> The Machinist and Tool Maker's INSTRUCTOR, CONTAINING AN EASY METHOD OF CALCULATING AND LAYING OUT DIFFICULT WORK IN THE MACHINE SHOP. THE CONSTRUCTION OF GEARING AND GEAR CUTTERS. THE UNIVERSAL MILLING MACHINE WITH PRACTICAL EXAMPLES. THE MAKING OF CUTTERS FOR VARIOUS PURPOSES. ALSO A METHOD of CALCULATING GEAR- ING FOR THE CUTTING of SPIRALS, Etc. THE UNIVERSAL GRINDING MACHINE WITH PRACTICAL EXAMPLES, The ELEMENTARY PRINCIPLES of MECHANICAL POWERS. SIMPLE AND COMPOUND GEARING. THE TREATMENT OF STEEL. TOGETHER WITH TABLES, RULES AND VALUABLE INFORMATION FOR MECHANICS GENERALLY. . ^ \ U9 FULLY ,L B L T RATED V^fcK-^-\ EDWARD GENUNG, * * Mechanical Engineer and Machinist, NEW YORK. &3 K COPYRIGHT, JUNK, l8 BY Edward Genung. KJ^ yy.% Price, $5.00 PRESS OP J. B. SAVAGH CLEVELAND. Preface. TN the preparation of this work I have taken into consideration the fact that as the average machinist has not had the education necessary to understand the more advanced mechanical works, and believing myself to be in a position to know the subjects least understood, (having had the manage- ment of machine shops for the past sixteen years) and also knowing the dislike generally for mathematical problems, more especially those in algebraic formulas, I have adopted arithmetic and plane trigonometry for this work, the former in as simple a manner as pos- sible, and in the latter treating of right angled tri- angles only. I would call the reader's attention particularly to that part of the work devoted to plane trigonometry, as, by its methods we are enabled to lay out geometric figures accurately and also to measure circular and angular distances of any description or magnitude. This subject, although new to most mechanics, can, by careful study be easily comprehended. The rela- tion of sines, cosines, tangents and secants in regard to their respective positions should be carefully studied. I trust the reader will not only recognize the value of this work, but that he will also appreciate its subjects. The writer gratefully acknowledges favors shown by Prof. De Volson Wood and R. H. Thurston; also to the Brown & Sharpe Manufacturing Co., and the Pratt & Whitney Company. The Author. New York, July, 1896. CHAPTER The following examples have been prepared to assist the reader not conversant with arithmetic : ADDITION OF INTEGERS OR WHOLE NUMBERS. The sign of Addition is marked thus (+), and, when placed between numbers means that they are to be added together, thus, 4 + 1 + 5 reads 4 plus 1 plus 5. The sign = means equal, and, 4 + 1 + 5 = 10; 6 + 2 = 8; 1 + 1 = 2. The sign of subtraction is marked thus (— ), and, when placed between two numbers, means that the one after it is to be taken from the one before it, thus, 7 — 3 = 4, or 7 minus 3 equals 4 ; 2 — 1 = 1 ; 70 — 50 = 20. MULTIPLICATION. The sign of multiplication is marked thus (X), and, when placed between two numbers, means that they are to be multi- plied together, thus, 4 X 2 = 8, or 4 times 2 are 8; 7X2X2 = 28;2X3X2X2X2 = 48, INTEGERS. A Vinculum , or bar, and a Parenthesis ( ), both have the same meaning; thus, 8X6 + 3 reads 8 times 6 plus 3 ; now 6 + 3 are 9, and 8 X 9 are 72. 4 X (6 + 2) : 6 + 2 are 8, and 4 times 8 are 32, Ans. 3 X (2 + 2 + 2) = 18, Ans., 5x1 + 1 + 1 + 1 + 1 = 25, Ans. 6 THE MACHINIST AND TOOI. MAKEE'S INSTRUCTOR. DIVISION. The sign of division is marked thus (-=-), and, when placed between two numbers, means that the number on the left is to be divided by the number on the right, as follows : 10 -7- 2 means that 10 is to be divided by 2 = 5 Ans., 40 -7- 20 = 2 Ans., 74 -T- 2 = 37 Ans. We sometimes place them in the following manner, thus 40/20, which means that 40 is to be divided by 20, which would be 2 ; 40/2 == 20 Ans., 1000/4 = 250 Ans., etc. DECIMALS. The word decimal means numbered by tens, and, in enumerating figures, we sometimes see a figure thus 4.6, which reads 4 and 6 tenths. The figure 6 to the right is called the decimal, or fractional number ; the figure to the left, or 4, is a whole number. The point between is called the decimal point because it separates whole numbers from parts of num- bers, or fractions, etc. ADDITION OF DECIMALS. What is the sum of 4 tenths, 5 hundredths, 52 thousandths? .4 reads 4 tenths, .05 " 5 hundredths, .052 " 52 thousandths. .502, Ans., reads 502 thousandths. Always keep the decimal point in a column, and as there are no whole numbers, consequently there are no units. Com- mencing at the left, enumerate as follows: tenths, hundredths, thousandths, etc. .001 = 1/1000, or, 1 thousandth, .021 = 21/1000, or, 21 thousandths. 1.001 reads 1 and 1 thousandth. Add the following numbers : 42.1, 421, 4, .04, .044. 42.1 reads 42 and 1 tenth, 421. " 421 4. " 4 .04 " 4 hundredths, .044 " 44 thousandths. 467.184 " 467 and 184 thousandths. THE MACHINIST AND TOOL MAKERS INSTRUCTOR. SUBTRACTION OF DECIMALS. From 23.1 take 19.9, or, 23.1 — 19.9. 23.1 reads 23 and 1 tenth, 19.9 " 19 and 9 tenths, 3.2 Ans., reads 3 and 2 tenths. From 45.04 take 20.004. 45.04 reads 45 and 4 hundredths, 20.004 " 20 and 4 thousandths. 25.036 Ans., reads 25 and 36 thousandths. Keep the decimal points in a column. 431.0095 — 47.000095. 431.0095 reads 431 and 95 ten thousandths, 47.000095 " 47 and 95 millionths. 384.009405 Ans., reads 384 and 9405 millionths. MULTIPLICATION OF DECIMALS. Multiply 3.7 by 2.6, or, 3.7 X 2.6. 3.7 reads 3 and 7 tenths, 2.6 " 2 and 6 tenths. 222 74 9.62 Ans., reads 9 and 62 hundredths. 840 X .004. 840 reads 840, .004 l< 4 thousandths. 3.360 reads 3 and 36 hundredths, or, 360 thousandths. .004 X .000004. .000004 reads 4 millionths, .004 " 4 thousandths. .000000016 reads 16 billionths = product. Always cut off as many figures in the product, or answer, as there are decimals in the multiplier and multiplicand together. There are nine in this last case. THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. DIVISION OF DECIMALS. Divide 4.5 by 7.2, or, 4.5 -+- 7.2. 7.2 ) 4.500 ( .62+ reads 62 hundredths. 4.32 180 144 36 + 4.5 in this case is the Dividend, and reads 4 and 5 tenths. 