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Fisher on Some Recent Improvements in the Construction of the Printing Press
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On Printing Pretses and their Theory. 31 1
MECHANICS AND ARTS, CHEMISTRY AND PHYSICS,
Art. IX.—On some recent improvements in the construction
of the Printing Press ; uith a particular notice of that
lately invented by Mr. John I. Wells, of Hartford, Ct.
by A. M. Fisher, Professor of Mathematics and Natu
ral Philosophy in Yale College,
The principal defect in printing presses of the ordinary
construction, so far as the mechanism employed to procure
a gain of power is concerned, consists in the want of adap
tation of this power to the variable resistance which is to be
overcome. The elastic substances interposed between the
form of types and the platen, present at first a compara
tively trifling resistance ; but it gradually increases as the
platen descends, and must finally be made immensely
great, in order to attach the ink with sufficient firmness to
the paper. But to overcome this resistance the mechanical
advantage furnished by the screw is perfectly uniform. To
make up for the want of an increasing advantage in the me
chanism, the pressman is obliged to place his body in such
an attitude that his weight shall conspire with the force of
his muscles, and to exhaust on the bar as much motion as
he can accumulate in a pull of three feet, in order to give it
a species of percussive effect. Hence the employment of
pulling at the common press has been always regarded as
one of the severest kinds of labour ; nor has it been repre
sented without reason as often " destructive of health and
life."
It has long been an object with those interested in the
improvement of the art of printing, to introduce into the
press a variable power, which shall increase with the resist
ance to be overcome, and thus render the pull on the bar a
nearly equable one throughout. The earliest contrivance
for this purpose which appears to have been in any degree
successful, was that of Mr. Roworth, a London printer.* In
his press the screw was dispensed with, and a plain spindle
substituted in its place. To the under side of the head or
* See art. Printing, Rees' Cyc. for a more fall account of this construction.
312 On Printing Presses and their Theory.
summer, where the spindle was inserted, was attached a
species of inclined plane, rounded off so as to have a varia
ble inclination. Through the spindle immediately beneath
ran a cross bar, which plied against this winding surface,
and forced the spindle down as it was turned round ; rapidly
at first, but more slowly as the inclination diminished, and
at last with a velocity as trifling as was shewn by experi
ence to put the press into the best working state. In a
press recently invented by Mr. Medhurst, an ingenious Eng
lishman, the power is gained by means of two iron rods,
one on each side of the spindle. These rods pass down
From the summer to the top surface of a circular enlarge
ment of the spindle, and rest at each end in hollows which
allow them a racking motion. The two extremities of each
are equidistant from the centre of the spindle, but are. pla
ced in a winding position when the platen is raised. The
bar turns the spindle partly round, and moves the lower
ends of the rods so that they come towards a vertical posi
tion, and bear down the platen with that kind of increasing
mechanical power which every one has seen exemplified in
bringing a prop erect by driving at right angles against the
bottom.*
But in most of the recent attempts to improve the con
struction of the printing press, a kind of mechanical power
has been resorted to in different forms, somewhat different
from either of the foregoing ; one which is well known to
every theoretical and practical mechanician, but which has
scarcely acquired a distinct name. To attempt to reduce it
to the head of the lever or wedge, as has been sometimes
done, appears an unwarrantable extension of the meaning
of these terms ; and yet I know not how to designate the
principle better than to call it that of combined levers. It is
the power of thrusting possessed by the outer extremities of
two straight rods, placed obliquely end to end or riveted to
gether, when the moving force is applied to bring them
straight. Let a pair of compasses, or a carpenter's rule the
two halves of which shut together with a joint, be opened
nearly straight and placed between two parallel obstacles :
on taking the rivet between the thumb and finger and vary
ing the angle, it will require no skill in mechanics to discov-* The principle of thii combination wijl bo investigated in the Supple
ment.
