Box 1, Folder 9: Notebook

Circle. Ellipsis. Surfaces.

The circumference of a circle to its diameter or a semicircle to its radius is as 3.141592653589793 to 1.000000000000000

To find the area of a circle, multiply the square of its diameter by 0.7854 for that fraction is the side of a square, the area of which is equal to a circle whose diameter is 1.0000.

To find the area of an ellipsis, multiply the product of the longest and shortest diameters by 0.7854.

To find the arch of any circle, the cord and versed line being given. From the cord of half the arch, take half the cord of the whole arch: add 3/8 ths. of their difference to the cord of half the arch: the same of which will be the curve length of half the arch; which doubled, of course the curvy length of the whole arch.

To find the area of the segment of any circle the cord and versed line being given. Divide the square of half the cord, by tho versed line and add the V.Sine to the quotient. which will give the diameter of the circle. Multiply half the diam, (or radius), by the length of the arch & half the product will be the area of the sector: from which deduct the area of the triangle the remainder will be the area of the segment. Note. The difference the v. sine + radius multiplied by half the cord, will give the arch of the triangle. The square of the cord of half the arch, divided by the v. sine will give the diameter: or the same divided by the diameter, will give the V. sine.

To find by another method, the diameter of any circle, the cord and perpendicular being given. If the perpendicuar is not at the

[right page] Circle. Surfaces. middle of the cord, then the two segments of the cord must also be given. With the two segments of the cord, as bases for two triangles, and the perpendicular common to both, find by calculation the length of the two cords, subtracting each portion of the arch. Then as the perpendicular is to one of those cords, so is the other cord to the diameter.

Or thus, If from the apex of any triangle inscribed in a circle, there be a perpendicular let fall on the opposite side; that perpendicular is to one of the sides including the angles, as the other side including the angles is to the diameter of the circle.

To find the superfices of a prism or cylinder. Multiply the perimeter of one end of the prism, by its length or height, and the product will be the surface of its sides. To which add to the area of its ends when required, for its whole superficial area.

To find the surface of a regular pyramid or cone. Multiply the perimeter of the base, with the slant height, or perpendicular from the vertex on a side of the base; and half the product, will evidently be the surface of the sides; or the sum of all the triangles which form it. To which add the area of the base if required. For the whole superficial content.

To find the surface of the freestrum of a regular pyramid or cone, Add together the perimeters of the two ends and multiply their sum, by their slant height, taking half the product for the answer

To find the surface of any sphere or segment. Multiply the circumference of the sphere by its diameter, and the product will be the whole surface of it. or

[left page] Surfaces. Solids. Or square the diameter, and multiply that square by 3.1416, which will also give the surface. Or square the circumference; then either multiply that square by the decimal .3183, or divit by 3.1416, which will also give the surface.

For all surface of a segment or frustrum multiply the whole circumference of the sphere be the height of the part required.

To find the solid contents of any prism or cyclinder. Find the area of the base or end, whever the figure may be: and mutiply by the length of the cylinder or prism for the solid content.

To find the solid content of any pryamid or cone. Find the area of the base, and multiply that area by the perpendicular height; then take one third of the product for the content.

To find the solidity of the frustrum of a cone or prymind. Add into one sum the area of the two ends, and the mean proportional be tween them; take one third of that sum for a mean area, which being multiplied, by the perpendicular height of the frustrum, will give its contents.

To find the solidity of a sphere or globe. Multiply the surfce by the diameter and take one sixth of the product for the content.

Or multiply the square of the diameter, by the circumferance and take 1/6 of the product. Or take the cube of the diameter and multiply it by the decimal .5236 for the content. Or cube the circumference, and multiply by 0.01688 for the solid content.

To find the solid content of the spherical segment. From three times the diameter of the sphere, take double the height of the segment

[right page] Solids. then multiply the remainder by the square of the height, and the product by the decimal 0.5236 for the content.

