its diameter 14 inch. Required the turns to each inch of roving.

RULE 2. The square root of the counts multiplied by 3.75 for twist, and 3.25 for weft, will give the number of turns per spindle, per inch of yarn.

(2) I am spinning 225's twist, how many turns per inch should the spindle make?

(3) In spinning 225's weft, required the number of turns of the spindle per inch of yarn.

(4) Required the number of turns per inch per spindle to spin 256's twist and weft.

(5) What ought the turns per spindle per inch to be to spin 289's twist and weft?

RULE 3. Square root of the

counts given

Square root of the

counts required f

number of turns given.

J*number of turns required.

(6) Suppose a mule spinning 64's with 30 turns to one

inch of yarn, how many turns in the same space will 81's

require?

(7) If the counts spinning be 144's with 45 turns, how many turns will 256's require?

(8) If the counts be 196 with 52 turns, required the turns to spin 324.

(1) If 196 hanks yarn have a bevil of 58, and a twist wheel of 54, what size of bevil and twist wheels must 256 have?

On Double Speed.

The Product of the small wheels will be the Numerator,

and the Product of the large wheels will be the Denominator, which reduce to the greatest common measure, the result will be the difference of speed.

(1) Admitting a mule to have on the axis of its rim a wheel of 40 teeth, that turns a wheel of 54 teeth, which has on its axis a wheel of 54 teeth, that turns a wheel of 40 teeth, giving motion to the quick speed. Required the difference of speed.

The drafts in Carding Engines.

RULE. The Product of the number of teeth in all the leaders, and the diameter of the delivery roller divided by the Product of all the followers, and the diameter of the feeding roller, will give the draft of the engine.

NOTE. The wheel on the main cylinder is a follower to the feeding roller wheel, and a leader to the doffing cylinder wheel; and the latter wheel is a follower to the main cylinder wheel, and a leader to the delivery roller wheel.

(1) What is the draft of a carding engine, the wheel on the feeding roller 140, pinion 14, wheel 140, main cylinder pinion 20, wheel 96, pinion 24, doffer wheel 96, delivery roller wheel 27, circumference of delivery roller 12·5, diameter of feeding roller 1 inch?

(2) The feeding roller is 14 inch diameter, wheel 140, pinion 14, wheel 140, main cylinder pinion 18, wheel 96, pinion 22, doffer wheel 96, delivery roller wheel 24, circumference of roller 118. Required the draft.

(3) The feeding roller is 1 inch diameter, the wheel 140, pinion 14, wheel 140, axle pinion 22, wheel 120, pinion 26, doffer wheel 120, delivery wheel 30, diameter of the roller 3 inches. Required the draft.

To find the weight of cotton to be put on the lap frame, spread a given length to produce a required No. of hanks yarn.

RULE The Product of all the drafts and length to be spread, divided by the Product of all the doublings, will be the length of yarn in inches, produced from the last process, which reduce to hanks, then find by proportion the weight required.

The draft of a spreading machine is 2, carding engine 108, first drawing 6, second 6, third 6, bobbin frame 5.33, jack frame 7.2, mule 9; 4 ends up at the first head of drawing, 6 ends up at the second head, 6 ends up at the third head, 2

ends up at the bobbin frame, 2 ends up at the jack frame, single roving in the mule. What weight of cotton must I put on the lap frame, 52 inches long, to produce 36 hanks yarn without allowance for waste?

Filleting required for the doffing Cylinder.

RULE. To the diameter of the cylinder add the thickness of the filleting, which sum take as the diameter; then the circumference of the cylinder multiplied by its length, and divided by the breadth of the filleting, will give the length required.

(1) Required the quantity of filleting to cover a cylinder 12 inches diameter, 20 inches long, the fillet with the wire inch thick and 11⁄2 broad.

(2) A doffing cylinder 16 inches diameter, 40 inches long, is to be covered with filleting inch thick and 11⁄2 broad, the length is required.