7.2 in the example is the Divisor, and reads 7 and 2 tenths. The Dividend being the smaller number we place several ciphers to the right, which does not change the number, but makes it more convenient to divide. It now reads 4 and 500 thousandths, as 5 tenths, 50 hundredths and 500 thousandths are the same, or, means one-half, it will be seen that it remains the same. The Dividend having three decimal points and the Divisor but one, subtract one from three, we have two places or figures to cut off. .4 -f- 142. 142) .40000(281 + 284 1160 1136 240 As the Dividend in this example contains five decimal figures, and none in the Divisor, the answer, or quotient, must have five decimal places also, and as we have but three figures, or as shown 281, we must annex two ciphers to the left, mak- ing it thus, .00281, Answer. Remember that if we continue the quotient one or more figures farther, we must also, for every figure thus carried, place a cipher in the dividend, which would leave the answer the same. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 17 and 1 tenth. 471.38 -r - 27.5. 27.5 ) 471.38 ( 275 17.1 + Answer 196 3 192 5 3 88 2 75 .0000050 1 13 -T-.02 .02 ) .0000050 ( 4 .00025 Answer. 10 10 In this example the Dividend reads, 50 ten millionths, and the Divisor 2 hundredths, and, as there are 7 decimals in the dividend and 2 in the divisor, subtracting 2 from 7, we have 5 decimals in the quotient, and, in a case of this kind, we have to add three ciphers, making the answer read 25 hundred thousandths. FRACTIONS. There are two kinds of fractions : Common and Decimal. Thus, .750 reads 750 thousandths, and is a decimal fraction, and if placed thus, 750/1000, it is a common fraction ; the meaning is not changed but still reads 750 one thousandths. There are also simple and compound fractions, thus, J4 of anything, 1/1000, 1/10, %, etc., are simple fractions. J4 of^£, or Yq of 1/5, is a fraction of a fraction, and is called a compound fraction. ADDITION OF FRACTIONS. Add the following sums : M + /^ + 1/6* The figures above the line are the Numerators, and, those below, the Denominator. We now find the smallest number (called the Least Common Denominator) that 4, 3 and 6 will 10 THE MACHINIST AND TOOI, MAKER* S INSTRUCTOR. divide without any remainder, which is 12; now one-fourth of" 12 is 3, one-third of 12 is 4 and one-sixth of twelve is 2, or thus, 12 -r- 4 = 3 12 -i- 3 = 4 12 -r- 6 = 2 9 Since these are all fractional numbers, less than one, 1/4 of 12 twelfths = 3/12 1/3 of 12 " =4/12 1/6 of 12 " = 2/12 or added together the sum is 9/12 = %. We can also do this in decimals, thus, 1/4 of 1000 = .250 1/3 of 1000 = .33333 + 1/6 of 1000 = .16666 + .74999 + Add together 2/5 + % + 1/7. In an example of this kind we multiply all of the lower figures, (Denominators,) together, thus, 5X3X7 = 105, then 2/5 of 105 = 42 2/3 of 105 = 70 1/7 of 105 = 15 127 which means 127/105, then divid- ing 125 by 105 = lyVs? Answer. Add together the following fractions : .1" + 1/5" + .3" + 34 7/ > or fractions of an inch. 1/10 of 1.000 = .100 1/5 of 1.000 = .200 5/10 of 1.000 = .300 1/4 of 1.000 = .250 .850, reads 850 thousandths of an inch. SUBTRACTION OF FRACTIONS. have 8 times 9 are 72, and % of 72 are 63 ; and 2/9 of 72 are 16. Then 16 from 63 = 47. As the denominators are 72, it will read, thus, 63/72 — 16/72 = 47/72, Answer. THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 11 4J£" — 2.1", reads, from 4^ inches take 2 and one- tenth inches. 41^ = 4.125 2.1 = 2.100 2.025, reads 2 inches and 25 thousandths. 12%" — 2.05" ? From 12%" take 2 and 5 hundredths of an inch. 12%" = 12.375" 2.05"= 2.05" 10.325" reads 10 inches and 325 thou- sandths of an inch. 10^ -4i. 10^ = 21/2 4| = 29/7, multiplying the denominators together we have 2 X 7 = 14, and we have 21/2 of 14 = 147/14 29/7 of 14 = 58/14 89/14, or thus, 14 ) 89 ( 6 T % Answer. 84 5 or carried out, thus, 14 ) 89 ( 6.35+ Answer. 84 50 42 80 70 or in decimals, 10^ = 10.5000 44 = 4.1428 7 ) 1,0000 6.3572, Answer. ,1428 Always remember that in decimals we try to reduce to thousandths, thus when we say 4-J, we divide 1000 by 6 ; if we say 4^, we divide 1000 by 9, etc. 12 THE MACHINIST AND TOO!L MAKER' S INSTRUCTOR. MULTIPLICATION OF FRACTIONS. }iXy 2 = Vs reads M multiplied by y 2 ; or y 2 X M = U multiplied by J^ = tne same. If we cut an apple into two equal parts, each of those parts is one-half, and, if we take one of those parts and cut it into four (4) equal parts, then there would be four (4) parts in the other half also, and as 4 and 4 are eight, or equal parts, •each one of those parts would be one-eighth (%) the answer, and, it will be the same as saying y 2 of J4> or > M °f Vk* ^et us take 34 °f % an inch for an example and see if we do not get y% of an inch. y^ of an inch = .500, reads 500 thousandths, M ) .500 .125, Answer, reads 125 thousandths, or ^' 1/2 X 1/6 = 1/12 Multiply the Numerator and Denomin- 1/7 X 1/8 = 1/56 ator together. 3/4 X 4/5 = 12/20 12 X 3/8 = 36/8 = 4f or 4J£, Answer. 12.5X .004 12.5 .00 4 .050 Answer, reads 5 hundredths, or, 500 ten thousand- ths. In multiplying decimals the most important thing to re- member is where to place the decimal point. 12.5 in the ex- ample is the Multiplicand. 12 is a whole number and .5 is the decimal; consequently, a decimal point must be in front •of it. .004 = 4/1000 is also a decimal and has three figures, to- gether with the one above, making 4 figures to cut off in the answer. Should there not be as many figures in the answer to cut off as in the Multiplier and Multiplicand together, then always place as many (Prefix) as required to do so and place Ihe decimal point in front as shown below. 5/12 X 7/9 = 35/108 Answer. 27 X y Q = 27/3 » 9 Answer. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 13 I .004 X .004 reads 4 thousandths multiplied by 4 thousand- ths. .004 .004 .000016 Answer. .000002 X .006. .000002 .006 .000000012 Answer. DIVISION OF FRACTIONS. 1 -r- 34 = 4 Answer. 34 -5- 1/5 reads 34 divided by 1/5. 34 -r- 1/5 = 5/4 = 134, Answer. 