Or to three times the square of the radius of the sigments base, add the square of its height, then multiply the sum by the height and the product by 0.5236 for the content.

[left page] Expansion of Metals + Fluids. Density of Water.

The following table shows the lineal dilatation of some of the metals for each degree of Fahrenheit thermometer, expressed in decimals of their length at the temperature of melting ice.

Copper 0.000096 Soft Steel 0.000059 Brass 0.000104 Tempered Steel 0.000069 Wrought iron 0.000068 Lead 0.000158 Cast iron 0.000061 Tin 0.000121

The cubical expansion of these bodies can be deduced from the above table, for it is a fact that is confirmed by theory and experiment, that the frac= tion which represents the expansion of any body in bulk; is just three times as great, as that which represents its lineal expansion; the solid contents being taken as unity in the first case, and the length in the second.

The whole dilatation of water between its boiling and freezing points is 1/22. Of Alcohol ..... 1/9. Of Mercury ... 1/56.

The densities of water at various temperatures are as follows. 32° 0.99989 79° 0.99682 34° 0.99995 100° 0.99299 39° 1.00000 122° 0.98753 44° 0.99995 142° 0.98182 49° 0.99978 162° 0.97552 54° 0.99952 182° 0.96891 59° 0.99916 202° 0.96198 69° 0.99814 212° 0.95860

When water congeals it suddenly expands increasing in bulk, one ninth past of its former dimensions.

[right page] Specific Heat - Expansive force of Steam.

Specific heat is the quantity of heat absorbed by different bodies in raising equal weights, an equal number of degrees.

The following table shows the specific heat of different bodies, between the temperatures of boiling and freezing water, according to Messrs Petit & Dulong.

Water 1.0000 Atmospheric Air 0.2669 Mercury 0.0330 Hydrogen 3.2936 Platina 0.0335 Oxygen 0.2361 Copper 0.0940 Steam 0.8470 Iron 0.1098

The expansive force of steam at various temperatures is very different - At 212° it just exceeds the pressure of the atmosphere - The general law of the tension or expansion force of aqueous vapour is, that while the heat increases in arithmetical progression, the expansive energy increases in a geometrical ratio - Its pressure doubles for every 40° of Fanrenheit as will be seen to be nearly true from the fol -lowing table

Table of the Elastic Force of Steam [TABLE HAS A FEW PROBLEMS] Tempe- Presure Pressure Tempe Pressure Press.per. Temper-Press in Tempe Pres rature in atmo- per. in. in rature in Atmos- Squ. in. Atmos rature in atmo sphere lbs phere in lbs. pheres

212 1 15 345 10 150 381 19 285 251 2 30 351 11 165 384 20 300 275 3 45 356 12 180 387 21 315 291 4 60 361 13 195 390 22 330 304 5 78 365 14 210 392 23 345 315 6 90 369 15 225 394 24 360 324 7 105 372 16 240 396 25 375 331 8 120 375 17 255 398 26 390 338 9 135 378 18 270 400 27 405

[left page] 22 Density of Steam - Radiation - Conduction

The density and volume of steam at different temperatures, may be ascertained by mians of the following table, in which the density and volume of steam, estimated in relation to water, taken as the unit, are given for elastic forces, estimated in atmosphere

Table of the density of steam under different pressures

Presure in Presure in Atmosphere Density Volume Atmosphere Density Volume 1 0.00059 1696 10 0.00492 203 2 0.000110 909 12 0.00581 172 3 0.000160 625 14 0.00670 149 4 0.00210 476 16 0.00760 131 5 0.00258 387 18 0.00849 117 6 0.00306 326 20 0.00937 106 8 0.00399 250

The following table shows the radiating power of different bodies. deducted from experiment Lamp Black 100 Ice 85 Water 100 Mercury 20 Writing paper 98 Brilliant Lead 19 Glass 90 Polished Iron 15 India ink 88 Tin, Silver, Copper 12

Of all substances examined, Lamp black and water radiate best, and polished metals, worst - When a metal is scratched or tarnished, or when it is covered with a coat of water, of varnish, or even of woollen stuff, its power of ratiation is increased.

