises that one can construct diagrams in which there also enter only points with numerical marks, but which by no means possess the indicated property with respect to the secant straight line. Therefore, in the end, d’Ocagne settled on the term—méthode des points alignés, characterizing the relative position of the points read simultaneously. We shall translate this term by the expression—method of points of rectilinear intersection.

We call the method of applying this method to the solution of hydraulic problems the method of conjugate scales, and the diagrams serving for this purpose—diagrams of conjugate scales of d’Ocagne’s type, or simply d’Ocagne diagrams.

The method of points of rectilinear intersection is applicable, as was indicated, to the graphical representation of equations having the form

\[ f_1(x) + f_2(y) = f_3(z). \tag{17} \]

Let us recall that equations of the form

\[ f_1(x)\cdot f_2(y) = f_3(z), \tag{20} \]

are reduced to them by transformation into

\[ \lg f_1(x) + \lg f_2(y) = \lg f_3(z). \tag{21} \]

The process of converting equation (17) into the form of a diagram of conjugate scales can be proved in various ways. The proof given by d’Ocagne is conducted with the aid of the so-called

... of parallel coordinates. In view of the unusual nature of the application of this peculiar coordinate system,¹ we shall not touch upon that proof here, but shall give our own, simpler derivation.

Take points A and B on an arbitrary straight line (Fig. 17), and through them draw two parallel lines AX and BY. On the latter set off \(f_1(x)\) and \(f_2(y)\) in the form of function scales, with moduli \(\lambda_1\) and \(\lambda_2\), on the basis of the equalities

\[ \begin{aligned} \lambda_1 f_1(x) &= x',\\ \lambda_2 f_2(x) &= y', \end{aligned} \tag{48} \]

where \(x'\) and \(y'\) are the lengths of the corresponding segments on the lines AX and BY. Replacing in equation (17) \(f_1(x)\) and \(f_2(y)\) according to (48), we obtain the equation

\[ \frac{x'}{\lambda_1}+\frac{y'}{\lambda_2}=f_3(z). \tag{49} \]

Let us give \(f_3(z)\) the constant value \(f_3(z_0)\). Then equation (49) becomes

\[ \frac{x'}{\lambda_1}+\frac{y'}{\lambda_2}=f_3(z_0) \tag{49'} \]

Suppose this last equation is satisfied by certain quantities \(x_0'\) and \(y_0'\), i.e.

\[ \frac{x_0'}{\lambda_1}+\frac{y_0'}{\lambda_2}=f_3(z_0). \tag{49''} \]

The quantities \(x_0\) and \(y_0\) in drawing 17 determine a certain straight line \(X_0Y_0\). This straight line, depending on the change in the values of \(x\) and \(y\), changes its position, rotating about a fixed point, whose position is easily determined.

Indeed, suppose that equation (49′) is satisfied by another pair of quantities \(x_1'\) and \(y_1'\), i.e.

\[ \frac{x_1'}{\lambda_1}+\frac{y_1'}{\lambda_2}=f_3(z_0), \tag{49'''} \]

whereby another straight line \(X_1Y_1\) is determined. Let us call the point of intersection of the straight lines \(X_0Y_0\) and \(X_1Y_1\) \(P\), and determine its position. From equations (49″) and (49‴) it follows that

\[ \frac{x_0'}{\lambda_1}+\frac{y_0'}{\lambda_2} = \frac{x_1'}{\lambda_1}+\frac{y_1'}{\lambda_2}, \tag{50} \]

\[ \frac{1}{\lambda_1}(x_0'-x_1') = \frac{1}{\lambda_2}(y_1'-y_0'), \]

\[ \frac{x_0'-x_1'}{y_1'-y_0'} = \frac{\lambda_1}{\lambda_2}, \tag{51} \]

¹ Those wishing to acquaint themselves with the proof given by d’Ocagne may find it in my work cited above; the general foundations of the system of parallel coordinates are set forth in d’Ocagne’s book, Coordonnées parallèles et axiales.

or, according to the drawing,

\[ \frac{X_0 X_1}{Y_0 Y_1}=\frac{\lambda_1}{\lambda_2}. \tag{51′} \]

Draw through the point \(P\) a straight line \(CZ\), parallel to \(AX\) and \(BY\), until it intersects \(AB\) at the point \(C\), and the straight line \(QR\), parallel to \(AB\). The triangles \(X_0PX_1\) and \(Y_0PY_1\) are similar to one another. Therefore

\[ \frac{PQ}{PR}=\frac{X_0X_1}{Y_0Y_1}=\frac{\lambda_1}{\lambda_2} \tag{52} \]

or, what is the same,

\[ \frac{AC}{CB}=\frac{\lambda_1}{\lambda_2}. \tag{52′} \]