10f -5- J4 lOf = 74 A. Now if we had said lOf -r- 3, we would take J£ of the 74/7 or 74/21 ; but we say divided by 3^3, which would be 9 times as much, since y% is only 1/9 of 3, so we place the figures thus, 74/7 -f- 3/1, or invert the Yq ; now multiply the numerators together and divide by the denominators, thus, 74 3 7) 222 31f Answer. If I had said divide 12" X Yz\ or divide 12 inches by 3^ inches, you would not say the answer was 6, but you would say 24 ; the answer is, however, the same, no matter what we are speaking of ; again 48-t-4 = 12 Answer. 48 -r- 34« As there are four fourths to one, in four whole numbers there would be four times four or sixteen times as much ; now let us invert (turn upside down) the figures, and see what we get. It should be 16 times 12 or 192. 48 -H 4/1 = 192/1 or 192, Answer. 8-rl/3 = 24 Answer. 1 -T- 1/10 = 10. 4 -f- 1/10 = 40. 7 -T- 1/7 = 49. 43^-M = 18. 2^-^ = 5. 14 THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. INVOLUTION, or, Squares, Cubes, etc. The word Involution means the raising of a quantity to any given power. The power of any number is obtained by using that num- ber one or more times as a factor, thus, 2 X 2 = 4, means that 4 is the square of 2, or the second power of 2 ; the first power of the number is 2 or the number itself. Again, 4 X 4 X 4 = 64. 4 =, the first power, 4 X 4 = 16, the second power. 4 X 4 X 4 = 64, the third power. The second power would then be the square of the num- ber, and the third power would be the cube of the number, etc. The squares, cubes, etc., are also indicated in another manner, thus, 4 2 means the square of 4, or 4 X 4 = 16, Ans. 4 3 means the cube of 4, or 4 X 4 X 4 = 64 Ans. 2 5 means the fifth power of 2, or 2 X 2 X 2 X 2 X 2 = 32 5 4 means the fourth power, or 5x5x5x5 = 625 Ans. The small figure at the top of the number is called the Index, and means the power. Remember the figure itself is always the fii st power. EVOLUTION. Square and Cube Root. The word Evolution means the reverse of Involution. The sign of square root is the symbol y/ and is always placed before the number, thus, \/3, which means that the square root of 3 is wanted; they are usually made thus, 2 \Zo> to distinguish them from cube root, which is also made in the following manner z y/2>. The small figure 2 means the square root and the small figure 3 means the cube root is wanted of 3. What is the square root of 4, or 2 \/4? I explained in Involution that the square of 2 was 4, and, Evolution being the reverse of Involution, the square root of any number is THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. 15 found by the following method. In the example the figure 2 multiplied by itself, or 2 X 2 = 4, now 2 is the square root cf 1, and 3 is the square root of 9 : 2^/16 = 4, means the square root of 16 is 4, 2 ^/64 = 8, means the square root of 64 is 8. The square root of 9/25 = 3/5; of 16/64 = 4/8 ; of T 2 D % = b/10 ; of 4/9 = 2/3 ; of 4/16 = 2/4, etc. Find the square root of 256. 2 56 ( 16 Ans. 1 156 26 156 In square root we commence by separating into periods ci two figures each; always remember to commence at the decimal point to cut of, thus, in 256 there is no decimal point, because it is a whole number, then commence from the right. Extract the cube root of 25.4. 25.40 (5.039+ Answer. 25 4000 1003 3009 99100 10069 90621 8479 Find the square root of 18. IS'00 (4.2426 + 16 82 ) 2 00 1 64 S44) 3600 3376 8482 ) 22400 16964 81846 ) 543600 509076 16 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. Find the greatest square that will go into 18, which is 4 ; 5 would be too much because the square of 5 is 25. Place the 4 in the quotient, as in division ; subtract 16 from 18, we have 2, and annex the next period, or add two ciphers in the absence of other figures. Now double the figure 4 and place it on the left, as shown, to form a new divisor. Now 18.00 ( 4 find how many times this 8, with sor*.e other 16 figure will go into 200, which is 2; place this 2 at the right of 8, making it 82, then multiply 8 ) 2 00 the 82 by the 2, which is supposed to go into the quotient with the 4, and continue as before. Remember to double all figures in the quotient as fast as you proceed, except, the last named figure. CUBE ROOT. As square root is separated into periods of two figures each, so is cube root separated into periods of three figures each. As the cube of 2 is 8, so the cube root of 8 is 2. The cube of 3 is 27, so the cube root of 27 is 3, and the cube root of 27/64 = %, etc. The cube of 1 = 1, the cube root of 1 = 1, The cube of 2 = 8, the cube root of 2 = 1 .259 + , The cube of 8 = 512, the cube root of 8 = 2. The cube root of 512 = 8. The cube root of 8 = 2. The cube root of 1 = 1. The cube of 1.259+ =2. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 17 Extract the cube root of 1420. 1'420 1 (11.23 + 12 x 300 = 300 1X30X1= 30 1 2 = 1 331 420 331 112 x soo = 36300 11 X 30 X 2 = 660 2 2 = 4 36964 89000 73928 112 2 X 300 = 3763200 112 X 30 X 3 = 10080 3 2 = 9 15072000 11319867 3773289 3752133 Fstract the cube root of 9840. 9.840 ( 21.43 nearly. 2 2 X 300 = 1200 2 X 30 X 1 = 60 1 2 = 1 1840 1261 21 2 X 300 = 132300 21 X 30 X 4 = 2520 4 2 = 16 1 261 579000 134836 39344 214 2 X 300 = 13738800 214 X 30 X 3 = 19260 3 2 = 9 13758069 39656000 41274207 TO EXTRACT THE CUBE ROOT. Commence at the Decimal point, if any; if there is none ommence from the right, and separate the figures into periods f three figures each, as shown in the examples ; find the lighest number that, when cubed, will go into the first period IS THE MACHINIST AND TOOT. MAKER'S INSTRUCTOR. at the left, which in the last example is 2; place this figure in the quotient, then cube it, which is 8, and place it under the figure 9 as shown ; subtract this, and we have 1840 for a re- mainder. We will now have to find a divisor for this 1840, which is done in the following manner : Place the same figure in the quotient, whatever it may be, to the left of the example, square it and multiply by 300; then take the same figure and multiply by 30, find how many times these figures (that is, the 2 2 X 300 + 2 X 30) will go into the divi- dend (1840 in this instance) and we find