MECHANIC S. 
63 i 
E will advance an inch. If we now turn the fcrew DE 
ten times backward, the point E will move downwards 
J-^-ihs of an inch, and the refult of both motions will be 
to lift the point E an eleventh of an inch upwards. But, 
if, while the fcrew C D is turned io times round, DE be 
kept from moving, the effeCt will be the fame as if it had 
moved io times round with CD, and been turned back 
again io times ; that is, it will advance of an inch. 
At one turn, therefore, it will advance -Jg of j-fs °f 
an inch. If now the handle be fix inches long, the power 
to produce an equilibrium mult be to the weight as i to 
iicX^X^7r;=4-i46'912. Thus, the force of Mr. Hun¬ 
ter’s fcrew is greatly fuperior to the common one; for a 
common one with a fix-inch handle niuft have iio threads 
in an inch to produce the fame effect, and this great num¬ 
ber of threads would render it too weak to refill any-con- 
iiderable violence. 
Of the Wedge. 
A wedge, in the common form, is like two inclined 
planes joined together at their bafes; and the thicknefs 
of thefe planes (oppofite their fharp edges) makes the 
back of the wedge, to which the power ot the fledge or 
hammer is applied in cleaving of wood. 
Definition. A wedge is a triangular prifrn, or a folid 
conceived to be generated by the motion of a plane tri¬ 
angle parallel to itfelf upon a flraight line which paffes 
through one of its angular points. The wedge is called 
ifofceles , reilangular, orfealenc, according as the generating 
triangle is ifofceles, riglu-angled, or fealene. 
It is very frequently ufed in cleaving wood, as repre- 
fented in the Frontifpiece; and often in railing great 
weights. The theory of knives, fwords, coulters, nails, 
&c. is generally reduced to that of the wedge. The doc¬ 
trine of the wedge is very imperfeCt, and can only be ex¬ 
hibited at all by making gratuitous afl'umptions ; fuch of 
thofe as are molt likely to obtain in practice are made the 
bafis of tlie three fucceeding propofitions. 
Prop. XIX. When a rfifling body is fujlained againjl the 
face of a wedge, by a force ailing at right angles to its direc¬ 
tion, in the cafe of equilibrium ; the power is to the refinance as 
the fine of the femi-angle of the zvedge is to the fine of the angle 
zukich the direilion of the refifiance makes with the face of the 
■zvedge-, aid the fufiaining force will be as the cofine of the latter 
angle. —Let ABC (fig. 4.5.) be a rectangular wedge, 
•whofe edge is C, face BC, and back AB. Let this 
wedge f ide freely along the plane LN ; let a body E be 
drawn or urged in the direction KE againft the face of 
the wedge, and let it be kept in that direction by a force 
acting in the direction D E, at right angles to K E. There 
tire now three forces aCting on the body E, viz. the refift- 
'ing force K E, the fuftaining force D E, and the re-aCtion 
of the wedge in the direction A E, perpendicular to the 
furface B C. On E D demit the perpendicular A G ; and, 
fince the three forces are in equilibrio, they will be to 
each other as the fides of the triangle AEG drawn pa¬ 
rallel to their directions. Draw EF perpendicular to 
A C, and the force A E will be refolved into two, one of 
which, EF, prefies the wedge perpendicularly againft the 
plane L N, and is balanced by the re-aCtion of the plane ; 
,the other, FA, endeavours to move the wedge upwards 
along the plane L N, and is balanced by the power on the 
back of the wedge. If, therefore, A G reprefent the 
force KE, EG will be the fuftaining force, and A F the 
ower applied on the back of the wedge, when thefe forces 
alance each other. Hence, making AE radius, A F is 
the fine of the angle A E F or A C B ; and A G is the 
fine of the angle AEG or KE C, thefe two angles be¬ 
ing the complements of A E K. 
If the wedge be ifofceles, or compofed of two rectan¬ 
gular wedges, the force E F, which in the former cafe 
was counteracted by the plane, will now be counteracted 
by the other half of the wedge; and the power, refift- 
.ince, and fuftaining force, will remain in the fame ratio 
*s before. 
