IM- PROVEMENTS IN APPARATUS FOR EFFECTING CALCULATIONS
Year: 1871

page_text: LETTERS PATENT to Henri Lefevre, of Rathbone Place, Oxford Street,
in the County of Middlesex, Gentleman, for the Invention of "IM-
PROVEMENTS IN APPARATUS FOR EFFECTING CALCULATIONS.”
Sealed the 26th September 1871, and dated the 24th June 1871.
.
PROVISIONAL SPECIFICATION left by the said Henri Lefevre at the
Office of the Commissioners of Patents, with his Petition, on the
24th June 1871.
I, HENRI LEFEVRE, of Rathbone Place, Oxford Street, in the County
of Middlesex, Gentleman, do hereby declare the nature of the said
Invention of "IMPROVEMENTS IN APPARATUS FOR EFFECTING CALCULATIONS,"
to be as follows, that is to say :—
This Invention relates to certain apparatus of a simple and economical®
construction for effecting various calculations in an easy and expeditious
manner. The general formula on which the main principle of the said
apparatus is based is as follows, videlicit:-When four lengths forming
parts of similar triangles aro in the proportion
H

page_text: if one of these lengths be taken for a unit, a for example, we shall have
whence d=cb.
If to 6 there is given a numerical value from zero to infinity the direct
measure of the other two terms of the proportion, by the aid of any rule
divided into equal parts, will give-
for d the product of two numbers,
for c or 6 the quotient of two numbers.
The construction of the improved calculating apparatus is based on
the following principle, videlicit :-If at the extremity of a right line O A
we draw another right line A B at any angle, and if we join the point 0 1
of the former to the point C of the latter, and then through a point P
in OA draw a line parallel to A B until it meets OC'at M, we have the
following proportion :—

page_text: Second. That the direct measure of OP gives the quotient of s
divided by r.
Third. That the direct measure of AO gives the product of the
multiplication of the two numbers fand r.
It thus follows that by the direct measurement of lines there may be
obtained the products or quotients of numbers the approximation of
which will be closer in proportion as a smaller unit is adopted.
or
If, now, be make AB equal and perpendicular to OA, and complete
the square, and if we set out on O A, starting from the point O, distances
proportional to the vulgar fractions from 0 to 1, then every fraction })
OP will have for its complement or OP¹, the points P and P¹ of
which will be equally distant from the extremities of OA; and if we
divide A B into 100, 1000, 10,000, or more equal parts, like an ordinary
rule, and then turn down, first, the right-hand half of the square on to
the left half; second, the upper half of these two parts already turned
down together on to the lower half, we shall have reduced the whole
figure to a quarter of its size, and this quarter will contain all the
inductions of the figure from which it is derived by observing the
positions that the right-hand half of OC takes in the successive folds.
It will be sufficient to remark that each point P belongs at the same
time to and to in the figure thus reduced, and consequently it
should admit of this double determination.

page_text: figures engraved opposite to it. From these divisions parallel straight
lines are ruled vertically across the tablet, which lines are crossed at
right angles by other horizontal parallel lines ruled at equal distances.
These horizontal lines are numbered on the right-hand side of the tablet
from, by preference, 1 to 50 inclusive from the bottom to the top, and
on the left-hand side from 50 to 100 from the top to the bottom, but
other numbers may be used. A thin elastic or other cord is extended
from the bottom left-hand corner to an adjustable slide working in a slot
on the right-hand side of the tablet, and another similar cord extends
from the last-named slide back again across the board or tablet to a
second slide, also capable of adjustment along a slot on the left-hand
side of the tablet. A third slide adjustable along a slot in the top side
of the tablet serves to deflect the second cord to the required angles
when the number to be read is higher than is contained on the right-
hand column, in which case the reading is effected from the left-hand
column, taking the intersection of the second cord with the ruled lines
in lieu of or in addition to the intersections of the first cord. In lieu of
a cord or cords a straightedge may be used. The object of using two
cords and two columns of equal divisions is to enable the apparatus to
be made in smaller compass, as in lieu of extending one cord in one
direction to double the distance; and causing it to indicate upon a single
column of figures of double the length, the same result is obtained by
simply employing a second reversed cord and second column of divisions
of half the length that would otherwise be required.
As an example of one mode of using this calculating tablet suppose
it be required to add together / ¼, ½, and ¼, which would under the
ordinary process involve a considerable amount of calculation, by the aid
of the tablet it would be simply requisite to follow the vertical line
corresponding to each fraction to its point of intersection with the cord
and one of the horizontal lines, and then to follow the horizontal lines
and add together the figures taken from the right-hand column corres-
ponding to each line. The sum of these figures is then to be found on
the right or left hand column, as the case may be, and the corresponding
horizontal line is to be traced to its point of intersection with one of the
cords and one of the vertical lines before mentioned. On tracing this
latter line .down to the bottom edge of the tablet the exact sum of the
several fractions will be found marked opposite to such line. The same
principle of subdividing a given length into any desired
number of