On Printing Prepsps and their Theory. &$.§
er that there is a gain of power, which gradually increases
as the two halves approach a straight line, and becomes im
mensely great at the moment this position is attained. The
thrust of two such arms is precisely the same, and varies ac
cording to the same law for different angles, as the pull in
the simplest case of the funicular polygon ; that is, when a
rope is tended by a certain force and is drawn aside from
a rectilineal position by pulling at the middle.
This principle is introduced in different forms into the
Ruthven, Stanhope, and Columbian presses. In that in
vented by Earl Stanhope it is employed to give a diminish
ing velocity to the screw : in the Columbian press of Mr.
Clymer, it is employed to give a diminishing velocity to a
large lever of the second kind, which is substituted for the
screw. These two presses, especially the latter, from their
durability, the neatness and uniformity of the impression
they produce, and the diminution of labour they occasion
to the pressman, have been justly held in high estimation.
To the excellence of the Columbian press, honorable testi
monies have been borne in foreign countries : among others
has been a present of six thousand rubles to the inventor
from the Emperor of Russia.
But of all the presses which act on the principle of com
pound leverage, the one recently invented by Mr. Wells,
of Hartford in this State, appears to possess the highest re
commendations. It has now been in operation in various
parts of the country more than two years,—a period suffi
ciently long to furnish an experimental test of its excel
lence ; and it seems due no less to the interests of the me
chanical arts in this country than to the ingenious and wor
thy inventor, that a more particular account of it than has
hitherto appeared should be given to the public.
A perspective view of this elegant piece of mechanism is
given, plate II, fig. 1. The frame is of iron, cast (with the
exception of the feet) in a single piece ; and is of such form
and dimensions as to be incapable of springing, while the
press is in operation. The platen (4) is of cast iron, and is
of the dimensions of an entire form. The circular projec
tion in the middle, with six radiating pieces, gives it an am
ple degree of firmness. The platen is immediately acted on
314 On Printing Presses and their Theory. ,
are fifteen inches each in length ; and in the position repre
sented in the figure, which is that of the greatest obliquity,
they want two and a quarter inches at their point of contact
of being straight. The lower end of each lever is four inch
es broad, and is rounded off into a portion of a cylindrical
surface of half an inch radius. A piece of steel fixed within
the circular projection in the middle of the platen has a hol
low bush or bed of corresponding figure : in this the lower
end of the lever (17) is set. The upper end of this lever is
hollowed out in the same manner to receive the lower end
of (6) ; and the upper end of (6) to receive a projection
from the under side of the top of the frame. At (5) there
is a provision for raising or lowering this projection by slips
of sheet iron or tin, and thus adjusting the position of the
levers to the best working state. The ends of the levers
and the beds in which they rest are overlaid with steel, and
the beds are so contrived as permanently to retain a small
quantity of oil. (9) is a spindle of wrought iron fastened at
the upper end by a screw and nut to the shorter arm of the bal
ance lever (7), and branching below into three parts, each of
which is attached by an adjusting screw to the platen. This
answers the double purpose of keeping the platen steady,
and enabling the weight (18) attached to the longer arm of
the lever (7) to lift, the platen and carry back the bar imme
diately after each pull. The platen is still farther guided
by lateral projections which run in grooves connected with
the cheeks of the press.
The mode in which the movement of the working bar(12) is transmitted to the main levers, will be best under
stood from Fig. II. which is a representation of the parts 11,
12, 13 and 15, as they would appear to an eye looking down
upon the press from above. The bar BA (the lever work
ed with the hand) is inserted into a strong cast iron roller(13), which turns in sockets secured to the right cheek of
the press. From this roller, about six inches above the bar,
proceeds an arm AC three inches in length, and to the ex
tremity of this is connected by a joint the driving lever CD,
twenty-one and a half inches long. The extremity D is
connected in a similar way with the iron rod EF, one end
of Which slides in a pewter guide (represented by 10, in
Fig. I.) while the other end is fastened by a hook and eye
to the upper main lever (6), at the distance of an inch from
On Printing Presses and their Theory. 315
the bottom. (16) is a bar check, which limits the revolu
tion of the bar to a precise arc. The carriage part of the
press, which stands in front of the upright iron frame, pre
sents nothing materially different from the Columbian press,
and will not require a particular description.
