Improvements in Maps to Facilitate the Measurement of Distances on them.
Year: 1888

page_text: Improvements in Maps to Facilitate the Measurement of Distances
on them.
WILLIAM LUCAS, 10, Caledonia Street, Engineer, King's Cross, London, THOMAS
ALEXANDER GARRETT Engineer 10 Caledonia Street King's Cross London do hereby
declare the nature of this invention to be as follows:-
Hitherto it has been usual in some cases, for example in maps of towns, to divide
the
map into squares each representing a quarter of a square mile or some other
convenient area in order to assist a person's judgement in estimating roughly the length
of a route.
The object of our invention is to enable any person to find quickly and easily the
approximate length of a route by a simple process of counting without having to use
his judgement at all.
We accomplish this object by inscribing on the map hexagons all the sides and
angles of which are equal, which are so situated that no space is left between any two
adjacent hexagons. The hexagons are made of such a size that the approximate
average length of a great number of lines drawn at random across one of them, which
may be determined by the help of experiment or calculation is some convenient part
or multiple of the length which represents one mile on the scale of the map, so that
by multiplying the distance in miles which this approximate average length represents
by the number of hexagons cut by any given route the approximate length of this
route is obtained in miles.
On any given map the smaller the hexagons are the more exactly can the length of
a route be obtained.
10uL .:ונות
T11000

page_text: COMPLETE SPECIFICATION.
Improvements in Maps to Facilitate the Measurement of Distances
on them.
WILLIAM LUCAS 128 Clerkenwell Road London Engineer THOMAS ALEXANDER
GARRETT 128 Clerkenwell Road London Engineer 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 :-
Hitherto it has been usual in some cases for example in maps of towns to divide the
nap into squares each representing a quarter of a square mile or some other
convenient area in order to assist a person's judgement in estimating roughly the
length of a route.
The object of our invention is to enable any person to find quickly and easily the
approximate length of a route by a simple process of counting without having to use
his judgement at all.
We accomplish this object by inscribing on the map hexagons the sides and angles
of which are equal which are so situated that no space is left between any two adjacent
hexagons.
It can be proved mathematically that if from a point taken at random within a
regular hexagon two straight lines be drawn at random to the perimeter of the
hexagon the mean value (M) of the sum of the lengths of these two straight lines will
πανα
where T =
3.14159 and a half the length of a side of the hexagon.
be
2
=
Hence the length of a side of the hexagon
-
4 M
T√3
From which if a particular
value be given to M the length of a side can be determined.
If a map be divided up into regular hexagons in the manner described above and a
line, fulfilling these conditions firstly that it is made up of straight lines secondly that
not more than one angular point occurs inside any hexagon, be drawn at random on
the map, the portions of the line in a hexagon may in some cases be less that the
mean value (M) given above and in some cases greater, and the sum of the excesses
above the mean value (M) will be approximately equal to the sum of the deficits
below it so that the whole length of the line will be approximately equal to the mean
value (M) multiplied by the number of hexagons through which the line passes.
The greater the number of hexagons through which the line passes that is to say the
longer the line or the smaller the hexagons the less is the probability that the error in
the result thus obtained will exceed any given value.
It will be found that although an actual route on the map does not fulfil the two
conditions mentioned above yet by making the hexagons small enough the length of
the route is given with sufficient accuracy for all ordinary purposes.
As an example if we take a map the scale of which is six inches to the mile, then
in order that the mean value (M) may be three quarters of an inch which represents

page_text: on this scale one eighth of a mile it will be found from the formula given above that
the length of each of the sides of the hexagons which are to be drawn on the map
should be 55 of an inch. Then the length of any route on the map will be
approximately given in miles by counting the number of hexagons through which the
route passes and dividing this number by eight.
Having now particularly described and ascertained the nature of our said Inven-
tion, and in what manner the same is to be performed, we declare that what we
claim is;
Describing straight lines on a map in such a way as to divide it into a system of
WILLIAM LUCAS.
T. A. GARRETT.