it to be 1; then place this figure 1, which is called the last figure of the root, in the divisor, which now reads 2 X 30 X 1 ; take the same figure 1 and square it as shown, add all together and we have 1261. Then as the last figure in the. quotient is 1, we multiply 1261 by 1 and place it in the dividend under the 1840, then subtracting, we have 579 ; add the next period, or, if none, three ciphers, and we now proceed to find a new divisor, which is done in the same manner as before. Remember when making a new divisor that we always take all the figures in the quotient, no matter how many there are and first square them, then multiply by 300 ; next take the same figures of the quotient and multiply by 30, and this again by the next figure that we will have to find the same as Defore explained, and proceed as shown in the example as far as desired. MENSURATION. The word mensuration means the process of taking dimensions, and includes lines, angles, surfaces and solids. Surface measurements have no thickness. Example : We want to find how many yards of carpet to cover a floor that measures twenty by twelve feet (20' X 12'): twenty times twelve is 240 ; this would be square feet, and as a square yard is three feet each way, which makes 9 square feet, dividing the 240 by 9 we have 26f or 26% yards. THE MACHINIST AND TOOIi MAKEB's INSTRUCTOR. 19 We have a table, the top of which measures two feet each 7Q.y> and two times two are four, which makes four square Ifeet. This can easily be proved by running a line across the Imiddle on each side and it will just divide it, leaving four [equal spaces, which will measure a foot each way. To find the length of one side of a right angled triangle, [when the length of the other two sides are given. Example : Figure 1 represents a right angled triangle, so called because there is one right angle, or 90°. In the figure one of the sides is marked 2" long, the other 3 /7 ; either one of these sides can be called the base and the other would then be the perpendicular. A perpendicular line always means a line at right angles to the object spoken of. If we drive a nail in the side of a building, we call it perpendicular to the wall. The hypothenuse of a right angled triangle is always the side opposite the right angle and is always the longest side. To find the length of the hypothenuse we pro- ceed as follows : We square the base and perpendicular, separately, and add them together, then extract the square root, thus, 2 2 or 2 times 2 are 4 3 2 or 3 times 3 are 9 13'00 (3".605 + Answer. 66 ) 4 00 3 96 72 ) 40000 36025 7205 ' Figure 2 is similar to figure 1. In this case we have decimals to deal with, otherwise it is the same. 20 THE MACHINIST AND TOOL, MAKER' S INSTRUCTOR. ' ** / A / / / / / / / / , / / ' ,0° / 1 a ■+--,756 — > THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 21 Example, Figure 2 : Let us call the bottom of the figure the base, then If" = 1.375" : 1.375 1.375 6 875 96 25 IJ" = 1.875, which squared 412 5 1375 equals 3.515625, and both added 1.890 625 3.515 625 together == 5.406 250 ( 2.325" 4- Ans. 4 43 ) 1 40 129 462 ) 1162 924 4645 ) 23850 23225 In Figure 3 we have given the base as .756 // in length, the hypothenuse as 1.190 in length. In examples of this kind, figures 3 and 4, we square both numbers as before, but sub- tract the less from the greater and then extract the square root; the quotient is then the answer. Thus, figure 3 = .756 1.190 1.416100 .756 1.190 .571536 Ans. 4536 10.710 .844564 ( .919 + 3780 1190 81 5292 119 181 ) 345 181 .571536 1.41610 1829 ) 16464 16461 Figure 4 is done in the same manner as figure 3. To find how long a piece of wire it will take to make a spiral spring, or, what is the same thing, to find the length of a spiral groove, sometimes called a helix. 22 THE MACHINIST AND TOOIL MAKFe'S INSTKUCTOE. Example : A spring 2 //f in diameter, 1/4" pitch = 4 to the inch, 12" long. Diameter 2", the circumference would be: 3.1416 2 1/4 " pitch = .250" .250 6.2832 squared 6.2832 12500 . 500 12.5664 188 496 5026 56 12566 4 376992 .062500 39.478602 39.541102 (6.288 + 36 39.4786 0224 122 ) 3 54 2 44 1248 )1 1011 9984 th of one coil 12568 ) 102702 100544 = 6.288" 48 coils 50 304 25152 12)301.824 // 25.152 Ans., in feet and decimal of a foot. To understand how to figure the contents of solid bodies is of the greatest importance to any mechanic. Frequently we wish to get some particular size and shape forged in the blacksmith shop and not having anything on hand of that particular shape, we have to take something smaller, larger, or, the shape may be quite different. It sometimes happens that we want a square block forged and we have to take a piece of round material to make it. The question then arises how long a piece will have to be cut off, so that we may get THE MACHINIST AND TOCXL MAKEE'S INSTRUCTOR. 23 what we want without waste of material, or without having it &9 5 too large, etc. Figure 5 is a block, we will say of steel, as shown, 5 // square and 3" thick, and, as shown before, the sur- face measure would be thus, 25 7/ this would be the 3 number of square "^7" inches, on the end only, and if it is three inches deep or thick, we multiply this by three inches^") giving 75. This is the number of cubic inches in the block. Or, the rule would be, to multiply the three dimensions to- gether ; if the figures given were inches, the answer would be inches, and if they were feet, the answer would be feet, etc. If the answer obtained were feet, multiplying by 1728 would give the number of cubic inches. Figure 5 therefore has 75 cubic inches. Now Figure 7 is a round piece of 4" u 4* ? *BifL *1 diameter ; let us see how many inches of this it would take to make figure 5. In any round piece we square the diam- eter and multiply by .7854; this gives the surface of one end only, thus, 4 2 = 4 4 16 .7854 16 47124 7854 12.5664 Always remember to cut off as many figures in the answer as there are decimals, which is four in this case. Now there 24 THE MACHINIST AND TOOX. MAKER'S INSTRUCTOR. are a little more than twelve and a half inches (12J£"), and dividing the seventy- five (75) by the twelve and a half, will give the length of a round piece of 4 // diameter, as shown in Figure 7. 