The operation of the mechanism will now, it is believed,
be sufficiently apparent. When the bar BA is brought
round, the roller A and the arm AC are made to turn with
it : this drives forward the lever CD, and this in its turn
gives motion to EF, which by means of the elbow at F
brings the two main levers (6) and (17) towards the posi
tion of a straight line. As the movement of the bar is con
tinued, the mechanical advantage not only increases from
the gradual approach of the two main levers to a vertical po
sition, but from the approach of AC and CD towards a
straight line. The combination is therefore one which is
eminently adapted to effect that rapid increase of power
near the end of the pull, which has been already mentioned
as the great desideratum in the construction of this part of
the printing press. To determine the actual gain of power
at the beginning and at the end of the pull, measures have
been taken from an individual press, of the lines necessary
for the computation. When the bar was thrown back, the
angle ACD (of the triangle ADC formed by joining the
three centres of motion with straight lines) was found to be
= 1 13° 52', CDA=7° 12', and the distance of the centre of
motion of the two adjacent ends of the main levers from the
straight line joining their outer extremities =2i inches.
The length of AC was 3|, and the distance from A to the
part of the handle where the hand was generally applied
was 24 inches. Hence, as will appear from the theorems
annexed to this paper, the gain of power will be found by
compounding the four following ratios : 24 to 3J, Cos. 7°
12' to Sin. Tl3° 52', 15 to 2x2^, and 14 to 15; which
gives a total of 20 to 1.
At the end of the pull, the angle ACD = 172°, the angle
CDA— 1° 3', and the distance of the vertical levers from a
straight line, according to the specification of the inventor,
which was found nearly exact, = half an inch. Hence the
gain of power will be found by compounding the following
ratios : 24 to 3£, Cos. 1° 3' to Sin. 172°, 15 to 2x$, and
14 to 15 ; which gives a result of 763 to 1.
3iB On Printing Presses and their Theory.
It thus appears that the power gained is about thirty-eight
times greater at the end than at the beginning of the pull.
While the re-action of the elastic substances which form
the tympan is small, the mechanical advantage is small, and
the platen is brought down rapidly ; but as the resistance
increases, the power gained undergoes somewhat more than
even a proportional increase, so that during the last mo
ments of the revolution, the pull actually grows somewhat
easier. In consequence of this, although the bar is stopped
suddenly by the action of the check, nothing of that violent
jar is produced on the arm, which is so serious an incon
venience in the common press ; and to relieve which most
pressmen find it necessary to sacrifice a part of the force
exerted by inserting an elastic heading over the tenons of
the sumnier. -^fe6-:,
Let us now compare for a moment the mechanical ad
vantage furnished by this combination with that furnished
by the screw of the ordinary press. In all presses alike,
the perpendicular motion of the platen may be regarded as
a constant quantity. It must necessarily rise a sufficient
distance to allow the thickness of the tympan frame to pass
freely under it. The distance allowed for this purpose ap
pears to be in general about £ of an inch. But in addition
to this, in the screw press, we must allow at least } of an
inch for the spring of the summer ; making the vertical dis
tance described by the interior relatively to the exterior
screw half an inch. Then supposing the length of the pull
to be no greater than in the Lever press, the mechanical ad
vantage gained will be uniformly 44 : 1. But if we sup
pose, as is generally the the case in fact, that the distance
described by the hand is greater by about a foot, (although
the increase of the distance is in reality only an exchange of
one disadvantage for another,) the power gained Will be
uniformly 66 to 1. Hence, on the most favourable hypothe
sis, the strength of the pull at the last point, independently
of the force already accumulated in the body, need be but
X as great in the Lever as in the screw press ; or £ as great,
if but half a form js worked at once with the latter.* It
* The force actually exerted at the last point of the revolution of the bar
j'n the Lever press, was found by measurement to be on an average, for the
lightest kinds of work 30 lbs. ; for the heaviest, 45.