page_text: fractions, and of subdividing another equal length into, say 100 equal
parts, by preference disposed for the sake of compactness in two separato
columns, figured respectively from the top downwards and from the
bottom upwards, may be carried out in the form of a sliding rule which
is combined with the tablet before mentioned. The sliding and stationary
portions of the rule are divided on the surface into any desired number
and variety of fractional parts of the length of the rule, and figured
accordingly. Along a longitudinal groove in the sliding portion of the
rule there is fitted a slide carrying a pointer or index, which is capable
of being adjusted to any of the divisions marked on the face of the rule.
In the back of the groove or channel along which the sliding part of the
rule slides there are two columns of equal divisions, the one on the left
being figured from the top to the bottom from 1 to 50 inclusive, whilst
the column on the right is figured from the bottom to the top from 50
to 100 inclusive. Other subdivisions may be added if desired according
to the degree of accuracy required, and any other number than 100 equal
parts may be used, but 100 will generally be found most convenient in
practice.
Supposing by this slide rule it is required to add together one-tenth,
ono-seventh, and one-fifth, the index is first adjusted along the slide to
the division corresponding to one-tenth; the slide is then drawn down
until the index coincides with the bottom of the rule. Whilst still
holding the slido in this position the index is moved to the division in
the slide corresponding to the fraction one-seventh, and the slide is again
moved so as to bring the adjusted index opposite the bottom of the rule.
The same operations are repeated with regard to the third fraction one-
fifth, when the exact sum of the addition of the three fractions will be
found to be indicated in decimals by the position of the top of the slide
on the two equally divided columns before referred to, whilst the vulgar
fraction will be shewn on the front of the slide by the index.
If it be desired to operate upon other amounts than hundreds the rule
may be provided with additional side wings or leaves, which for con-
venience may be hinged so as to fold into small compass, such wings or
leaves having a series of vertically divided columns marked thereon,, say
from ten to ninety, the divisions on one side of the rule being figured
from the top downwards and on the other from the bottom upwards.
The double columns 10 are divided in this manner into ten equal parts:
or divisions, the columns marked 20 into twenty divisions, the columns

page_text: marked 30 into thirty divisions, and so on with any subdivisions that
may be desired.
A special index for indicating on these additional columns is fitted to
the upper end of the sliding portion of the rule, one end of the said
index being in one column and the other end in the corresponding &
column on the opposite side or wing. Supposing it be required to divide
thirty by four the small index is adjusted in the groove in the slide
to the fractional division corresponding to one-fourth, the slide is drawn
out till the small index corresponds to the lower end of the rule, it will
be found that one end of the long index will indicate 7½ in one column,
whilst its opposite end will indicate 22½ in the opposite column, 7½
being the one-fourth of thirty, and 22%, the complement of that sum.
Although I have only given as examples the above form of apparatus
constructed on the principles herein-before laid down, it is obvious that
the same principles may be embodied in apparatus of other forms, as for
example, parallelopipeds, cylinders, or cones; and although I have only
given one example of the mode of using my apparatus it is applicable.
to all arithmetical operations. Thus not only can fractions having
different denominators be reduced to the same denominator and added
together, but the relative proportions of fractions having different deno- !
minators can be obtained, and any given number or sum can be divided
into fractions having different denominators, and the extraction of the
square root of any number effected.
SPECIFICATION in pursuance of the conditions of the Letters Patent,
filed by the said Henri Lefevre in the Great Seal Patent Office on 28
the 23rd December 1871.
TO ALL TO WHOM THESE PRESENTS SHALL COME, I, HENKI
LEFEVRE, of Rathbone Place, Oxford Street, in the County of Middlesex,
send greeting.
WHEREAS Her most Excellent Majesty Queen Victoria, by Her 30
Letters Patent, bearing date the Twenty-fourth day of June, in the year
of our Lord One thousand eight hundred and seventy-one, in the
year of Her reign, did, for Herself, Her heirs and successors,
give and grant unto I, the said Henri Lefevre, Her special license that
I, the 'said Henri Lefevre, my executors, administrators, and assigns, or