12.56) 75.000 (5.97" + Ans. 62.80 12 200 11304 8960 8792 Now if we were to get this forged to finish we would have to change our sizes or we would not get what we wanted; in- stead of calling the block that w T e intended to finish 5 /7 , we would say 5.2 7/ square, or more, and 3}^ // , or more, in length, depending upon the accuracy of the forging. There is also a little waste in the forging of anything in the fire, but that does not usually amount to much on small pieces. Figure 6 is 1%; " X m" and w long; what is the contents in cubic inches ? 3.5 2.5 17 5 70 8.75 5.5 square inches, long. 4375 4375 5^6 48.125 cubic inches. Figure 8 represents a round ball, say 4%" in diameter What is the number of cubic inches in the ball ? THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 25 Cube the diameter and multiply by .5236 : 4.25 4.25 s& s 2125 85 1700 18.06 25 4.25 903125 361250 722500 76.765625 .5236 460593750 230296875 153531250 383828125 40.1944812500 Answer. Figure 9 represents a ring of round steel or iron, say 12" inside diameter and made of 2J# '' iron ; what is the number of cubic inches in the piece ? To the inside diameter of the ring, which is 12 // , add the thickness of metal 2^ // which makes 143^ // , and this sum multiplied by 3.1416 : m? 26 THE MACHINIST AND TOOL MAKER S INSTRUCTOR. 3.1416 14.5 2.5 Diameter of 2.5 ring 157080 125664 31416 the length of ring 125 50 6.25 .7854 6.25 39270 15708 47124 45.55320 = 4.908 + 36442560 40997880 18221280 9r>Q K7R1ftRfiA i 4.908750 sq. inches at end of ring; if cut open and this multiplied by the length of ring, it will give the answer in Cubic inches. Figure 10. This is an eliptical ring, 8" between centers, 4 7/ inside diameter, and 8 2 r/ iron; what is the cubic contents of link ? 4 + 2 = 6", the average diameter of link^ **- which multiplied by 3.1416 will give the length of the two ends if placed together with the center (8 /7 ) taken out. 3.1416 18.8496 Now add the two center pieces of 8" or 16 / 16 34.8496 = the whole length of piece. The area of the ring is 2 X 2 X .7851 = 3.1416 and this again multiplied by the length, thus, 34.849+ X 3.1416 = 109.427 cubic inches in the ring. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 27 Figure 11 represents a cone 6 /7 diameter at trie base and 14" high. What is the number of cubic inches in the piece ? 6" Diameter of base. The height is 14" 6 .7854 3 ) 14 — 36 36 47124 23562 28.2744= area of base. 4.666 28.27 32662 9332 37328 9332 %// ¥_ 131.90882 Ans. Multiply the area of the base by one-third (J£) the perpendicular height. The following two examples are similar (in shape) al- though not round, to figure 9. Suppose we want to make a ring gear that is to be shrunk on a cast iron center, or flange. The ring is to finish 241^ inches outside diameter and 22% inches inside diameter and \\^ inches wide. In a ring as large as this there should not be less than J£ of an inch all over to finish. The best way to do a piece of work of this kind is to cut from a bar a piece that, when split through the center and opened up round, there will be no weld, leaving the ring solid throughout. The question now is, how ~arge a piece shall we use to make this ring? If we make it y%" large all over, then the ring forged would be 24^ /7 diameter outside and 22%" diameter inside and 1J£" wide ; then 22% from 24J£ leaves 1%", and one-half of this will be the right thickness of the ring when forged. But if we can get a bar 1%" X lj^", it will be just right when opened up, for there is very little waste in this kind of work. The ring when forged will be about |f" X 1J£" (in section). We will now have to find the solid contents or cubic inches in this ring before we cut off the bar, for we might make a mis- take and cut it off an inch or two too long or too short. We first find the diameter of the ring midway between the inside 28 THE MACHINIST AND TOOL, MAKER' S INSTRUCTOR. and outside diameters, which would be about 23H'' or 23.812 inches diameter, and multiplied by 3.1416 = 74.8 + inches long. As a section of the ring be ff" by 1J^ 7/ , we multiply this together, thus, j-f = .938 and 1%= 1.5, then .938 X 1.5= 1.4 + square inches of section in the ring. We then multiply the length by the area or section, thus, 74.8 X 1.4 = 104.7 cubic inches in the forged ring. Now, as the rough bar is 1% 7/ X lj^ inches, we will have to find the area (or cross sec- tion), thus, 1%= 1.875 and 1^ = 1.5, then 1.875 X 1.5 = 2.8 + inches area. We now divide the cubic contents of the forged ring by the area of the rough bar, thus, 104.7 -5- 2.8 = 37.2 + inches in length, or, say 37^ // that we would have to cut from a bar 1% X 13^ to make the ring. We want to make a steel ring that will finish 16^2 // out- side diameter and 14% /r inside diameter and 1J4" wide. What size bar shall we use and what should be its length ? Allowing y%" all over to finish, then the forged ring will be 16«% X 14 J£ inches, and, measured as before through the center of its section, will be about 15% /7 diameter; then 15.875 X 3.1416 = 49.87 + inches for the length of ring. As the forged ring should be lj^" wide, the cross section of this ring would be one-half (J£) the difference between the inside and outside diameter, multiplied by 1%, or thus, 1.12 -f X 1.5 = 1.68 + square inches of section. Then, 49.87 X 1.68 = 83.7 + cubic inches in the ring when forged. Our bar should then be the difference between the outside and the inside diameter, or 2.25 inches by ]J£"; then the area will be 2.25 X 1.5 = 3.37 + inches. Then 83.7 -r- 3.37 = 24.8" Answer. TRIGONOMETRY. Trigonometry is the science of determining the sides and angles of triangles by means of certain parts which are given. It is of the greatest importance that every machinist, and more especially those who work to close measurements, should fully understand this science. In this work I propose to treat of right angled triangles only, as I believe this will be sufficient for all purposes. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 29 An angle is the opening between two lines that meet each other. The point at which they meet is called the Vertex; thus in Figure 12, A is the vertex of the angle. In Figure 21, when speaking of the angle A, or whose ver- tex is at A, remember to always say the angle B A C or C A B ; in other words, the letter that is placed between the other two is the angle to be considered. If at any time we see on a drawing an angle similar to Figure 21, with the words written, the angle A C B, or B C A, we would know with- out asking any questions that the angle was 90 degrees, as shown. The size of an angle does not depend on any length, but the rapidity with which the lines separate from each other. Adjacent angles are so called from their having a common vertex ; thus, in Figure 17 B is the vertex of both angles and B D a side of both angles ; consequently either angle is ad- jacent to the other, that is, the angle C B D is adjacent to the angle DBA. Angles are measured by making their vertex the center of a circle, and computing the arc that is included between its sides. The circumference of any circle is divided into 360 equal parts, called degrees, each degree into 60 equal parts, called minutes, aud each minute into 60 equal parts, called seconds. TANGENTS. A tangent is a straight line that touches a curve. In connec- tion with an angle, the tangent of an angle is a straight line that touches the circumference of a circle at one end, and ex- tending out to and meeting a secant at the other. It is always at right angles (or perpendicular) to the radius. A secant is a straight line that is drawn from the center of a circle and, proceeding through the circumference, meets with a tangent to the same circle. A sine is a straight line drawn from the end of an arc and SO THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 57* /7 Ji-QlS THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 31 touching a line perpendicular to the diameter at the other end. The cosine of an arc or angle is a line extending from the sine to the center. The versed sine of an arc or angle is a continuation of that part of the radius from the cosine to the circumference. Figure 18 will more fully explain the positions of the tan- gents, sines, cosines, secants, etc., with reference to the arc and angle, and always refers to the angle at the center of the circle, or as shown at C, which in this case is 50 degrees. In any triangle, right angled or otherwise, there are always two (2) right angles; consequently in Figure 18 the angle A C B being 50 degrees, then the other acute angle (there are always two in every right angled triangle), whose vertex is at B, is the difference between 50 and 90 degrees, which is 40 degrees, and this is always called the complement of the angle; that is, in speaking of any right angled triangle, suppose we find one of the acute angles to be 20 degrees, then the complement of that angle would be 70 degrees. A Chord is a straight line joining the extremities of the arc of a circle, and when passing through the center is equal to the diameter of that circle. The Chord of an angle is found by doubling the sine of half that angle; that is, if we want to know the chord of 50° we find the sine of 25° and double it. A tangent is always outside of the arc or circle, as shown in Figure 18, and the sine is always within the circle. In Figure 19 1 have shown a right angled triangle, in which the radius of the circle corresponds to one of the sides of the angle. Now, let it be remembered that when the distance C B (which is the longest line) of any right angled triangle is known, then the line A B is always the sine, and A O is always the cosine of the angle. Remember, also, that if the arc was removed that either C or B may be used as the vertex of the angle. In the figure (19) C is the vertex of the angle AC B, then the lines 32 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. A S "\ *s° , \ /,'■ THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. do marked sine and cosine are correct, but if we were to figure the complement of this angle, then the vertex of the angle would be at B, the sine, and cosine would then exchange places, because the sine never touches the vertex of the angle while the cosine always does. Remember that when the longest line of the angle is known that it is then called radiu9, and that the other two sides are always sine and cosine. In Figure 20, which is similar to Figure 19 (as regards the angle), when the distance A C is known, then A B becomes tangent, and B C the secant of the angle. Remember that when the longest line is not known it is always the secant of the angle. Remember, also, that if the arc of the circle was removed that either A B or A C can be called radius (that is, if the distance of either one of them was known). The length or distance of the one known is called radius ; thus, when the distance A C is known it is called radius, and A. B. becomes the tangent of the angle, but if A. B. was known it would be called radius, and A C the tangent of the angle ABC. In other words, where we figure tangents, the longest line of the angle is always the secant. And when we figure sines, the longest line of the angle is always the radius. Also that radius and tangent are perpendicular, or at right angles to each other. And that the sine and cosine of an angle are always at right angles to each other. In the trigonometric tables that will follow this chapter we take the radius of 1 and, for convc ience, we usually say one (1) inch. In Figure 21 the radius of the angle is marked 3 inches, and the angle is 33° 40'. In the table of tangents we find that the angle 33° 40 / corresponds with the decimal .666 + , which means that for every inch in length of the radius, that the tan- gent is .666+ inches in length; therefore, multiplying this decimal by 3, we have 1.998+ for the length of the tangent. 34 THE MACHINIST AND TOOL MAKER'S INSTEUCTOE. "WnillBeT :>6 .- A'— > \B f | <^/> 27 THE MACHINIST AND TOOL MAKEK's INSTRUCTOR. 