On Printing Presses and their Theory. 317
must not be supposed however that this ratio is a fair crite
rion of the total strength exerted. This is probably about
half or 4 as great, in the former as in the latter. When a
pressure is to be produced between the paper and form of
types of from 25 to 35,000 pounds, it is not in the power of
mechanism to supersede the application of a considerable
aggregate force to the bar. The superiority of the Lever
press lies much more in the equalization of the force which
it occasions, than in the reduction of its total amount. It is
true at the same time that the Lever press does considerably
diminish the total force of the pull ; but it is chiefly by per
mitting a diminution in the thickness of the elastic substan
ces which form the tympan, and dispensing with the spring
of the summer,—not from the peculiar nature of the mechan
ism which effects the gain of power.
By admitting the two main levers (6) and (17), or the
two horizontal ones AC and CD to come much nearer to a
straight line, a far greater mechanical advantage might have
been obtained ; but it would have been of no practical use.
The inventor has rightly judged that it is time to stop the
bar when it begins to move sensibly easier. If it were per
mitted to go further, the platen could descend but an ex
tremely minute interval, and consequently the elastic re
action of the tympan and blankets would remain nearly sta
tionary. At the same time, the positive disadvantage would
be incurred of rendering it impossible for this elastic force
to produce the return of the bar.
There are a variety of circumstances relating to the Lever
press, aside from the peculiar nature of the power it em
ploys, which recommend it to the attention of the owners of
printing establishments.
1. The force exerted being exactly gauged by the pin
which stops the bar, the impression of different successive
sheets will be absolutely uniform, except the trifling and
scarcely perceptible difference which may arise from the
variable thickness of the paper.
2. For the same reason, the pressman will find it much
less easy, if disposed, to do his work imperfectly. Indeed
from the superior facility with which the press is worked,
the temptation to slight his lask is in a great measure re
moved.
3. The whole of a form being worked at once, and the
platen admitting a superior evenness of surface and exact.
318 On Printing Presses and their Theory.
ness of movement, the different pages of the same sheet will
present a neater and more uniform appearance than when
worked with a wooden platen and two pulls. This remark
is especially applicable to the duodecimo page.
4. By admitting a less thickness to the tympan and its
contents, it produces a less rapid wear of the hair strokes of
the letter.
5. The ribs on which the carriage runs have the peculiar
construction seen in the figure, by which the friction is much
reduced, and the waste of oil diminished.
6. From the best estimate which can be made, this press
,will in a course of years be attended with an actual saving of
money to the purchaser.*
Many of the foregoing advantages, it is readily conceded,
are such as this press possesses in common with that of Mr.
Clymer; but without detracting from the merits of the lat
ter, there is little danger in hazarding the prediction that its
use will be speedily superseded ; and that as it has thrown
* The grounds of this conclusion are the following. In the first place, the
wear is almost nothing. With the exception of the main roller (13) which
with its sockets is of cast iron, and the joint C, (Fig. II.) all the moving
parts slide over an extremely small arc ; and these are made of hardened
steel. The writer has examined the parts most exposed to wear in a press
which has been in constant operation nearly two years, and the effects of
friction were found wholly insignificant. The slight roughnesses which had
been left on the surfaces by the manufacturer were scarcely affected. It iy
obvious, however, that the wear of many of the parts might become very
considerable before the action of the press would be sensibly impaired ;
and that others might be replaced at a trifling expense. The original cost
of a common wooden press is about one hundred and seventy dollars ; and
the annual expense of maintaining one will consist of the following items : in
terest on the original cost $10,20 ; principal to be replaced, supposing the
average time of wearing out to be twenty-five years, $6,80 ; repairs, inclu
ding accidents and insurance, $10,00. The interest on the first cost of the
Lever press is from 18 to 21 dollars ; principal to be replaced in all proba
bility not $2 ; repairs &c. perhaps $4,00. This estimate is made out from
inquiries addressed to different printers who have been conversant with both.