On the other hand, the similarity of the triangles \(X_0PQ\) and \(Y_0PR\) gives

\[ \frac{QX_0}{RY_0}=\frac{PQ}{PR}=\frac{\lambda_1}{\lambda_2}. \tag{53} \]

Taking into account that

\[ \begin{aligned} QX_0 &= x_0' - CP,\\ RY_0 &= CP - y_0', \end{aligned} \tag{54} \]

we have

\[ \frac{x_0' - CP}{CP - y_0'}=\frac{\lambda_1}{\lambda_2}, \tag{53′} \]

whence

\[ CP=\frac{\lambda_2 x_0' + \lambda_1 y_0'}{\lambda_1+\lambda_2}. \tag{55} \]

But equation (49′) may be represented in the form

\[ \lambda_2 x_0' + \lambda_1 y_0'=\lambda_1\lambda_2 f_3(z_0). \tag{49IV} \]

Hence the final value for \(CP\) is obtained in the form

\[ CP=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}\,f_3(z_0). \tag{56} \]

Relations (52′) and (56), for a known length of the line \(AB\), completely determine the point \(P\). Relation (52′) shows that the point of intersection of the straight lines \(X_0Y_0\) and \(X_1Y_1\) lies on the fixed axis \(CZ\). Relation (56) proves, on the other hand, that the distance \(CP\) of the point \(P\) from \(AB\), for a constant value of \(f_3(z_0)\), is constant, and consequently the point \(P\) also remains constant, as was required to prove.

The position of the point \(P\) is determined if we draw the straight line \(CZ \parallel AX\) and \(BY\) at such a distance between them that the relation

\[ \frac{AC}{CB}=\frac{\lambda_1}{\lambda_2}, \tag{52′} \]

is satisfied, and on this straight line set off from \(AB\) the length

\[ CP=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}f_3(z). \tag{56} \]

On this basis, if one lays off respectively on \(AX\), \(BY\), \(CZ\) the quantities \(f_1(x)\), \(f_2(y)\), \(f_3(z)\) entering into equation (17), in the form of functional scales, with moduli \(\lambda_1\), \(\lambda_2\), and \(\dfrac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}\), then the points obtained will lie on one straight line intersecting all three scales. From this it follows, evidently, that if the values of two of the functions mentioned are given, it will be sufficient to draw the straight line joining the corresponding points of the two scales in order to read, at its intersection with the third scale, the value of the third function.

We shall now say a few words about the practical methods of constructing a diagram of conjugate scales. Owing to the freedom in the choice of the moduli \(\lambda_1\) and \(\lambda_2\), the two principal scales may always be taken of equal length. In the case of constructing three conjugate scales for the equation

\[ f_1(x)+f_2(y)=f_3(z) \tag{17} \]

one must first of all choose arbitrarily the length of the scales. The moduli \(\lambda_1\) and \(\lambda_2\) are obtained according to the usual rules for constructing scales¹ on the basis of the formula

\[ \lambda\,[f(x_n)-f(x_1)] = L, \tag{57} \]

where \(\lambda\) is the required value of the modulus, \(f(x_1)\) and \(f(x_n)\) are the extreme values of the function, and \(L\) is the adopted length of the scale. The modulus \(\lambda_3\), on the basis of (56), is determined by the formula

\[ \lambda_3=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}. \tag{58} \]

After this we lay off on two parallel axes the scale \(f_1(x)\), beginning from point \(A\) toward \(x\) (Fig. 18), and the scale \(f_2(y)\), from \(B\) toward \(y\). Since the inclination of the lines \(AX\) and \(BY\) to \(AB\) may be arbitrary, we choose the points \(A\) and \(B\) on a line perpendicular to the direction \(AX\) and \(BY\). The distance of the scales \(AX\) and \(BY\) is arbitrary. Therefore we may choose it in such a way that the smallest angle formed, when reading with a ruler or with the indicating line of a transparent sheet,² with the direction of the scales should not be too small and should have the magnitude we desire (for example, \(\dfrac{\pi}{4}\)).

Now draw the line \(CZ\), parallel to the other two scales, in such a way that

\[ \frac{AC}{BC}=\frac{\lambda_1}{\lambda_2}. \]

and we choose a pair of values of the variables \(x\) and \(y\), for which the values of the variable \(z\) would be obtained from the basic equation in the most convenient manner. Having drawn the corresponding lines on the diagram and found these latter values of \(z\), it is not difficult to construct the scale of the function \(f_3(z)\), within the limits of the straight lines AB and Xy.