Ratio of two Pulleys.

The Product of the drivers will be the Numerator, and the Product of the driven will be the Denominator, the Quotient will be the ratio of the lesser pulley. The circumference of the doffing cylinder multiplied by the space over which it passes before the doffer strikes, will give the ratio of the greater pulley.

A carding engine has on the axis of the main `cylinder a pinion of 16 teeth, that turns a wheel of 136 teeth, on its axis a pinion of 44, that turns a wheel 138 teeth, placed on the axis of the doffing cylinder the diameter of which is 12 inches. Required the ratio of two pulleys, one on the axis of the main cylinder, the other on the axis of the doffer crank, that may cause the doffer to strike once while the doffing cylinder moves over of an inch.

To produce the same counts of yarn from an extra doubling in the preparation.

RULE. Find the draft of the rollers through which the cotton has to pass, commencing at the first head of drawing to the place of the intended increase of doubling, which multiply by the counts then spinning, and divide by the decrease in the counts caused by the additional doubling; divide the Quotient by the increased draft in the fly frame, the latter Quotient will be a power, the root of which must be extracted

according to the number of heads of drawing through which the cotton has to pass; that is, if 2 heads of drawing, extract the square root; if three heads, the cube root; if 4 heads the biquadrate root.

There are 3 heads of drawing, the draft in each head is 7-5, the draft in the fly frame 10-5, the counts spinning 80's. I intend to put 2 ends up at the fly frame with a draft of 12.5. Required the proportionate drafts in the three heads of drawing to produce 80's as before.

To find the draft of a tube frame which is to supply the place of a can frame and jack frame or stretcher.

RULE. Find what portion of a hank a given length of the last drawing will produce, (see No. 9. in Proportion) which, multiply by the doubling in the tube frame; this Product multiplied by the counts to be spun will give the amount of draft to be apportioned to the tube frame and mule, to produce which, work by double position, (see Question 9) the result will be the draft for the mule, which multiplied by 22 will give the draft of the tube frame.

ON BLOCKS AND PULLEYS.

PROBLEM I.

To find the power to raise the weight.

RULE. Divide the weight to be raised by twice the number of moveable pulleys, and the Quotient is the power to raise the weight.

Two blocks having 3 pulleys or shieves each, the one fixed and the other moveable, required the power to raise 216.

PROBLEM II.

To find the weight to be raised.

RULE. Multiply the power by twice the number of movesable pulleys, and the Product is the weight to be raised.

What weight will a power of 64 lbs. lift when applied to a 4 and 5 shieved block and tackle, the 4 shieved block being moveable?

DUODECIMALS,

Or Multiplication of Feet and Inches.

NOTE. A foot is divided into 12 inches, an inch into 12 parts or primes, a prime into 12 seconds.

RULE. Write the Multiplier under the Multiplicand; feet under feet, inches under inches, &c.

First. Multiply the Multiplicand, beginning at the lowest denomination, by the feet in the Multiplier, and place each Product under that denomination of the Multiplicand from which it arises, always carrying at 12.

Next. Multiply by inches and set each Product a place farther to the right hand.

Lastly. Multiply by the parts and set each Product another place towards the right hand.

NOTE. Feet multiplied by feet give feet.

Feet multiplied by inches give inches.
Feet multiplied by primes give primes.
Inches multiplied by inches give primes.
Inches multiplied by primes give seconds.
Primes multiplied by primes give thirds.

(23) What is the price of a marble slab at 6s. per foot, measuring 5 feet 7 inches, by 1 foot 10 inches?

(24) There are 8 windows to be glazed, each measuring 3 feet by 1 foot 6 inches, what will the glazing cost at 7åd. per foot?

(25) A house contains 3 tier of windows, three in a tier, the height of the first is 7 feet 10 inches, the second 6 feet 8 inches, and the third 5 feet 4 inches, the breadth of each is 3 feet 11 inches, what will the glazing cost at 14d. per foot?