Cor . 1. When EK is parallel to B A, A G becomes 
equal and parallel to E F ; and F. G equal and parallel to 
A F ; and the power is to the refiftance as A F to E F, or 
A B to A C, and equal to the fuftaining force. 
Cor. 2. It E K be perpendicular to B A, the direction 
of the refitting force will be parallel to AB; there¬ 
fore, the refitting and fuftaining forces changing deno¬ 
minations,. this will be a cafe corrtfpondinsj with the 
former. 
C«r. 3. When KE is perpendicular to B C, the fine 
of the angle K F. C ts radius ; and its cofine, which repre- 
lents the fuftaining force, vanithes ; therefore, the power 
is to the refiftance, as the fine of the femi-angle of the 
wedge to radius. 
Prop. XX. When the refinance is made againjl the face of 
a wedge by a body which is not fujlained, but will adhere to the 
place to which it is applied zvithout fliding, the power is to the 
ref fiance, in the cafe of equilibrium, as the cofine of the differ¬ 
ence between the femi-angle of the wedge and the angle which the 
direilion of the refifiance makes with the face of the wedge, to 
radius. —From any point K (fig. 46.) draw ths line^KE 
through the middle point of the back, meeting the face 
of the wedge in E ; let E be the unfiiding body, which 
aCts in the direction E K, and let the magnitude of the 
force with which it is urged be reprefented by A E. 
From E let fall the perpendicular EF upon A C ; and 
the force A E will be refolved into two, one of which, 
E F, will he balanced by the oppofite half of the wedge, 
and the other, A F, will be counteracted by the power ; 
therefore the power is to rhe refiftance as AF to A E, 
that is, making AE radius, as the cofine of the angle 
E A F to radius. 
Cor. 1. When KE is perpendicular to B C, the power 
is to the refiftance as A F to AE i that is, as the fine of 
the femiangle of the wedge to radius. 
Cor. 2. When K E is parallel to A B, AF vanifhesj 
that is, the power is indefinitely lefs than the weight. 
Cor. 3. When K E is perpendicular to AB, EFva* 
nifhes, and A F and A E, which reprefent the power and 
refiftance, become equal. 
Prop. XXI. When the refiJUng body is neither fujlained, 
nor adheres to the point to which it is applied, but Jlidcs freely 
along the face of the wedge, the power is to the refifiance, as the 
produil of the fines of the femiangle of the wedge and the angle in 
which the refifiance is inclined to its face, to the fquare oj' radius. 
•—Let AE (fig. 47.) be perpendicular to B C, and let t he 
body E be urged againft the face of the wedge in the di¬ 
rection KE; and let KE reprefent the magnitude as 
well as the direction of that force. On AE produced 
let fall the perpendicular K O, which will be parallel %o 
C B ; thus will the force KE be refolved into two, one 
of which, KO, will carry the body down along the face 
of the wedge, and the other, O E, will propel it perpendi¬ 
cularly againft it. Now in the cafe of equilibrium, the 
power is to O E, that part of the refiftance which aCts 
perpendicularly againft the face of the wedge, as the 
fine of the angle A C B to radius ; and O E is to the 
whole refiftance, asOEtoKE; that is, making K E ra¬ 
dius, as the fine of the angle OKE, or its alternate 
KEB, to radius. Therefore, ex aquo & componendo, the 
power is to the refiftance as fine A C B X fine KEB to 
the fquare of radius. 
Cor. 1. When KE is perpendicular to B C, the fine of 
the angle in which the refiftance is applied is radius ; 
therefore the power is to the refiftance as the fine of the 
femiangle of the wedge is to the radius. 
Cor. 2. When K E is parallel to A B, the angle of in¬ 
clination is the complement of the femiangle of the 
wedge ; and therefore, the power is to the refiftance as 
the produCt of the fine and cofine of the femi-angle of 
the wedge to the fquare of the radius. 
Cor. 3. When K E is perpendicular to A B, the angle 
of inclination is equal to the femi-angle of the wedge, 
and the power is to the refiftance in a duplicate ratio of 
the fine of the femi-angle of the wedge to the radius. 
The theory of the equilibrium of the wedge has en¬ 
gaged the attention of man y philofopbers, as Mr. Lud- 