Such is the theoretical type of the diagram of conjugate scales. In practice one may depart from it to a greater or lesser degree. It is clear, for example, that if the calculated values of the moduli \(\lambda_1\) and \(\lambda_2\) are not expressed by simple numbers, they have to be rounded off, which introduces an inequality in the lengths of the scales. On the other hand, the format of the diagram may make it necessary to reduce the calculated limiting angle of inclination of the index transparency to the scale lines. It happens that in practice the limiting values of one or another variable relate only to part of the scale of another variable. Then the scale \((z)\) may be correspondingly reduced.

Diagrams of conjugate scales constructed by the method of the points of rectilinear intersection (d’Ocagne diagrams) are applied only to formulas of logarithmic form. Up to the present time the formulas of Flamant and Lévy-Vallot have been subjected to graphical representation by this method. For Flamant’s formula the French engineer Bertrand constructed a diagram of conjugate scales in the form of a system of nine scales (Beschmann. Salubrité urbaine, II, Assainissement). Dariès (Calcul des conduites d’eau) simplified this diagram, reducing it to the usual type with four scales (discharges, slopes, diameters, and velocities). The diagrams mentioned were constructed for metric measures. Prof. M. M. Cherepashinsky, in his course on water conduits, also gives a diagram of the d’Ocagne type for Flamant’s formula, borrowed from American sources, which is constructed for English (and Russian) measures. Finally, Dariès (Application de la Nomographie au calcul des conduites d’eau d’après la formule de M. Lévy. Nouv. Ann. de la Constr., 1897) constructed a diagram of the same type for the Lévy-Vallot formula.

As an example, we shall dwell on the diagram for Flamant’s formula constructed by Dariès.

Prof. Flamant’s formula¹ is usually applied in the form

\[ D^{5/4} i = a v^{7/4}, \tag{58} \]

or

\[ D^5 i^4 = a^4 v^7. \tag{58′} \]

For the coefficient \(a\), when applied to pipes slightly covered with deposits, such as water-supply pipes usually are after several years of service, Flamant gives the value

\[ a = 0.00092. \]

Homographic Methods of Calculation

Flamant’s formula, in whatever form it may have been taken, represents a special case of an equation of the form

\[ f_1(x)\cdot f_2(y)=f_3(x) \tag{20} \]

and therefore can be represented in the form of a diagram of conjugate scales, by the method of points of rectilinear intersection.

To represent this formula in the form of a diagram with scales of diameters, hydraulic gradients, and discharges, it is necessary to transform the formula

\[ D^5 i^4 = a^4 v^7, \tag{58'} \]

on the basis of the relation

\[ v=\frac{4Q}{\pi D^2}. \tag{59} \]

Then we obtain

\[ D^{19} i^4=\left(\frac{4}{\pi}\right)^7 a^4 Q^7, \]

or, solving with respect to \(D\),

\[ D=\left(\frac{4}{\pi}\right)^{\frac{7}{19}}\cdot a^{\frac{4}{19}}\frac{Q^{\frac{7}{19}}}{i^{\frac{4}{19}}}, \tag{60} \]

which, for the coefficient \(a\) for pipes that have been in service, becomes

\[ D=0.251\frac{Q^{\frac{7}{19}}}{i^{\frac{4}{19}}}. \tag{60'} \]

Applying logarithms to formula (60′), we obtain:

\[ \lg D=\lg 0.251+\frac{7}{19}\lg Q-\frac{4}{19}\lg i. \tag{61} \]

To represent this equation in the form of a diagram, take two parallel axes \(Q\) and \(J\) (Fig. 19), situated at an arbitrarily chosen distance \(s\) from one another, and take the first of them as the scale of discharges, and the second as the scale of gradients. Then, on each of them respectively, using a logarithmic ruler, lay off the scales of the functions \(Q\) and \(i\).

For choosing the modulus of plotting, we shall take the following into consideration. If we were to take, for both scales without change, the modulus \(\lambda\) of the logarithmic ruler, then, in order to plot the functions \(\frac{7}{19}\lg Q\) and \(\frac{4}{19}\lg i\), we would have either to multiply the whole \(\lg\) by \(\frac{7}{19}\) and \(\frac{4}{19}\), or to change the scale of the logarithmic ruler in the ratio \(\frac{7}{19}\) and \(\frac{4}{13}\). In order not to do this, it is more convenient to take the modulus

\[ \text{for the scale } Q \;—\; \lambda_1=\frac{19}{7}\lambda \]

and for the scale \(i\),

\[ \lambda_2=\frac{19}{4}\lambda \]

(or multiples of them, depending on the dimensions of the diagram and the mutual arrangement of the scales). Then we shall have to lay off along the scales \(Q\) and \(i\), in order to express the functions \(\frac{3}{8}\lg Q\) and \(\frac{3}{16}\lg i\), the quantities \(\lg Q\) and \(\lg i\) to the scale of the logarithmic rule.

