Year: 1898

page_text: I, ALFRED JULIUS BOULT, of 111, Hatton Garden, in the County of Middlesex,
Chartered Patent Agent, do hereby declare the nature of this invention and in
what manner the same is to be performed, to be particularly described and
ascertained in and by the following statement:-
This invention applies to a rule modified so as to enable it to be used for
tacheometric calculations, giving sines and cosines with a much greater exactitude
than hitherto and the coordinates with an approximation hitherto not attained and
this by a single position of the slide or movable part of the rule, without any
advantages inherent to the rule being lost.
The rule according to this invention as regards the fixed part differs from the
ordinary rule only by a scal! D of equal divisions being added to the edge of
the rule.
The front face of the fixed rule has therefore marked on it in the usual manner
along the upper edge a logarithmic scale of numbers, extending through two units or
consecutive intervals, of the powers of 10, and along its lower edge another scale or
of numbers extending through a single interval or through a single logarithmic
unit but of double length.
On the front side of the slide-which is also the same as usual-the same
logarithmic tables of numbers are repeated arranged identically with those of the
fixed rule.
To read off, a glazed slider C is provided with an indicating line continued on to
the metallic part along the edge of the latter and serving to transmit the
indications on to the scale of equal parts (Fig. 3) on the side.
The essential modifications of the rule according to this invention are
follows:-
as
On the other side of the slide B (Fig. 2) instead of the usual scales giving the
values of log. sin., log. tan. and of equal divisions, there are engraved or marked
the following scales.
1. Along the upper edge, the series of the values of log. sin. A cos. A the value
of A varying between 0.64 and 50 if the graduation is the centesimal, or between
equivalent angles, if sexagesimal graduations be adopted, the intervals between
consecutive lines being as small as is consistent with clearness for reading..
2. Along the central line of the slide, the values of log. sin.2A are marked between
the limits of A 0.64 and A = 6.38 for centesimal graduation or corresponding
limits for sexagesimal graduation.
=

page_text: 3. Along the lower edge there are marked the values of the same log. sin. 2A
between the limits of A
= 6.30 and A = 100 for centesimal graduation or between
corresponding limits for sexagesimal graduation.
At the ends and in the middle of the scales are three transverse lines, so as to
subdivide the three scales into six half scales.
At the right hand end of the right hand half scale, and at the left hand end of
the left hand half-scale, there are marked numbers which may be just as well the
characteristics as the numbers of figures of the natural values of the functions within
the semi-scale at the side of which they are arranged. In the example illustrated
(Fig. 2) these numbers are the characteristics.
Manner of using the rule. The rule according to this invention is especially
adapted for calculations required in surveys made by the method of tacheometry,
which calculations are as follows:-
I. Given the generating number N of a point surveyed and the angle A of the
depression or elevation of the line of collimation to the horizon to find:
a) The correction to be made in N in order to obtain D, i.e., the horizontal
distance of the point surveyed.
b) The difference in the level h between the centre of the instrument and the
point of collimation on the levelling staff.
Answer. The two determinations are based on two formulae
=N. 81n. A cos. A
and are obtained by a single position of the slide.
The left hand (or the right hand) end of the slide is brought under N read on
the left hand (or right hand) scale of the upper fixed part. The slider is placed
on the angle A read on one of the two lower scales of the slide, and there is
obtained at the corresponding point of the upper fixed scale, the correction N. sin.²A
which is to be subtracted from N in order to obtain D.
Leaving the slide in the same place, the slider is placed on A, read on the upper
scale of the slide, and at a corresponding point of the upper scale of the fixed rule,
there will be obtained h.
For most values of A it is possible to start from N read either on the right or left
hand scale of the fixed rule. For certain values of A, however, it is not immaterial
but it is necessary to use one or the other of the scales so as to have A both in the
upper and in the lower scale of the slide comprised between the fixed rule (which
is always possible).
Then there will be obtained by means of numbers on the side of the half-scales
the figures or the characteristics of the results.
Instead of corrections of N, it is possible to find directly the distance D if, instead
of placing the slider on A it is placed on 100 - A, as in this case there will be
obtained N. cos.2A = D.
2. Given the distance D of a point from the surveying station and azimuthal
angle A of its radius vector, to find the rectangular co-ordinates D. sin. A
and D. cos. A of the point.
This can be done by means of the new rule in two different ways.
The first is the same as for other rules and consists in using the two lower scales
of the rule by referring them to the lower scale of the fixed rule relatively to which
they represent not the log. sin.2A but log. sin. A, as the logarithmic unity of the base
of this scale is the double of the upper scale of the fixed rule.
If one of the ends of the slide is brought to D on the fixed rule, and the slider
on the angle A, there will be obtained on the corresponding end of the lower fixed
scale D sin. A. By moving the slider to 100-A in a similar manner D. cos. A
will be obtained. Very often the two determinations require two positions of the
slide as 100-A may lie outside the fixed rule.

page_text: For small angles A or near 100, one of the co-ordinates becomes inexact as the
slider comes near the end 100 of the slide where the angles A are crowded in a
small space.
The second system of finding the co-ordinates is peculiar to the rule according to
the present invention and obviates the above mentioned drawback and at the same
time has the great advantage of giving the two co-ordinates with great approxima-
tion and with a single position of the slide. This system is based on the formulae D
sin. A, D cos. A transformed as follows:
These formulae reduce the determination of coordinates to the use of the scales
of the slide, combined with the upper scales of the fixed rule, by following the
same method as described for the correction of N and determination of h, except
that D is doubled and the angle A is halved.
Moreover, by taking care to render A less than 50 which is always possible-
the reading of A always remains in the first divisions of the scale, i.e. in a position
very favourable for obtaining a great approximation to exactness. 3. Although
the scale of the tangents has been suppressed it is always possible to find the
tangent of any angle by the formula
which can be determined by a single position of the slider for any angle.
All the described determinations are obtained without appreciable error for
angles less than 0.64, if care is taken to operate in an analogous manner, as well as
for angles A 10,100 or 1000 times greater, but, of course, the variations in the
characteristics must be taken into consideration.
The advantages of the new rule are as follows.
The rule can be used for all calculations for which an ordinary Mannheim rule
can be used, and serves for calculations involving trigonometric functions. Sines,
cosines, tangents, cotangents, the approximation for sin. and cos. being much
greater than in the ordinary rules, as the scales for them are of double length.
The new rule can be moreover used for all calculations required in rapid
measuring, the operation being very simple and quick.
Finally, this rule gives, by a single position of the slide, the co-ordinates with an
approximation hitherto unattained in apparatus of the same length.
By means of a rule 25 cm. long such as the one shown in the accompanying
drawing, and for distances D up to 400 metres, the co-ordinates are obtained with
an error which never exceeds 10 cm. and as a rule is within 5 cm.
Having now particularly described and ascertained the nature of the said
invention as communicated to me by my foreign correspondent and in what
manner the same is to be performed, I declare that what I claim is :-
-
1. A slide rule provided on the reverse side of the slide instead of with the
ordinary divisions, with values of log. sin., log. tan. and equal parts, the upper edge
being marked, in the manner described with divisions for log. sin. A cos. A along
the central line, and along the lower edge the divisions for log. sin.2 A, whereby
the co-ordinates can be calculated with great exactitude by a single position of
the slide, the instrument being suitable for calculations in tacheometric surveys
without losing any of the advantages of the ordinary rule, substantially as
described.

page_text: 2. A slide rule substantially as described and illustrated in the accompanying
drawings.
BOULT & WADE,
Agents for the Applicants.