page_text: such others as I, the said Henri Lefevre, my executors, administrators,
or assigns, should at any time agree with, and no others, from time to
time and at all times thereafter during the term therein expressed,
should and lawfully might make, use, exercise, and vend, within the
United Kingdom of Great Britain and Ireland, the Channel Islands, and
Isle of Man, an Invention for "IMPROVEMENTS IN APPARATUS FOR EFFECTING
CALCULATIONS," upon the condition (amongst others) that I, the said Henri
Lefevre, my executors or administrators, by an instrument in writing
under my hand and seal, should particularly describe and ascertain the
nature of the said Invention, and in what manner the same was to be
performed, and cause the same to be filed in the Great Seal Patent
Office within six calendar months next and immediately after the date of
the said Letters Patent.
NOW KNOW YE, that I, the said Henri Lefevre, do hereby declare
the nature of my said Invention, and in what manner the same is to be
performed, to be particularly described and ascertained in and by the
following statement, reference being had to the accompanying Drawings
and to the letters and figures marked thereon, that is to say:-
My said Invention relates to certain apparatus of a simple and
economical construction for effecting various calculations in an easy
and expeditious manner.
The general principles on which the construction and arrangement
of the said apparatus is based is as follows:-If we take a length AB,
(Figure 1) of the annexed Sheet of Drawings for unity, and commencing
at the point B take distances BC, BO¹, BO", BC¹¹¹, &c., &c., pro-
portional to the various fractions comprised between zero and unity,
it is evident that if wo require the sum of two, three, four, &c. of
these fractions it will be simply necessary to add together the lengths
corresponding to each of them and look on the same line AB, for
the fraction corresponding to this total length. If then a slide
rule AB (Figure 2), divided in the manner just described be provided
with a groove in which slides another rule KL of the same length
as AB, and bearing the same divisions, the slide KL being also
provided with a central groove in which a small pointer P moves,
and if we place this pointer successively at the divisions corresponding
to the fractions which it is desired to add together, and drawing KL
by its end L bring back the pointer P every time to the lower and
of AB, the division of KL corresponding to B will naturally.

page_text: .
.
indicate the sum of the fractions which. it was required to add
together. With respect to the difference of two fractions this may be
found by bringing the pointer P to the division of AB, corresponding
to the greater, and drawing the slide KL until we bring the pointer
to the division corresponding to the smaller. The result of the
operation will be given by the division KL corresponding to the lower
extremity of A. B.
If we now take into consideration that every fraction ½ smaller than
has for its complement a fraction greater than and consequently
that the extremities of the lengths corresponding to two complementary
fractions on AB, commencing from the point B will fall at equal
distances from the centre of AB, so that if we fold AB, the divisions
of the right-hand half will fall on the complementary divisions of
the left half we arrive at the result that the same division may
represent at the same time and whence it follows that the rule
J
apparatus or A B may be reduced to one half the length. For example,
on the left side from B to A will be placed the divisions and distances.
corresponding to the fractions from zero to one half and on the right
hand side from A to B the complements of those fractions from one
half to one.
The same divisions will be marked on the slide KL on
the right and left hand side of the central groove in which the pointer P
moves. When then the slide K L has arrived at the extremity of the
groove, that is to say at 1, it is raised again for the new fractions
which it is required to add together, and the indications are read on
the right-hand side which contains the fractions greater than .
In order to enable the same operations to be performed on very small
fractions, say for example, on the fractions comprised between zero
and, or to, or ½, a corresponding portion of the rule commencing
at B would be taken and enlarged in such a proportion as would admit
of the divisions being rendered appreciable. If the line AB be curved
to a circumference of the same length as in Figure 3 the rectilineal
distances BO, BƠ¹³, BC", BC¹¹, &c. will become arcs of a circle which
will represent fractions of the circumference, the latter being taken
as unity. If this circumference he provided like the face of an ordinary
watch with two hands OK and OR, and the long hand OK by a
suitable arrangement of catches be capable of conducting with it the
short hand QR in the direction of the arrow F, but also of returning