Since the function \(\frac{4}{19}\lg i\) has a negative sign, while \(\frac{7}{19}\lg Q\) is positive, the increase of the numerical values of the divisions on the scales \(Q\) and \(i\) must proceed in opposite directions. This must be kept in mind when choosing the initial points from which to lay off the divisions. In the present case it is more convenient to place approximately in the middle of the diagram the divisions \(O\) and \(O'\), corresponding to certain mean values of \(Q\) and \(i\), and then from them to carry the divisions in both directions.

Having thus constructed the scales of discharges and slopes, it is necessary, on the basis of relations (52) and (58), to determine the position, modulus, and divisions of the scale of diameters \(D\).

Since, as before, the moduli of the scales \(Q\) and \(i\) are

\[ \lambda_1=\frac{19}{7}\lambda, \]

\[ \lambda_2=\frac{19}{4}\lambda, \]

then, denoting the distance from the axis \(Q\) to the axis \(D\) by \(x\), we must have, according to (52),

\[ \frac{x}{s-x}=\frac{19}{7}:\frac{19}{4}=4:7, \tag{62} \]

whence

\[ x=\frac{4}{11}s. \tag{62'} \]

The modulus of the scale of the function \((\lg D-\lg 0.251)\) is determined, on the basis of (58), by

\[ \lambda_3=\frac{\lambda_1\cdot\lambda_2}{\lambda_1+\lambda_2}=\frac{19}{11}\lambda, \tag{63} \]

where \(\lambda\), as before, is the modulus of the logarithmic rule.

To put the divisions on the scale of diameters, connect the previously chosen points \(O\) and \(O'\) (or any others) of the scales \(Q\) and \(i\). The intersection of \(OO'\) with the scale \(D\) gives the point \(O''\), the mark of which, corresponding to the values of \(Q\) and \(i\) at the points \(O\) and \(O'\), is determined by calculation. Then, having constructed a logarithmic scale with modulus \(\lambda_3\) and applying its corresponding division to the point \(O''\), we mark the other divisions of the scale of diameters, extending it in both directions as far as necessary.

If, for convenience, other moduli are chosen (multiples of \(\lambda_1\) and \(\lambda_2\)), then the position of the \(D\) axis must be changed accordingly. For example, with the scale of discharges 3 times greater than the scale of slopes, from equation (52) we obtain:

\[ \frac{x}{s-x}=\frac{19.3}{7}:\frac{19}{4}=12:7, \tag{64} \]

whence

\[ x=\frac{12}{19}s. \]

Usually, on diagrams of hydraulic formulas constructed by the method of points of rectilinear intersection, another axis is also drawn, serving as a scale for determining velocities \(v\). To determine the position and divisions of the velocity scale on the basis of the positions of the \(Q\) and \(D\) scales, one could proceed in the same way, starting from the relation

\[ v=\frac{4Q}{\pi D^2}, \]

taking the logarithm of this expression and representing the resulting equation in the form of a system of conjugate scales in which the \(Q\) and \(D\) scales coincide with those constructed earlier. But it is not necessary to resort to this, since the required positions and divisions can be determined purely graphically. For this it is only necessary to take into account that a velocity of \(1.00\) m is developed in a pipeline of diameter \(0.30\) m at a discharge of \(70.7\) liters, and in a pipeline of diameter \(0.60\) m at a discharge of \(282.7\) liters. On this basis we can draw the corresponding intersecting straight lines and thus find the point of the \(v\) axis marked \(1.00\) meter. Repeating the same operation with the data \(Q=7.07\) liters, \(D=0.30\) m and \(Q=28.27\) liters, \(D=0.60\) m, we obtain the point corresponding to a velocity of \(0.10\) m. These two points must lie on a line parallel to the other scales. It remains only to graduate the distance between \(0.10\) m and \(1.00\) m and to continue the divisions in both directions.

Drawing 20 represents a diagram of conjugate scales for the graphical calculation of pipelines of circular cross-section by the Flamant formula, constructed by Daries. It consists of four scales arranged in the following order, counting from the left: the scale of discharges (in cubic meters per second), the scale of diameters (in meters), the scale of head losses (in the form of decimal fractions determining the ratio of the height of head loss to the length of the pipeline), and the scale of velocity (in meters per second). The scale of discharges includes discharges from \(0.5\) meter to \(1000\) liters per second, the numbers increasing from bottom to top. The scale of diameters covers diameters from \(0.01\) meter to \(1.00\) meter, with—

than which the numbers also increase from bottom to top. The scale of slopes (head losses) contains slopes beginning with \(1:1000000\) up to \(1:1\), the numbers increasing from top to bottom. Finally, the scale of velocities contains velocities within the limits from 0.05 meters to 10.00 meters.

