fisher on some recent improvements in the construction of the printing press

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.


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.


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.


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


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


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.


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,)


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'.


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.

fisher on some recent improvements in the construction of the printing press

Documents

printing presses

printing press uith

art of printing

printing pretses

variable power

variable resistance

common press

variable inclination