Of all conductors of heat, gold and silver and the best, and all the metals are good conductors. Among the solid substances, on which experiments have been made, the following relative powers of conductivity heat have been observed. See Table on next page.

[right page] 23 Table of the conducting power of different Bodies

Gold 1000 Tin 304 Silver 973 Lead 180 Copper 898 Marble 24 Iron 374 Porcelain 12 Zinc 363 Fire Brick 11

Fuel The value of wood for fuel, as ascertained by the experiments of Count Rumford is as follows.

Species of Wood lbs. of water heated 1' by 1 lb. of fuel. Oak Seasoned 4590 do. dried on a Stove 5940 Maple, or sugar tree, dried on a Stove 6480 Fir, Seasoned 5466 do. dried on a Stove 7150

This results is considered too high for practical purposes, as wood in common use cannot count always be made perfectly dry. Under ordinary circumstances, about 4500 lbs. of water, is the quantity which 1 lb. of hard wood is capable of heating 1'; and 5000 lbs. for the quantity heated 1' by Pine wood.

The experiments of M. Bull, give the following as the relative value of wood of different kinds.

Kind of Wood Might of Comp. Value Cord in lbs per cord Shellbark Hickory 4469 100 Pignut Hickery 4241 95 Red-heart Hickory 3705 81 White Oak 3821 81 Red Oak 3254 69 Hard Maple or Sugar tree 2878 60 Jersey Pine 2137 54 Pitch Pine 1904 43 White Pine 1868 42

Fuel. Linacite &c. of materials for Boilers.

By experiments on the different kinds of coal for generating heat, these results are obtained.

Kind of Coal Pounds of water heated by 1 lt of Fuel Bituminous Coal 9200 Coke 8600 Anthracite 7800

The tenacity of the metals of which Steam Boilers are usually made, and the comparasive thickness to bear the same strain, are as follows Comp Numbers representing the strengthens Kind of Metal thickness of equal thickness of those metals.

Copper 3. 6000 lbs the stress which a cubic Sheet Iron 2. 9000 lbs inch, at a red heat, will Cast Iron 6. 3000 lbs bear without breaking.

Their respective densities and cost, are.

Metals & their density Metals & their Cost per lb. Copper 8.878 Copper 34 cts Sheet iron 7.788 Sheet Iron 16 " Cast Iron 7.207 Cast Iron 6 "

The product of these three elements, give their relative cost, as follows. Copper 906 Sheet iron 250 Cast iron 260 Or taking Sheet Iron as the unit Copper 3.60 Sheet iron 1.00 Cast iron 1.04

[right page] 25 Boilers

The rules for estimating the thickness of the plates, of which cylindrical boilers are made, are as follows

Multiply the radius in inches, by the pressure on each square inch in pounds, and divide the product by the number representing the strength of the metal on p.24; the quotient is the thickness in inches.

The rules for the ends, are a follows. If of the same material with the body of the cylinder, the ends, if hemispherical, need only be half as thick; if a portion of a sphere whose radius is equal to the diameter of the cyclinder, the two thicknesses are equal. If of any other radius, multiply half the radius in inches by the pressure in pounds, and divide by the cohesive force of the material.

If the ends be plates of cast iron; multiply the pressure on each sqr. inch in lbs. by the square of the diameter in inches, and divide the product by twice the cohesive force of the material, the sqr. root of the quotient is the thickness in inches. -

As a general rule the diameter of cyclindrical boilers, for high presure steam, should not excede 30 inches - except in locomotive engines and those steam boats which have their flues within the boilers.

The following table shows the thickness of sheet iron boilers, with cast iron heads, calculated by the foregoing rules, as compared with the practice of some of the best engineers of this country - The boiler sustaining a pressure of 100 lbs. per. sqr. inch. -