M E C H 
the axe A Z, fo regulated, that, while the triangle makes 
a complete revolution, the point B is moved to G, and 
the whole triangle is found in the pofition G F'G', and 
lo on : the folid thus generated is called the interiorfcrezo ; 
and the height G B is called the diflance of the threads. 
The exterior fcrezo is fo adapted to the other, as if it were 
its mould ; and is nothing elfe than the folid generated 
by the polygon H G F B C, fuppofing it to partake of the 
fame motions as the triangle B G F. For rhe fake of dif- 
tinffion, we lhall apply the name of fpindle to the inte¬ 
rior fcrevv, calling the exterior one only the fcrezo. The 
fpindle then is a cylinder inverted with a fpiral band of 
uniform thicknefs, and of which the inclination with re- 
fpect to the axe of the cylinder is conftant: the fcrew, 
on the contrary, is a folid, having a correfponding fpiral 
hollow. In foine cafes the fpindle is fixed in a block, as 
A B, fig. 4.0. while the fcrew E is moved upon it by means 
of a lever D C. At other times the fcrew is fixed, and 
the fpindle moveable, (fig. 41.) but this caufes no diffe¬ 
rence in the theory. 
The curve which any one of the points of the gene¬ 
rating polygon, (fig. 39.) as N for inrtance, defcribes about 
A Z, is obvioully traced on the furface of a right cylin¬ 
der whofe axis is A Z, and radius of its bafe E N. If we 
develope this, then d c (fig. 41.) being the circumference 
which has EN (fig. 39.) for its radius, and taking the 
perpendicular be (fig. 42.) equal to the diftance between 
two contiguous threads, the hypothenufal line db will be 
the developement of an entire revolution of the point N, 
(fig- 39-) I n effeft, the helix being throughout of con¬ 
ftant inclination with refpect to any pofition whatever of 
the generating line of the cylinder, every parallel to A D 
will make with the developement of that curve the fame 
angle : thus, the developement will be a right line, as db , 
(fig. 42.) and in like manner the right line af will be the 
developement of a fecond revolution. This being granted, 
we may demonltrate, in a very fatisfaftory manner, the 
truth of the following propofition. 
Prop. XVIII. There zoill be an equilibrium in the fcrezo 
zohen the pozoer is to the refjlance, as the dijlancc betzocen tzoo 
contiguous threads in a direction parallel to the axis, to the 
circumference deferibed by the pozoer .—Let us fuppofe the 
fpindle AB (fig. 40.) to be fixed, and that the fcrew is 
moveable by the aid of a power P applied to the extre¬ 
mity C of a lever C E= R, afting horizontally perpendi¬ 
cular to the lever. Let W be the weight of the fcrew, or 
that which the fcrew fupports, or the refinance oppofed 
by the fcrew to the power P. If the fcrew preffed only 
on one of the points of the fpindie, fuppofe it to be at 
the dilfance r from the axis, and that its pofition on the 
developement db of the fpiral be at n ; then will the prtf- 
fure on the fpindle be exactly the fame as on the inclined 
plane d b (fig. 42 ,) The power M, which we fuppofe ap¬ 
plied horizontally in diredlion Mr, muft, to retain the equi¬ 
librium, fatisfy this proportion: M : W :: cb : cd ; 
cb It 
whence M ™ W. —■— W.-: where h be, and 7r — 
cd 2®r 
3’141593. The force M, which is fuppofed applied in n 
when the helix is not developed, is perpendicular to the 
edge of the cylinder, or acts in a tangential direction to 
the cylinder, and of confequence always parallel to the 
power P. 
Now, fubrtituting for this fubfidiary power M the power 
P, acting at the dilfance R, we have, Irom the principles 
of the lever, P R=M r, the lengths of the arms being R 
and r. For M in this equation, fubftitute its value in 
•the former one, and there arifes a w R P =: W/i, This 
equation, not containing r, is entirely independent of the 
diftance at which the point n is fuppofed from the axis ; 
it will therefore be the fame if we fuppofe that point any 
where elfe on the fpindle. Hence we deduce a general 
rcfult ; for this equation will even be true, if the fcrew, in¬ 
stead of touching the fpindle in a fingle point, as we have 
hitherto fuppofed, touch it in any number of points what- 
Vol. XIV. No. 1001. 