35 In the table of secants we find the number 1.201 + oppo- site 33° 40', which means that for every inch in length of the radius that the secant is 1.201+ inches; therefore, multiply- ing this number by 3, we have 3.603+ inches for the length ofAB. As the angle C A B is 33° 40', the complement of this angle must be 90° — 33° 40' = 56° 20'. Figures 22, 24 and 25 are respectively 60°, 20° and 45°. Find the lengths of their tangents and secants. You will find in the table of tangents for 45° that 1 is the number given, which always means that the tangent is equal in length to radius. In Figures 26 and 27 we have two pieces of taper work, which is a very common thing in the machine shop. Figure 26 represents a piece 10 inches long (through its axis), and tapers from 1%" to l%" t which is just % 7/ taper in its whole length. Then, dividing %" or .500" by 10 inches, we have .050" for one inch long ; but we only want one side of the taper and, consequently the taper for one inch would be .025". In the table of tangents we find the angle 1° 26' to correspond with this number (.025), and this would be the angle to turn or grind the piece. In Figure 27 the taper is from lj^" to 2J£ // , or one inch to 8 inches in length; this would equal .125 to each inch, or .0625 inches for one side, and the tangent .0625 corresponds to the angle 3° 35 r , which is the correct angle to set over the table on the Universal Grinding machine, or the taper attach- ment on the lathe, as the case may be. It must not be forgotten that if told to turn or grind a piece of work 2° taper, that in the machine you would set it for 1° only; the taper of Figure 27 is, therefore, 7° 10'. Now, if you were given a piece of work, say like figure 27, to turn 8 inches long and lJo // diameter at the small end, the other end not being given, you would naturally want to know, the first thing, how large the piece would be on the other end. As the angle given would be 7° 10', take % of this, or 3° 35'. In the table of tangents opposite 3° 35' you will find the 36 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. THE MACHINIST AND TOOL MAKEK's INSTRUCTOR. 37 decimal .0626, which, multiplied by the length, 8", = .5008, then double this and it will tell you how much larger the piece must be on the other end. Figure 28 is also a piece of taper work quite common in the machine shop. In this case the taper does not extend the whole length of the piece. In turning with a taper attachment or grinding by swivel- ing the table of a Universal Grinding machine, the angle of the piece is all that is required ; that is, if the piece is longer or shorter, it makes no difference in the angle. But if turned in the lathe by setting the tail stock over, then, if a number of pieces are turned taper, they will all have to be the same length, otherwise their tapers will vary. The tail stock should not be set over (out of alignment) if possible, as it is hard to get a piece of work accurate in this manner, for the reason that the center has but little bearing surface. If such is re- quired, however, Figure 28 will show a very good way to do it if accuracy is desired. The drawing shows that the taper is to be y^" to 3% 7/ in length, or %" for one side; then, dividing J4" or «250 by 3.75, we have the decimal number .0666 + for the taper to one inch in length, and multiplying this by the whole length of the piece, which is 9 inches, we have ,5994 // , which is the distance that the tail stock center will have to be moved over. Now file a piece of wire to the micrometer caliper of this length, and, placing a tool with a smooth end in the tool post, with this piece between it and the center, then slack the screws and first take out the piece of wire (.odd"), then move the center over until it just touches the tool, then clamp the center firmly, and you will have but little trouble in fitting the piece by filing. POLYGONS. We now come to a class of work that is almost an every- day occurrence, at least in some shops. Suppose that we want to turn a piece of work that will be large enough to mill a hexigon S}^ // across the flat sides. In Figure 29 the distance AB is, of course, 1% inches, and is the radius of the angle A B C of 30°; then B C is the secant, and in the table of secants we find that the secant of 30° is 1.154+ , which, multi- plied by the radius and doubled, or by the diameter, thus, 1.154 X 3.5 = 4.039 inches, the diameter across the corners. In 38 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. <- MZ > u ^r V N V \ s T" ' \ /. ^ o / ° / s CO .,..-<- 1 „L..* :.>cq $ - * • o S » ° / \ cr \ \ '' \ ^ \ / \ \ / i \ \ • \ \ t \ THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 39 other words, to find the diameter across the corners of any size of hexagon (six sides), multiply the secant of 30° by the diameter across the short side (or flat). Or, if you have a round piece of steel and want to know how large a hexagon you can mill on it, multiply the cosine of 30° by the diameter of the piece, and the result is the width across the flat side. Figure 30 represents a piece 2J^g inches square. To find the diameter across the corners of any given square, multiply the secant of 45° by the diameter required across the flat side. Thus, in Figure 30 the square required is 2V& inches, and the secant of 45° is 1.414, which, multiplied by 2^, = 3.003, the answer across corners. To find how large a square we can make from any piece of round material: Multiply the cosine of 45° (.707+) by the diameter of the piece given, and the result is the width across the flat side, or the size of square. Figure 31 represents an octagon or eight sided figure (or polygon). ^ To find the diameter across • *****.. the corners of any size octagno: Multiply the secant of 22^ degrees = 1.082+ by the diameter required across the flat side. Thus, suppose that Figure 31 was to be 3 ins. across the flat side, then 1.082 X 3" = 3.246 inches across corners etc.; or, sup- pose that we have a round piece of stock and we want to mill it to the shape of an octa- gon. Then mul- tiply the