Improvements in Sheets Ruled with Co-ordinates for Mathematical Purposes.
Year: 1892

page_text: Date of Application, 12th Oct., 1882.
Complete Specification Left, 1st May, 1893-Accepted, 2nd June, 1893
PROVISIONAL SPECIFICATION.
Improvements in Sheets Ruled with Co-ordinates for Mathematical
Purposes.
I, EDWIN HUMAN 8 Princes Street Westminster London Civil Engineer do
hereby declare the nature of this invention to be as follows:-
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This invention relates to sheets prepared for geometrical drawing purposes
having ruled lines on their surface for rectangular co-ordinates. The invention
consists in arranging the co-ordinates in a special manner by which particular
properties are secured for curves and drawings made on the sheets.
The arrangement of the co-ordinates is such that the distance of any ordinate
or abscissa from a given point taken as origin on the sheet is proportional to the
logarithm of the number which would represent its distance if measured on a sheet
of Cartesian co-ordinates drawn to the same scale: In a slicet ruled on the Cartesian
system of co-ordinates each line stands at a distance from the origin proportional
to the number assigned to that line but this invention provides for sheets ruled
with a system of co-ordinates in which each line stands at a distance from the
origin proportional to the logarithm of the number assigned to that line measuring.
from left to right for integers and right to left for fractions, horizontally and
upwards for integers and downwards for fractions, vertically.
•
The surface ruled in the manner stated may be prepared of any suitable material.
The spacing of the co-ordinates may be logarithmic dimensions of any convenient
scale of measurement; and the ordinates and abscisse may be ruled to the same or
different scales respectively.
Dated this Twelfth day of October 1892.
EDWIN HUMAN.
COMPLETE SPECIFICATION.
Improvements in Sheets Ruled with Co-ordinates for Mathematical-
Purposes.
I, EDWIN HUMAN, Civil Engineer, formerly of 8 Prince's Street, Westminster,
London and now of Halifax House, Cinnamon Gardens, Colombo, Ceylon, 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 relates to sheets prepared for geometrical drawing purposes,
having ruled lines on their surface for rectangular co-ordinates. The invention
consists in arranging the co-ordinates in a special manner by which particular
properties are secured for curves and drawings made on the sheets.
The arrangement of the co-ordinates is such that the distance of any co-ordinate
or abscissa from a given point taken as origin on the sheet is proportional to the
logarithm of the number which would represent its distance if measured on a sheet
of Cartesian co-ordinates drawn to the same scale. In a sheet ruled on the Cartesian
system of co-ordinates, each line stands at a distance from the origin proportional
to the number assigned to that line, but this invention provides for sheets ruled
with a system of co-ordinates in which each line stands at a distance from the
origin proportional to the logarithm of the number assigned to that line, measuring
from left to right for integers and right to left for fractions horizontally; and
upwards for integers and downwards for fractions vertically.

page_text: The surface ruled in the manner stated may be prepared of any suitable material.
The spacing of the co-ordinates may be logarithmic dimensions of any convenient
scale of measurement, and the ordinates and abscissæ may be ruled to the same or
different scales respectively.
In order to explain more clearly the use of such sheets I will now 'describe
various operations, which will be understood by reference to the accompanying
drawing in which a sheet is shown upon a reduced scale. The origin is marked a
and the sheet is ruled with rectangular co-ordinates. The ordinates are figured at
the bottom of the sheet and the abscisse at the left side.
It will be seen that the ordinates are figured with integers on the right of the
origin and with fractional values on the left of the origin, similarly the abscissæ
are figured with integers above the origin and with fractional values below the
origin. The distance of any ordinate or abscissa from the origin is proportional to
the logarithm of the number placed upon the particular co-ordinate, thus the
distance from the ordinate marked 1 passing through the origin to that ordinate
marked 10 may be taken as unity since log. 10 is equal to unity, log. 2 is equal
to 0:30103 so that the distance of the ordinate marked 2 from the ordinate
marked is three tenths of the distance from the ordinate 1 to the ordinate 10,
again the ordinate 100 is twice as far from the ordinate 1 as is the ordinate 10 and
this is so because log. 100 is equal to 2, the same law applies to the abscissæ and
in all cases the distances of the lines are to be measured from the co-ordinate
passing through the origin a..
I will now illustrate the use of the sheets by a few examples. Any of the curves
included in the order y= ac" can be drawn with only one numerical computation.
Thus parabolas & hyperbolas of every degree and a vast variety of other curves
in daily practice with and without specific names can be easily and quickly obtained
by means of these sheets. All curves included in the order of equations named
above become straight lines on the sheets; hence only two points are required to be
known (one is always given by the terms of the problem in hand) to draw the
straight line which represents the required curve. All other points on the curve
can be read off directly and being plotted on ordinary squared paper, the actual
required curve can be obtained at once. I will now illustrate the use of the sheets
for a few special cases. Suppose y 4 is an equation to a curve required then
if the value of r bo unity the value of 4 a= 4, therefore y = 4, it is thus obvious
that one point on the curve is at the junction of the abscissa 4 and the ordinate 1,
again suppose = 5 then y = 100 and a second point on the curve is therefore at
the junction of abscissa 100 and ordinate 5. Again in the case of y = 2. if
x= 1, 2 and one point of the curve is at the junction of abscissa 2 and
ordinate 1, if e 3, y=54 and a second point is at the junction of abscissa 54 and
ordinate 3, the same method applies to all the other straight lines shown on the
sheet. It is of course obvious that for accurate reading it is preferable to again
divide the distances between the several co-ordinates and this is done upon the
actual sheets which are much larger than that shown in the drawing.
=
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In the general equation y ax the exponent n may be positive or negative, a
whole number or a fraction. The co-efficient a may be any positive number, whole
or fractional and a must be always a positive quantity.
These sheets may be used for various purposes such as to draw curves of adiabatic
and isothermal expansion in fluids, curves of expansion of saturated and superheated
steam, curves of variation of bending movements, shear stress, and deflection in
beams, values of load for different spans in girders and joists, curves of variation
of horse power, thrust, speed etc. in ships, curves for values of discharge for water
flowing in pipes and channels, variation of loss of heat due to friction in pipes,
calculations in electrical engineering, and for curves in ballistics.
It is to be noted that the origin of the diagram denoted by the letter a in the
drawing is the intersection of the axes of co-ordinates and these axes are marked 1,
since log. 1 = 0 and the boundary of each square group of crossed lines is to be
marked a multiple of ten in due order of progression along the axes from left to

page_text: right and from below upwards. Co-ordinates to the left of and below the inter-
section of the axes at the origin are to be marked with fractional numbers and
although the value zero cannot be represented, values as near zero as may be
required can be represented also it may be noted that negative quantities cannot
be represented on these sheets.
The sum of two co-ordinates will give the product of the numbers they represent
and their differences the quotient. The sheets may be produced by printing,
lithography or other suitable means and I prefer to use a good quality of drawing
paper but other convenient materials may be employed.
Having now particularly described and ascertained the nature of my said inven-
tion, and in what manner the same is to be performed, I declare that what I
claim is:-
1. The production of sheets ruled with co-ordinates for mathematical purposes
in which the co-ordinates are spaced to logarithmic dimensions substantially as
described and illustrated.
2. The production of sheets ruled with co-ordinates for mathematical purposes
in which the co-ordinates are spaced to logarithmic dimensions, the said sheets
being prepared by any suitable process upon the surface of any convenient material
substantially as described and illustrated:
Dated this lot day of May 1803
JOHN P. O'DONNELL,
70 & 71, Palace Chambers, London, S.W., Agent for the Applicant.