This diagram is constructed in the following manner. The distance between the scale of discharge and the scale of hydraulic slopes is chosen with such a calculation that the logarithms of the discharges are laid off on a scale 3 times larger than the logarithms of the slopes. In this case, if we denote the distance between the scales of discharges and slopes by \(l\), and the distance between the scale of discharges and the scale of diameters by \(x\), then the following relation must be satisfied:

\[ \frac{x}{l-x}=\frac{3.19}{7}:\frac{.19}{4}=12:7, \tag{65} \]

whence

\[ x=\frac{12}{19}\,l. \]

The value \(x\) was taken equal to 70 mm. Therefore

\[ l=\frac{95.19}{12}=110.8\ \text{mm}. \]

The modulus of the scale of hydraulic slopes is to be taken so that one unit of logarithms corresponds to 20 mm. The scale of discharges, as stated, is 3 times larger, i.e. 60 mm per unit of logarithms. Proceeding from this relation, on the lines spaced at the above-indicated distance, the logarithms of the numbers are laid off: for discharges from 1 to 1000, and for slopes from 1 to 0.000001. In this way the scales of discharges and hydraulic slopes are obtained. To obtain the scale of diameters, there was chosen (on the basis of the table for Flamant’s formula) such a relation of \(Q\) and \(i\) that with it \(D\) would be equal to 1.00 m, and the points corresponding to these values of \(Q\) and \(i\) were joined by a straight line. Its intersection with the line of the diameter scale gives the point marked 1.00. In the same way the point for 0.10 m was found. The distance between these points was divided proportionally to the divisions of the logarithmic ruler, and the divisions were continued in both directions. In this way the scale of diameters was obtained. The scale of velocities is obtained in a similar manner.

The method of using the diagram for Flamant’s formula, shown in drawing 20, follows naturally from the preceding. The corresponding values of the four elements characterizing the flow—discharge, diameter, hydraulic slope, and velocity—are found on one straight line crossing the scales, which is determined by two of these given quantities. The other two unknowns are read on

at the points where the secant line meets the corresponding scales. In practice, they avoid drawing the secant line on the drawing itself, which would cause it to become quickly soiled and damaged. Instead, it is much simpler, quicker, and more convenient to use either a stretched thread or a transparent strip of paper, celluloid, etc., on which a straight line has previously been drawn (called a tracing straightedge). In doing so, one may move this strip either with the fingers or, better, by means of two pointed pins fastened at its ends. In the latter case it is convenient, having placed one point on a known division, to rotate the straight line about the axis of the point, without danger of slipping or displacement.

The graphical methods for calculating pipelines that may be called nomographic are exhausted by those described above. But there exist graphical methods and procedures, also applied to the calculation of a water-supply network, of another type, essentially different. At the beginning of this work I indicated the fundamental distinction between these two methods of graphical calculation. Nomographic diagrams represent one or another mathematical relation for all possible values of the elements entering into it, within specified limits. The other method of graphical calculation consists in replacing numerical calculation by the drawing of a diagram (épure) as applied to the data of each particular case, so that each time, for a new set of data, a new épure must be prepared. Such a method may be called graphical calculation in the proper sense of the word. I call it, in distinction from the nomographic methods of pipeline calculation, the idiographic method, and the diagrams belonging to it—idiographic diagrams.

An example of the application of such an idiographic method to the calculation of water mains is the attempt at graphical calculation of the simplest elements of a water-supply network, due to M. S. Yasyukovich (“A Graphical Method for Calculating a Network of Water Pipes”).

I shall not dwell in the present work on the idiographic methods of calculation defined above, intending to devote another work to this interesting question. I shall say only that the application of this method to the calculation of individual pipelines is entirely possible, i.e., to the solution of that single problem which is solved directly by means of nomographic methods. Among others, I too have had to arrive at one such method, and it must be thought that this method is not the only one. But the role of idiographic methods of hydraulic calculation is far more important. They make it possible, in essence—something inaccessible to nomographic methods of calculation—to calculate more or less complex combinations of pipelines. There is therefore reason to think that precisely these methods, when brought into connection with the principles of graphi-

cal integration, may provide a path toward the solution of the difficult problem of calculating an entire network of water-supply pipes, one less conditional than that employed at the present time. It seems to me, in general, that in the matter of the hydraulic calculation of water mains the next questions in order are precisely the development of idiographic methods for calculating pipelines, on the one hand, and, on the other, the graphical calculation of entire networks, or at least of their elements—that is, combinations of individual pipelines—by means either of the indicated methods or, perhaps, of other methods.