The foregoing particulars are such as chiefly interest the proprietor of a
printing establishment; but no inconsiderable additional advantage has
been conferred on the public by Mr. Wells, in lightening the task of the
journeyman. Not only is the pull rendered far easier, as has been already
shewn at length, but the number of pulls is reduced one half. Those who
might apprehend a reduction of wages from an acknowledged reduction of
their labour, can scarcely be expected to be among the most forward to pro
claim the extent of their obligations to the inventor ofthe Lever press; but if,
as the writer has been credibly informed, it has in some instances been hired
by individual journeymen at an annual expense of fifty dollars, in prefer
ence to using the common presses offered them by their employers, a stronger
testimonial to its superiority from this class of persons could not be desired.
On Printing Presses and their Theory. 319
prior inventions into the back ground, it must in its turn
yield to the progress of improvement. The points of supe
riority in the Lever, over the Columbian press, appear, to be
the following. 1. It is afforded at two thirds of the expense.
2. The mechanism is lighter, and more compactly stowed.
3. From the greater simplicity of structure, it is less liable
to get out of repair, and is more easily put in order when
„ out of repair by a person of common mechanical skill. 4.
The surfaces which move in contact are so contrived as to
be kept oiled without being taken in pieces. Accordingly,
those who have had trial of both, so far as the writer can
learn, both owners and workmen, give the preference to the
Lever press.
High as is the perfection to which this press has been
brought by its inventor, it would be strange if it were abso
lutely incapable of improvement, or if farther experience
should not point out some changes for the better. Among
the infinite variety of which the adjustment of the levers is
capable, there can be but one which is absolutely the best;
and it is scarcely supposable that this one has been yet at
tained. A slight variation of the position and form of the
working parts in different successive castings, promises more
effectually than any thing else to make known those slight
improvements of which they may still be capable. Several
of the parts appear to possess superfluous strength. The
cheeks might probably be reduced to one half their present
size with advantage. The top and bottom of the frame
must be made strong, because they require to be incapable
of springing as well as of breaking. But while this is the
case, the strain on the sides so far as it is produced by the
two main levers is wholly a longitudinal one, and the re-ac
tion of the driving lever against the right side is comparative
ly small. Admitting the re-action of the platen, in perform
ing the heaviest work, to be thirty-five thousand pounds, the
two sides would possess sufficient strength to prevent their
being drawn asunder if made of sound cast iron three-fifths
of an inch square. But if reduced to one half their present
size, they would possess sixteen times this degree of strength.
The driving lever also, if made very nearly straight, as it
might be without interfering with the main levers, might be
diminished one half or two thirds in size. On other im
provements of a more problematical character which have
suggested themselves, it will not be necessary to enlarge.
320 Qn Printing Presm and their Theory.
SUPPLEMENT.
As the power gained by different combinations like those
referred to hvthe preceding pages seems to have scarcely
attracted the notice of writers on Mechanics, I shall subjoin
an investigation of such as are most likely to occur in prac
tice, for the information of those who may be concerned in
the invention or improvement of machines which contain
such combinations, and to whom it may be sometimes im
portant to know with precision the mechanical advantage
they gain by different supposed arrangements of machinery,
and the strain to which the different parts are subjected.
Prop. I. Let CB (Plate III. Fig. I.) be a straight rod,
moveable about C, and BA another rod, connected by a
joint with CB at B, and with its other extremity A confined
to move in the line CA produced : it is required to deter
mine the power which applied at B, in a direction at right
angles to CB, will overcome a given resistance acting on
the point A, in the direction AC.