page_text: to its starting point (which is also the zero point) whilst leaving the
short hand in the position given to it in its first motion, it will be
readily understood that the hand OK being brought successively to
the divisions corresponding to the fractions which it is required to add
together the hand OR will indicate the sum of these fractions, thus
performing the same functions as the pointer P; and the lower edge
of the rule herein-before described. What is true of the rule in regard
to the complementary fractions - and and is also true of the circle,
and the same may also be said with respect to the small fractions of
the unit the representing lengths of which require to be enlarged in
a certain proportion in order to be appreciable. These principles are
carried out in the instrument which I shall now proceed to describe
and which is but a modification of the rule herein-before described put
into a more convenient and portable form.
1
This calculating instrument has the form of an ordinary watch,
Figure 4, but with two faces, the upper one A B of which is equal to },
and its circumference is divided, commencing at A, for example, pro-
portionally to the vulgar fractions comprised between zero and }.
The lower face CD equals N' and its circumference is divided com-
mencing at the point O, for example, proportionally to the vulgar
fractions comprised between zero and . The short hands of these 2
dials are so connected together by internal gearing that when the hand
of the dial A B makes one circumferential turn the short hand of the
dial CD makes turns of its own circumference. These two short
hands moreover should turn in the same direction on their respective
faces. With regard to the long hands which are actuated either by the
aid of knobs K. and K', passing through the glasses of the two faces,
or by the aid of remontoirs placed in the case, these are independent of
one another. When the knob K or K³, for example, is turned from left
to right the long hand will cause its respective short hand to partake
of its motion, but when the long hand is brought back to its starting
point the short hand remains in the position it has acquired by the pre-
ceding movement, and does not return from right to left in the same
manner as watches are wound up with the keys on Breguet's system.
It follows from this arrangement that the hands of the face OD
acting as a vernier perform. additions of fractions on the face A B,.

page_text: I am enabled to obtain immediately the product or quotient of two
numbers by forming a rectangle, the base OA of which is divided,
(commencing at the point O) into lengths O P, &c., proportional to the
vulgar fractions comprised between zero and unity, and the height of
which is divided into equal parts like an ordinary metrical scale. By
joining the commencing point O of the fractions to a number expressed
on the side A B and drawing the parallel MPI have the three elements
of a multiplication or division.
What has been herein-before stated with respect to the slide rulo
on the subject of complementary fractions below and above equally
applies to the rectangle OA B O, which may be folded over along a
line Y, Y, perpendicular to O, A, passing through the centre of the
same, so as to reduce the figure to one half of its size. Moreover what
is true for O A, is also true for A. B, videlicet, that this length being
divided into M equal parts, a number of these parts represented by a
M
M
has for its complement +a so that the rectangle may be again folded
over around an axial line XX perpendicular to A B, and passing
through the centre of the same so as finally to reduce the original
rectangle to a quarter of its size without reducing the. indications which
it is capable of supplying. In these motions the line drawn from the
point O to any point of A B, will have been bent as many times as it
has met axes of revolution; thus OK is bent but once at E whilst
O G is bent first at E¹ and again at F¹. Now it is important in dealing
with these calculators to be able to determine the points F¹ in function
of the points E¹ and G³. Let us find their equation in relation to the
axes of the rectangle reduced to a quarter of its size, call a the base,
6 the height, and find the equation of the points x y, x¹ y¹, (Figure 6).
We hown than for the first

page_text: which they are parallel, such slides being provided with a central groove
in which work the tenons x, y, and x¹, y', which are caused to follow
the curves to which they respectively belong, the reciprocating motion
of any one of these slides in a direction parallel to the plane of the
table will determine all the respective positions of the three points E¹, F¹,
G', and consequently all the positions of the broken line O, E¹, F¹, G¹.
"
Finally if x y and x¹ y' on arriving at the end of their course in Y
and H be caused to descend again, the length of the diagonal Y X
and the line H X the point F¹ vanishing, the two ends of the two slides
parallel to the axis of X X (Figure 5) would determine the reciprocal
relations of the points E and K', when the line drawn from the point O
falls on a point K at a distance from A less than one half the height
of the rectangle.
To sum up, if a line A, D, (Figure 7) at right angles or at any other
angle be added to the rule A B divided as firstly herein-before described,
such line A. D being divided like an ordinary meter, for example, com-
mencing at A, we shall have the results above mentioned and nothing
more will be required to render the apparatus practicable than to reduce
it to the simplest dimensions by the aid of the principles and deductions
herein-before enunciated.
If in lieu of taking a straight line I had taken a circle, I should have
obtained a cylinder possessing the same properties as the rectangle by
determining the novel conditions of its construction which I reserve to
myself the right to do if desirable.
On these principles various instruments may be constructed differing
from one another only in their dimensions, and in the Figures which
will correspond to the special.application in view.