As for the nomographic methods proper for calculating water mains, which have been discussed in this work, it is apparent from the foregoing that the number of methods for the graphical representation of hydraulic formulas and equations is rather large; they have been especially widely developed in application to pipelines of circular cross-section. All of them represent applications of various principles falling within the domain of nomography and worked out by Descartes, Pouchet, Lalanne, and d’Ocagne. The guiding aim of applying the methods of nomography in the indicated methods of hydraulic calculation, and the reason for the successive replacement and greater or lesser dissemination of the various methods—that is, of the various types of diagrams—have been (apart, of course, from the accuracy of the formulas that were subjected to graphical representation) the simplicity and convenience of the practical use of the diagrams, and the consequent saving of labor and time. In this respect it may be said that the nomographic method of hydraulic calculation at the present time, with the introduction into use of diagrams of conjugate scales and logarithmographic tables, has achieved such success that it is difficult to go further.

To illuminate this question, which determines the raison d’être of all graphical methods of calculation, let us say a few words about the advantages of such calculation over numerical calculation, even when facilitated by tables, and about the comparative merits of the separate methods of graphical calculation.

The value of all means that accelerate and facilitate the conduct of the numerous and complex computations associated with technical calculations is generally recognized. All aids directed toward this end in the field of hydraulic calculation strive to eliminate entirely the need to perform any computations, and to make it possible to carry out the entire calculation of a pipe network with only this aid and a form for recording the obtained results at hand. Such aids are various kinds of tables, numerical or graphical.

Numerical tables, allowing one to have only two input variable quantities along two coordinates, compel—

Nomographic Methods of Calculation

make it possible to break up a calculation formula, which usually contains a large number of variables, into component parts, to compile a separate table for each such part, and to use several tables jointly. In this way the number of tables, the working time, and the possibility of error in using them are increased. In graphical tables, in one and the same place, lines are placed for all the variables, in the form of one or several systems. In this case, even when passing from one system to another, the operation is carried out very easily, without memorizing complex numbers, but merely by a simple movement of a finger, ruler, or transparent straightedge.

Numerical tables, in calculation, give a series of numbers that say little to the mind and, moreover, do not allow confidence in their freedom from error, if one takes into account the difficulty of proofreading numerical tables of every kind. For a graphical table, which gives results in a regular order to which the eye quickly becomes accustomed, the error is limited only by the limits of accuracy of reading (not counting the possibility of error in writing down the result obtained). A gross error, with correct use of the diagram, cannot occur.

Numerical tables, offering a direct answer only for numbers written along the coordinates and inevitably differing from one another by larger or smaller amounts, require, for intermediate values, either interpolation (not always exact when the intervals are large) or the performance of only an approximate calculation; and this, in formulas divided into several parts, is undesirable. A graphical table, by giving a continuous series of values embodying the dependence between the variable quantities, offers a direct answer for any assigned quantities whatever.

Only the advantages just listed are inherent in graphical tables in general. But, as was seen above, there are two types of graphical tables. In some, for constructing the lines expressing the equations of calculation, the variables themselves are laid off along the coordinate axes (diagrams of isopleth curves and straight lines); in others, along axes arranged in one way or another, the logarithms of the variables are laid off (logarithmo-graphical tables and diagrams of conjugate scales). It must be said that these latter, being newer types of diagrams, by adopting the logarithmic character of the divisions, thereby provide new advantages that distinguish them from diagrams with normal graduation of the axes. These advantages consist chiefly in the following.

Owing to the law of variation of the logarithms of successive numbers, the range of the table, i.e. the limits of the values entering into it,

may be very broad, with sufficient accuracy of subdivision. Indeed, in order to fit, on a normal scale, values from 0.00001 to 1, while still making it possible to read them off, either an enormous size of drawing would be required, or divisions and subdivisions differing so greatly that they could not be caught by the unaided eye. In diagrams of the logarithmic type, for the same purpose only an extremely limited size of drawing is required, with quite sufficient clarity and accuracy. This is the valuable quality of logarithmic diagrams. Further, the percentage of accuracy in calculations when read from diagrams with logarithmic subdivision, as was shown above, is always one and the same. In operations with small numbers the absolute accuracy is greater; in operations with large numbers it is smaller; but the magnitude of the possible error in relation to the reading remains the same.