The power will be to the resistance, in this, as in all oth
er cases, in the inverse ratio of the velocities of their re
spective points of application. We have therefore only to
investigate, for any given position of B and A, the velocity
with which B moves, or the circular arc DB increases,
compared with that of A's motion on the line CA. For
this purpose, draw the perpendicular BP, putAB=a, C
B=b, AC= x, BP=y, and DB=z. Then x— y/7^yl+
—y Ay ^-y dy
,,/bi-yz ; and taking the fluxions, das= ^—-=r+
But dz= ,—== ; hence by division— = JV —4- a
This expression gives the ratio of the velocities of A and B ;
and hence is to unity as the power is to the resistance. But
,by reinstating the values for which x, y, Sic. were substitu
ted, it admits of simplification. Reducing the terms to a
common denominator and restoring their values, it becomes
_BP AC sinBAC sinABC sinABC . f. „„_=AP- CB=—bac -^-bac=^7bac That ,s' the Pow'
,er is to the resistance overcome, as the sine of the angle
On Printing Presses and their Theory. 321
made by the two rods, is to the cosine of the angle, made
by the rod to which the resistance is opposed and the direc
tion of the resistance.
Cor. 1. If the power, instead of being applied at B, is ap
plied at any other point X in CB or CB produced,—pow
er : resistance : : CB'sin ABC : CXxos BAC.
Cor. 2. If the rods CB, BA become equal in length,
y/b*—y2=\/ai>~y2, and the general expression is reduced to
~. That is, the power is to the resistance as twice the
cosine of half the angle contained by the rods is to radius,—
or as twice the distance of their point of junction from the
line joining their outer extremities is to the length of either.Cor. 3. If the power, instead of acting in a direction
perpendicular to CB, act in the direction of BP, it is easily
inferred that power : resistance : : tan BAC+tan BCA : 1.
Prop. II. It is required to determine the ratio of the for
ces which keep each other in equilibrio when the point A
(Fig. 2.) is confined to move in any other given line AH.
From C draw Cb equal, and infinitely near to CB, and
from b as centre with BA as radius, intersect AH in a. Join
b, a, and draw the perpendiculars ar and bs. Ar is ultimate
ly equal to Bs. For Ar—Bs=AB—rs = ab—rs= (as may
be easily shewn) , , This being an infinitesimal of
the second order, is ultimately evanescent in respect to
Ar, and consequently Ar—Bs=0, or Ar=Bs. It follows
that Aa : Tib : : sec BAH : sec bBs : : sec BAH : cosec
CBA: : sin CBA : cos BAH. But Aa and Bb measure
the velocities of the points A and B ; hence power : resist
ance : : sin ABC : cos BAH. This result includes that of
the last Prop, as a particular case.
Prop. III. Let the extremity A, instead of moving in a
straight line, be confined to move in a circle, by being con
nected with the rod AC', moveable about the fixed point
C : the power applied to B will be to the resistance acting
at A with which it is in equilibrio, as the sine of CBA is to
the sine of CAB.
For draw through A the line AH perpendicular to AC :
then the initial motion of A will be in the line AH, and by
the last Prop, power : resistance : : sin ABC : cos BAH.
Vol. Ill No. 2. 41
322 On Printing Presses and their Theory.
But, BAH= comp. BAC; therefore power : resistance : :sin ABC : sin C'AB *
Cor. 1. When the the radii stand in opposite directionsas C'A, CB', ABC becomes a reflex angle of more than
1805; but the sine of any arc is the same (except in regard
to its sign) as the sine of its supplement to 360° ; hence, as
before, the two forces applied at k and B' will be in equili-
brio when they are to each other as the sines of the angles
AB'C, CAB' to which they are respectively applied.
Cor. 2. The same result may be extended to the casein
which A and B are confined to move in any lines whatever,
straight or curved, to which a tangent can be drawn. For
let AH and B6 be the tangents at the points A and B ; and
power : resistance : : cos AB6 : cos BAH, or (drawing the
normals CB, C'A,) : : sin CBA : sin C'AB.
Remark.—The foregoing results will equally apply when
the rods CB, BA, &c. are curved, and when in con
sequence of being inserted into different parts of the same
roller, they are not in the same plane ;—provided that CB,
BA, he. are taken as the perpendicular distances from the
central line of one roller to that of another.
