Special graduations, marking the position of constant factors which
frequently enter into engineering calculations, are found on most slide
rules. Usually the values of π = 3·1416 and (π)/(4) = 0·7854—the “gauge
points” for calculating the circumference and area of a circle—are
marked on the upper scales. The first should be given on the lower
scales also. Marks _c_ and _c_^1 are sometimes found on the lower scales
at 1·128 = √((4)/(π)) and at 3·568 = √((40)/(π)). These are useful in
calculating the contents of cylinders and are thus derived:—Cubic
contents of cylinder of diameter _d_ and length _l_ = (π)/(4)_d_^2_l_;
substituting for (π)/(4) its reciprocal (4)/(π), the formula becomes
(_d_^2)/(1·273 × _l_), and by taking the square root of the fractional
part we have (_d_)/(1·128)^2 × _l_. This is now in a very convenient
form, since by setting the gauge point _c_ on C to _d_ on D, we can read
over _l_ on B the cubic contents on A. This example indicates the
principle to be followed in arranging gauge points. Successive
multiplication is avoided by substituting the reciprocal of the
constant, thus bringing the expression into the form (_a_ × _b_)/(_c_),
which, as we know, can be resolved by one setting of the slide. The
advantage of dividing _d_ before squaring is also evident. The mark
_c_^1 = _c_ × √(10) is used if it is necessary to draw the slide more
than one-half its length to the right.
A gauge point, M, at 31·83 = (100)/(π) is found on the upper scales of
some rules. Setting this point on B to the diameter of a cylinder on A,
the circumference is read over 1 or 100 on B or the area of the curved
surface over the length on B.
As another example of establishing a gauge point, we will take the
formula for the theoretical delivery of pumps. If _d_ is the diameter of
the plunger in inches, _l_ the length of stroke in feet, and Q the
delivery in gallons, we have
Q = _d_^2 × (π)/(4) × _l_ × (12)/(277). (N.B.—277 cubic inches = 1
gallon.)
Multiplying out the constant quantities and taking its reciprocal, we
readily transform the statement into Q = (_d_^2_l_)/(29·4) or
((_d_)/(5·42))^2 × _l_. Hence set gauge point 5·42 on C to _d_ on
D and over length of stroke in feet on B, read delivery in gallons per
stroke on A; or over piston speed in feet per minute on B, read
theoretical delivery in gallons per minute on A.
Several examples of gauge points will be found in the section on
calculating the weights of metal (see pages 59 and 60). In most cases
their derivation will be evident from what has been said above. In the
case of the weight of spheres, we have Vol. = 0·5236_d_^3, and this
multiplied by the weight of 1 cubic inch of the material will give the
weight W in lb. Hence for cast-iron, W = 0·5236 × _d_^3 × 0·26, which is
conveniently transformed into W = (_d_ × _d_^2)/(7·35) as in the example
on page 60.
With these examples no difficulty should be experienced in establishing
gauge points for any calculation in which constant factors recur.
_Marking Gauge Points._—The practice of marking gauge points by lines
extending to the working edge of the scale is not to be recommended, as
it confuses the ordinary reading of the scales. Generally speaking,
gauge points are only required occasionally, and if they are placed
clear of the scale to which they pertain, but near enough to show the
connection, they can be brought readily into a calculation by means of
the cursor. Usually there is sufficient margin above the A scale and
below the D scale for various gauge points to be marked. Another plan
consists in cutting two nicks in the upper and lower edges of the cursor
near the centre and about ⅛ in. apart. These centre pieces, when bent
out, form a tongue, which are in line with the cursor line and run
nearly in contact with the square and bevelled edges of the rule
respectively. A fine line in the tongue can then be set to gauge points
marked on these two edge strips, the ordinary measuring graduations
being removed, if desired, by a piece of fine sand-paper.
For gauge points marked on the face of the rule, the author prefers two
fine lines drawn at 45°—thus, ✕—and crossing in the exact point which it
is required to indicate. With the “cross” gauge point the meeting lines
facilitate the placing of the cursor, and an exact setting is readily
made.[7] All lines should be drawn in Indian ink with a very sharp
drawing pen. For a more permanent marking the Indian ink may be rubbed
up in glacial acetic acid or the special ink for celluloid may be used.
If any difficulty is found in writing the distinguishing signs against
the gauge point, the inscription may be formed by a succession of small
dots made with a sharp pricker.