If one compares the two most modern types of diagrams for nomographic calculation, namely the method of logarithmic-graphical tables and the method of conjugate scales, it must be said first of all that both of these types possess such general advantages and bring hydraulic calculation to such simplicity that it is difficult to draw a serious distinction between them. However, one may note in the method of logarithmic-graphical tables certain shortcomings which are avoided in the method of conjugate scales.

In fact, the method of using logarithmic-graphical tables is reduced each time to finding the point of intersection of two lines and reading the mark of the straight line passing through this point. Since the marks, as necessary, are written either at the ends of the lines, or in any case at some usually considerable distance from one another, one has, whether starting from the mark and seeking the point of intersection, or conversely, to follow the line over a known extent. In applying this operation to three lines at each reading, there is always the danger, through the slightest inattention, of passing from the given line to a neighboring one and thereby making an error in the reading. In this respect, the use of a diagram of conjugate scales, where the sought point and its reading are indicated by the intersection of a ruler or the line of a transparent strip with the corresponding scale, appears somewhat simpler. Further, if the values of the variables with which one has to deal are not those to which the lines actually drawn on the drawing correspond, then, when using a logarithmic-graphical table, one has to perform a mental interpolation between these lines. This operation, for all its ease, is still not as simple as remembering by eye an intermediate point between the divisions marked on one of the scales of d’Ocagne’s diagrams.

The number and mutual interlacing of lines in logarithmic-graphical tables do not, of course, present a serious inconvenience, since the eye quickly becomes accustomed to it. However, this circumstance, as well as the rather considerable attention required in working, may in the end tire the eye, and the sooner, the more systems of lines the table contains. In a diagram of conjugate scales, with its usual composition, there can be no question of interference by one system of scales with another; even with several systems of scales there can be no question of darkening, which is quite possible for a logarithmic-graphical table. It must be supposed that, thanks to this, the use of a diagram of conjugate scales is also less fatiguing.

Here it is appropriate to emphasize, with respect to both kinds of diagrams, that their chief advantage lies in simplicity. Therefore, in compiling and using diagrams, one should avoid everything that may be superfluous and that may make reading difficult. In addition, they must be drawn on a sufficiently large scale; otherwise the approximation may be inadequate.

The foregoing does not prevent us, of course, from considering calculation with the aid of logarithmic-graphical tables very convenient and valuable. We may say that both of the newest methods of nomographic calculation, both logarithmic-graphical tables and diagrams of conjugate scales, achieve, to perfection, the aim pursued by all methods in general directed toward simplifying hydraulic calculation, and bring the process of calculation, one may say, to ideal simplicity and speed.

From this it is clear that at present there is hardly any practical need to develop new methods for the graphical calculation of individual pipelines, and specialists in water-supply hydraulics may direct their work to other fields. As regards work on existing nomographic methods of calculation, one may perhaps wish only for their further development in application to the various forms of cross-section (besides the circular) used in practice, and to different degrees of filling. But it is quite natural to wish that these graphical methods should receive due attention from specialists and should become more widely known and applied, as an exceedingly valuable means of calculation, reducing in the best cases, one may say, to a minimum the labor and time required, without any detriment to practical accuracy, provided, of course, that the computational hydraulic formulas represented by the diagrams are properly chosen.

1910.

Literature on This Subject

Akulov. The service of old water-supply pipes and the application of the graphical method to the solution of hydraulic problems (Proceedings of the 5th Water-Supply Congress, 1901).

D’Aubrive et Villerupt. L’album des abaques pour le calcul des conduites d’eau. 1891.

Baumeister. Städtische Strassenwesen und Städtereinigung. 1887.

Bechmann. Salubrité urbaine. II. Assainissement. 1899.

Wehner. Ein Betrag zur Berechnung des Rohrwiederstandes in der Praxis. (Gesundheits-Ingenieur, 1897).

Vrublevskii. A graphical method for calculating water drains (News of the Society of Civil Engineers, 1907).

Hobrecht. Die Kanalisation von Berlin. 1882.

Gorbachev P. F. On the calculation of flow velocities and carrying capacity in water mains and drains. 1904.

Dariès. Calcul des conduites d’eau. 1900.

Kashkarov N. A. Calculation of pipelines by Bertrand’s graphical method. 1907.

Coffin. The graphical solution of hydraulic problems. 1900.

Collignon. Cours de mécanique. II. Hydraulique.

Lalanne. Mémoire sur les tables graphiques et sur la géométrie anamorphose appliquée à diverses questions qui se rattachent à l’art de l’ingénieur (Annales des Ponts et Chaussées, 1846).

Lampe. Untersuchungen über die Bewegung des Wassers in Röhren (Civil-Ing., 1873, Bd. XIX).

Van-Muyden. Abaque logarithmique pour le calcul des conduites d’eau sous pression. 1905.