page_text: I have described the instrument in its general form, and all that is
required is to apply the principles of its construction to the special
objects in view.
In the apparatus last described, as in the case of the slide rule and
the dial, any fraction of the table may be taken commencing at the 5
point O for the purpose of being enlarged in a determined proportion.
Only in this case but one slide would be used, videlicit, the one which
traverses the side A B, as the others are not susceptible of being
employed here. The various portions of the thread may be worked
simultaneously by the aid of the arrangement of slides and guides 1
herein-before described. In lieu of drawing the number of lines on the
table which may be inserted therein a slide rule may be caused to
traverse the side O A, such rule bearing on each side of its centre
line the same divisions 1₁ 21 3₁
N, and N-1, N-2,
N-3
•
N
.
•
as the sides A B and OC. According to the 1
various positions in which it is placed this rule supersedes the use of
the parallel lines drawn through the points,,, &c., and the inter-
section of the centre line with the elastic thread gives the indications
which in the ordinary apparatus are read on the side of the table.
The same principles may also be applied to the construction of scales 2
and steelyards by substituting weights for lengths or by the use of lover
arms suitably determined.
Figure 9 represents an apparatus in its simplest form adapted to the
calculation (on the decimal system) of interest, discount, arbitrations of
exchange, and continuations or prolongations. This apparatus or tablet 2
is divided into two triangular tables by the diagonal line A B, our
triangular table shewing decimal interest and the other the interest
for any number of days.
6 In calculating interest three numbers are constantly involved, vide-
licet, the principal, the rate per cent., and the amount of interest :
produced. When two of these numbers are known it is required to find
the third, which operation may be performed immediately by the aid
of this apparatus. The lower edge of this apparatus is divided into equal
parts from 0 to 110 which may represent at will continue, decimes,
francs, tens, hundreds of francs, &c., and the left-hand seal contains the
rate of interest from 0 up to 10 per cent.

page_text: for fifteen days may be determined by the aid of the upper dotted line
marked prolongations fifteen days..
In order to calculate interest for any number of days the right-hand
table is employed, the top row of figures representing the interest and
the right-hand row the number of days. Suppose that the interest for
one year amounts to sixty francs (which may be calculated by the left-
hand table) and that it be required to ascertain the interest for twenty-
four days. Place the end of the cord (which is attached to the upper
hook) at sixty and observe the point where the cord intersects the
horizontal line opposite the number twenty-four on the right-hand side
of the table, then the vertical line passing through this point gives
forty on the upper row of numbers i.e. 4'0 (one figure being a decimal)
the interest for twenty-four days.
Figure 10 represents a modification of the tablet which is more
especially adapted to the calculation of interest, discount, &c., on the
English monetary system. The tablet is divided diagonally in a similar
manner to that described with reference to Figure 9, but in lieu of
having only one row of figures at the base and on the top of the table
two additional rows are employed. The first row at the base of the
table from 0 to 240 may represent either pounds, shillings, or pence,
and it may also denote one pound sterling, which is equal to two hundred
and forty pence, and the second row will then represent shillings. When
the first row is taken as equal to one pound the diagonal line A B, which
divides the table into the parts, will indicate at once without moving the
cord the interest on one-pound sterling at any desired rate. Thus for
example in order to ascertain the interest on one pound at five per cent.
observe the point where the horizontal line opposite five per cent.
intersects the diagonal A B, and follow the vertical line passing through
this point to the bottom. The number given on the first row of figures
is 120 striking off one decimal figure we have twelve pence or one
shilling for the answer, or if we prefer to take the second row the number
given is 10, whence striking off one decimal figure we have 1s. as before.
The third row from 0 to 12 represents pounds, with their equivalents or
shillings, in the first row, but it may also denote one shilling divided
into 12 pence, which by the aid of the upper divisions could be divided
into farthings with facility. The interest on one shilling at any required
rate will be found by means of the diagonal in the manner herein-before
described. When the first and third rows are taken as representing

page_text: pounds the cord described in reference to Figure 9 must be employed
in the same manner as before. The figures on the top of the tablet
aro arranged similar to those at the base, their object being to effect
the calculation of interest for any number of days as in the tablet,
Figure 9.
Having now described and particularly ascertained the nature of
my said Invention, and the manner in which the same is or may be
used or carried into effect, I would observe in conclusion that what I
consider to be novel and original, and therefore claim as the Invention
secured to me by the herein-before in part recited Letters Patent is, the
construction and use of calculating apparatus in which the peculiar
mathematical principles herein-before set forth are applied substantially
in the manner specified.
In witness whereof, I, the said Henri Lefevre, have to this my
Specification set my hand and seal, the Fourteenth day of