ANICS. 633: 
ever. In this latter cafe every point on the thread of the 
fpindle bears a portion of the weight W ; thefe portions, 
being denoted by W, W", W", Sec. give W'+W^-j-W"', 
&c.=W. But, on the other hand, the force P, which, 
fupports the weight of the fcrew, may be confidered as. 
the fum of as many forces, P', P'', P''', &c. as there are 
points of contadi, each of which is employed in fupport- 
ing the weights W 7 , W", W'", See. To each of thefe the 
laft equation applies ; we have, therefore, W'/rmawR P', 
W'h=zz 7 rRF", W ,/, /i=:27rRP' / ', See. their fum manifeftly 
producing the equation W^=2wRP 5 whence P : W :: 
h : 2?rR. 
Cor. 1. If the fcrew had a fquare or reffangular fillet 
inliead of a triangular one, the conclufion would be the 
fame, for it is independent of the form of the generating 
polygon. 
Cor. z. In the fame fcrew the effeft is always the great¬ 
er, as the power is applied farther from the axis. 
Cor. 3. In two different ferews, a force afting with the 
fame diftance of lever produces a greater effect in pro¬ 
portion as the threads of the fcrew are nearer together. 
Cor. 4. In the endtefs or perpetual fcrezo B C, (fig. 43.) 
which drives the teeth of the wheel F D, we lhall, in the 
cafe of an equilibrium, have P X A B >< rad. of FD = 
W x dift. of threads X rad. of axle. For the endlefs 
fcrew is a combination of the axis in peritrochio and the 
fcrew-. 
The fcrew is of very extenfive ufe in mechanics, its 
great power rendering it more eligible for comprefiing 
bodies together than moll other machines, and the great, 
dilparity betwixt the velocity of the handle and that of 
the threads of the fcrew rendering it proper for dividing 
fpace into an almoft infinite number of parts. Hence, in 
the conftrmffion of many mathematical inftruments, fucli 
as telefcopes, where it is neceffary to adjult the focus to 
the eyes of different people, the fcrew is always made ufe 
of in order to move the eye-glafs a very little farther from 
or nearer to the objeft-glafs. 
In the Philofophical Tran factions, vol.lxxi. a new me¬ 
thod of applying the fcrew, lo as to make it a£f with the 
greateft accuracy, is deferibed by Mr. Hunter. This me¬ 
thod depends upon thefe general principles, applicable to 
molt machines. 1. That the ftrength of the feveral parts of 
the engine he adjufted in fucli a manner to the force they 
are intended to exert, that they fliall not break under the 
weight they ought to countera< 5 f, nor yet encumber the 
motion by a greater quantity of matter than is neceffary 
to give them a fuitable degree of ftrength. 2. That the 
increafe of power by means of the machine be fo regu¬ 
lated, that, while the force is thereby rendered adequate 
to the effeft, it may not be retarded in procuring it more 
than is abfolutely neceffary. 3. That the machine be as 
fitnple as is confident with other conditions. 4. That it 
be as portable, and as little troublefome as pollible in its 
application. 5. That the moving power be applied in 
fucli a manner as to adl to the greateft advantage ; and 
that the motion ultimately produced may have that direc¬ 
tion and velocity which is 1110ft adapted to the execution 
of the ultimate defign of the machine. 6. Of two ma¬ 
chines, equal in other refpefts, that deferves the prefer¬ 
ence in which the friction leaft diminifiies the effect pro- 
pofed by the whole. 
To attain all thefe advantages in any one machine is 
perhaps iinpcllible; but Mr. Hunter’s method of apply¬ 
ing the fcrew certainly combines a great portion of them. 
Let A B (fig. 44.) be a plate of metal in which the fcrew 
C D plays, having a certain number of threads in an inch, 
fuppofe 10. Within the fcrew CD there is an exterior 
fcrew, which receives the fmaller fcrew D E of 11 threads 
in an inch. This fcrew is kept from moving about with 
the former by means of the apparatus at A F G B. If 
the handle CICL be turned 10 times round, the fcrew 
CD will advance an inch upwards; and, if we fuppofe 
the fcrew DE to move round along with C D, the point 