Prop. IV. Let there be three levers CA, CB, CD,
(Fig. 8.) moveable about C, C, and C , as centres, and hav
ing their other extremities connected by straight rods AD
and DB : the power applied to A will be the resistance act
ing at B perpendicularly to CB, as sin CAD X sin C"DB
is to sin ADC" x sin DBC.
This proposition evidently' follows from the first Cor. to the
last, and is equally true for all possible positions of the cen
tres C, C, C", and of the rods CA, C'B,C 'D.
Cor. When CA and AB come into the position of a
straight line, sin CAB vanishes, and the power gained will
be infinite. If the rods be so disposed that CD and DB
come into the position of a straight line at the same time,
the power gained at the moment of attaining this position
becomes infinite upon infinite.
Prop. V. If any number n of equal rods be connected
by rivets at their middle and ends as in Fig. 4, the end C
* This proposition determines the mechanical advantage gained at any
given part of the revolution of the bar, in the Stanhope press.
On Printing Presses and their Theory. 323
being fixed, and A being moveable along CA ; the power
applied at B" and acting perpendicularly to CA is to the
resistance overcome at A, as twice the tangent of BAC is to
n—1.
Suppose in the first place that the power is applied atB :
by Prop, h Cor. 3. it will be to the resistance : : 2 tan
BAC: 1. But the nature ofthe combination requires that the
rods should in all states be parallel to each other ; hence
velocity of B'= 3 X vel. B ; vel. B" =5x vel. B, Sic, and
power at B "= J- power at B ; so that power at B'' : resist
ance at A : : 4 X 2 tan BAC : 1 : : 2 tan BAC : 5. In the
the same manner it may be shewn, whatever be the num
ber of rods combined, that power : resistance : : 2 tan
BAC : n—1.-.,..v,
Prop. VI. (Fig. 5.) Let the two levers of Prop. I. instead
of being united at B, act on each other by means of circu
lar cheeks BD, B'D, having equal radii BO, B O , less than
BA : it is required to determine the power which, applied
to B at right angles to BA, shall overcome a given resist
ance acting at A in the line AC.
To simplify the investigation of the relative velocities of
B and A, let it be supposed that when B suffers an indefi
nitely small change of position, A and C move equally in
opposite directions. Then the centres O, O', of the arcs B
D, B'D, will describe lines perpendicular to AC ; and if
the, motion of B be continued, the point of contact D, the
centres 0,0', and the points B, B' will all fall upon AC to
gether. Let ab (Fig. 6.) be the position which AB assumes
when it has moved an indefinitely small distance : the point
o will be in the perpendicular OP, and ao will be equal to A
O. Drawing the perpendiculars, bg, oh, Ae, and placing f
at the intersection of AO and ao, it may be shewn as in the
demonstration of Prop. II. that Bg=Oh=ea. Hence Ae :
ho : : tan OAP : cot OAP : : sin* OAP : cos2 OAP : : O
Ps : AP*. By sim. tri. ae :ho :: A/:/o. But/O may be
taken as=/<>; therefore A/:/0 : : OP* : AP2. By compo
sition, fO : AO : : PA2 : AO2; hence, putting A to denote
the angle BAP,/0=AO cos2A. Likewise oA=Ae.-^^^,
and Ae —Aa. sin A : so that oh = Aa. By sim. tri. gb :sin A °
oh ; : Bf: Of; that is, (by substituting the values already found,)
324 On Printing Presses and their Theory.
gb : Aa. ^± : : OB+AO. cos2 A : AO. cos 1 A. Divi
ding the second and fourth terms by we obtain, gb :" * sinA
Aa: : OB+AO. cos* A : AO. sin A. But gb represents
the velocity with which B moves, reduced to the direction
of a perpendicular to BA, and Aa denotes the cotemporane-
ous velocity of A. Hence the power is to either of two
equal resisting forces applied to A and C as AO "sin A :
AO -cos* A+OB.—If the angle BCA be not too large,
we may suppose C immoveable, and the whole resistance ap
plied at A. We shall then have power : resistance applied
at A : : 2AO sin A : : AO -cos 2 A 4 OB.