Nikolin Ya. I. Logarithmic-form formulas for the calculation of water mains. 1910.

Nikolin Ya. I. Graphical methods for calculating water supply and sewerage. Issue I. Theory and applications of the method of conjugate scales. 1910.

D’Ocagne. Coordonnées parallèles et axiales. 1885.

D’Ocagne. Nomographie. 1891.

D’Ocagne. Traité de Nomographie. 1899.

D’Ocagne. Exposé synthétique des principes fondamentaux de la Nomographie. 1903.

II

Satkevich V. A. Calculation of a water-supply pipe network by means of a logarithmic-graphical table (Stroitel, 1898).

Schilling. Ueber die Nomographie de M. d’Ocagne.

E. B. and G. M. Taylor’s diagrams of the discharge of pipes in accordance with Kutter’s formula. 1891.

Thiem. Ueber graphische Durchmesserbestimmung bei Wasserleitung (Journ. für Gasbeleucht. und Wasserversorg., 1885).

Flamant. Hydraulique. 1900.

Cheremashinsky M. M. Water supply of cities. 1905.

Chizhov N. K. A mechanical method for calculating head loss (Stroitel, 1897).

Yasyukovich M. Calculation of gutters by means of logarithmic-graphical tables. 1906.

Yasyukovich M. Graphical method for calculating a network of water-supply pipes. 1905.

Table 1.

Drawing 1.

[Figure: coordinate axes \(x\) and \(y\), with origin \(0\); a decreasing curve is plotted, and a point on it is projected to the axes, marking the corresponding values \(x\) and \(y\).]

Drawing 2.

[Figure: graph on a rectangular grid. The vertical axis is labeled \(c\), with marks from \(0\) to \(90\). The horizontal axis is labeled \(\rho = \frac{\Omega}{\omega}\), with marks \(0.01,\ 0.025,\ 0.05,\ 0.075,\ 0.10,\ 0.125,\ 0.15,\ 0.175,\ 0.20,\ 0.25,\ 0.30,\ 0.40,\ 0.60\). Three rising curves are shown and labeled at the right:]

\[ b = 0.27 \]

\[ b = 0.35 \]

\[ b = 0.45 \]

Plate II.

Drawing 3.

A horizontal \(x\)-axis marked with points:

\(0,\ x_0,\ x_1,\ x_2,\ x_3,\ x_4,\ x_n,\ x\).

Drawing 4.

Graph with vertical axis \(y\) and horizontal axis \(x\). The \(y\)-axis is marked from \(0\) to \(1000\) in hundreds; the \(x\)-axis is marked \(1,\ 2,\ 3,\ 4\).

Table III.

Drawing 5.

Drawing 6.

Table IV.

Drawing 7.

Drawing 10.

Table V.

Drawing 8.

Drawing 9.

Table VI.

Drawing 11.

Drawing 12.

Hydraulic slopes (in mm per meter)

Diagram (for asbestos-cement)

Diagram (for cast-iron)

Plate VII.

Drawing 13. Logarithmic-graphical table by Eng. A. van Muyden for calculating water-supply pipes.

(According to Lévy’s formula, for pipes in long-term service.)

[Graphical chart]

Visible chart labels:

• Hydraulic gradients (in millimeters per meter).
• Units marked on the scales: mm, cm.
• The chart consists of a logarithmic grid with families of oblique curves and numerical markings for pipe calculation.

Table VIII.

Prof. N. K. Chert. 15. Diagram for calculating water-supply pipes (Lotz’s formula).

\(Q\) — desired water discharge (in cubic feet) in fractions of the filled cross-section, for circular pipes of diameter from 2 to 30 inches.

\(R =\) hydraulic radius = wetted area / wetted perimeter.

\(V =\) mean flow velocity.

\(j =\) slope of the drop line / length of conduit.

\(K =\) roughness coefficient.

or:

Lamer’s formula:

\[ j = K \frac{V^{1.8}}{R^{1.25}} \]

The value \(K\) for water pipes is taken from Kutter’s formula.

Table IX.

Drawing 16.

Drawing 17.

Drawing 18.

Drawing 19.

Table X.

Drawing 20. Dariès diagram (Bertrand’s abridged version) for calculating water-supply pipes (Flamant formula).

Diagram headings:

• Discharges (liters per second).
• Diameters (meters).
• Hydraulic slopes.
• Velocities (meters per second).

Visible scale annotations include metric subdivisions for the hydraulic-slope scale: millimeters, centimeters, decimeters, meters, units, tens.

1. Details are in my work “Formulas of logarithmic form for the calculation of water conduits.” ↩↩↩↩↩↩

2. The methods of reading will be discussed at the end. ↩