Cor. When A is so small that cos* A may be consider
ed as = 1 , power : resistance : : 2AO sin A : AB. When
the two ends B,B', are immediately applied to each other,
as in Prop. I. the ratio is that of 2 sin A : 1. Hence when
two levers act by circular cheeks, they furnish a mechanical
advantage greater than that of two simple levers of equal
length at the same angle of obliquity, in the ratio of the
length of the lever to the excess of this length above the ra
dius of curvature of the cheek.*
Prop. VH. Let two equal rods AB, A'B' (Fig. 7) resting
on the immoveable points A,A', support the circular plane
BEB' at two opposite points of the circumference B,B'; and
let this plane be capable of turning round and sliding parallel
to itself on the fixed line CF, which passes through its centre
and rises perpendicularly from C the middle of the line AA' :
if DE be drawn parallel to CA, the power acting at the cir
cumference B, is to the weight resting on B EB which it
will support, as the sine of BDE to radius CA, is to CD.
* This principle (which may be called that of excentric rollers) has the
farther advantage ofentirely avoiding friction between the adjacent surfaces.
It deserves an inquiry whether the immense power which can be thus com
manded does not admit ofbeing advantageously introduced into certain kinds
of machinery. Possibly it would be an improvement to construct the adja
cent ends of the two main levers of Mr. Wells' press in this form.
It would be easy to assign the ratio of the power to the resistance for differ
ent obliquities of AB and CB', when BD, B'D, are elliptic arches having
AB, CB' for their semitransverse axes. But the result would be of no prac
tical importance, on account ofthe difficulty ofgrinding the surfaces BD B'D
to an exactly elliptical form. Whatever be the nature of these curves, the
proportion given in the Cor. will be true for very small obliquities, if BO,
B'D' be taken equal to the radii of curvature at the points B,B'.
On Printing Presses and their Theory. 325
Draw BP perpendicular to DE and join AP. Then the
plane AGDP is perpendicular to the plane BEB'; and BP
drawn perpendicular to their line of common section is also
perpendicular to the plane ACDP, and therefore to the line
PA which it meets in that plane. Hence APB is a right
angled triangle, and AP3 +PB * =AB 2 . If BD be put=
r, AC=r; AB=a, CD=^x, and ;BE=z, BP will be-
come= sin s, DP= cos z, and AP» (=DCJ +AOPD*)
=x!+(rVcos s)1. Hence sins z+x2 + (rVcos z) 2 a'.
Expanding (r' > cos z)2, and substituting r" for sin *z4-cos*
z, We have xt+r2— 2 r' cos z+ r2 =a*. Taking the flux
ions, 2 xAx= 2 r' d (cos z) =—- 2 sin z Az ; and by resolu-
r' r ,,'
tion, da: : — Az : : - sin z : x. But da? and — Az express the
velocities of the points to which the weight and the power are
respectively applied ; so that power : weight : : - sin z : x : :
sin BDE to rad. AC : CD.
Cor. I, When BDE=0 or 180°, sin BDE vanishes, and
the gain of power becomes infinite. But DE is evidently
the position which BB' assumes when AB and A B' come
into the same vertical plane. Hence the weight infinitely
exceeds the power necessary to support it, when the two
rods come into the same vertical plane. When the angle
EDB is so small, or the line DC so large, that the variation
of DC may be neglected, the power necessary to support
a given weight will vary as the sine of EDB, the angular
distance from the position at which it becomes evanescent.
Cor 2. Every thing else being the same, the gain of pow
er from this combination will increase in the same propor
tion as the distance of the lowest points of the rods from C
is diminished. 1
Cor. 3. If, as will generally be the case, the two extrem
ities of each rod are equidistant from the central line CF,
orAC=BD, the power will be to the weight simply as
BP: CD.