cosine of 22^° = .9239,bythe diameter of the stock, and the result is the diam- eter across the flat (some- times called the short diameter. Thus, if we have a piece of round stock 2 /7 in diameter, .9239 X 2 = 1.847+ for the size of octagon that could be made from a round piece 2 ins. in diam. 40 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. Figure 32 is also a polygon of ten sides, generally called a decagon. Example : We want to mill a piece 2J^ // across the flat side (the short diameter). How large a piece of round stock will be required to make it ? Multiply the se- cant of 18° = 1.051+ by the flat diameter required; thus? if we want it to measure fyz' across the flat sides, 1.051 X 2.5 = 2.628, for the diameter across the corners, etc. To find the diameter across the flat side of any decagon that many be re- quired, multiply the cosine of 18° = .951 by the diameter of the round piece; thus, a piece of round stock two inches in diameter will make a decagon .951 X 2 = 1.902 inches for the short diameter. Figure 33 is a polygon of twelve sides (called a dodec- agon. Having the short diameter (across the flat sides), to find the diameter across the corners multiply the secant of 15° = 1.035 by the diameter across the flat side. To find how large a dodec- agon we can make from a round piece of any given size, multiply the cosine of 15° == .966 by the diameter of the round piece required. EHE MACHINIST AND TOOl, MAKER'S INSTRUCTOR. 41 42 THE MACHINIST AND TOOL MAKER 5 S INSTRUCTOR. SCREW THREADS. There are three forms of screw threads now generally used in making taps, dies, machine screws, etc., and it is very im- portant at times to know the exact depth of the thread. In Figure 34, I have shown the common sharp V thread, in which the surfaces are inclined toward each other at an angle of 60°. A section of this thread, ABC, forms an equilateral triangle, each side of it being equal to the pitch of the thread (that is, in the figure the pitch A D is of the same length as A B or B C), and its depth, measured perpen- dicular to the axis, is found by multiplying the cosine of the angle of 30° = .866 by the pitch of the thread ; thus, in a tap of eight threads per inch the pitch of the thread (or distance between two consecutive threads) is }/% inch, or .125", then .866 X .125 = . 10825" ; this, then, is the depth on one side only, so that in any thread of J^" pitch, or eight threads per inch, the depth of both sides together would be twice this, or .2165". The most convenient way, however, to find the depth of both sides is to double the cosine .866, and we then have the constant number 1.732, and this number divided by the num- ber of threads per inch of any V shape threads, will be the answer ; thus, 1.732 -?- 10 threads per inch = .173" for ten threads ; 1.732 -4- 40 = .043 for 40 threads. Suppose we have a screw 6 inches diameter and we want a thread of 1 inch pitch cut on it (sharp V style), what will be the diameter of the screw at the bottom of thread ? 1.732 -r- 1 = 1.732; then 6" — 1.732 = 4.268", Answer. If we have a tap one inch in diameter and ten threads per inch, then 1.732 -*- 10 = .1732 ; and 1" — .1732 = .826, the an- swer (or diameter at root of thread). Figure 35 is the shape of the United States standard, or the Franklin Institute thread (sometimes called the Seller's system), and the depths of threads are found in the same man- ner as the sharp V, and then subtracting J4 of the whole :he machinist and tool maker's instructor. 43 I I CO * I I I I "K 44 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. depth ; thus, in the sharp V thread, four threads per inch, the depth on both sides is .433 // , and for the U. S. standard it will be .433" — X = .325", Answer. Figure 36 represents the Whitworth or English Standard shape of thread, the angle of which is 55°, as shown. We can- not find the depth of C B in this shape of thread in the same manner as shown in the previous examples, for the reason that the distance, A B, is not the same as A D and, conse- quently, we have only the angle and distance C A (which, of course, is one-half the pitch) to work from ; we therefore call the distance C A of the right angled triangle the tangent, and C B the radius. Now, in the table of tangents for 27° 30' we find the decimal .5205 for the length C A, when C B or radius is one inch, and as C A in the figure is just one inch, then we know that C B is nearly two inches, or just what .5205 will go into 1. Thus, 1 -r- .5205 = 1.921 for the depth C B. In the Whitworth threads 1/6 is rounded off on the top and bottom of the threads, so that we will have to deduct y Q of this full depth; thus, 1.921 — J£ = 1.281 inches for one side, as shown in the figure. In Figure 37, which is a departure from the original in- tention of confining to right angled triangles, I have shown a class of work that you may be called upon to perform at any time, that is, to key a pair of levers on a shaft so that they wrill stand at an angle of 80° to each other (or anything similar). Now, if the angle A B D is 80°, then we can easily see that by making the small acute angle B A C we will have a right angled triangle, A C D. The object now is to find the length of A D (between centers). If we know this distance, by placing a pin in each of the holes A D we can make an end gauge by means of the micrometer caliper or vernier, and get it almost exact to the proper angle. We know the lengths A B and D B, but as we are now confined to a right angled triangle we must know the distances A C and DC, and to do this we must calculate the sides AC and B C of the right angled triangle B A C. As the THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 45 46 THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. longest side, A B, is given as 4", then B C is sine, and C A cosine of the angle of 10°. In the table of sines we find the number .1736 opposite 10°, then multiplying thus, .1736 X 4 the radius = .694, the distance B to C; and in the table of co- sines we find the number .985 nearly which, multiplied by the radius (A B) 4" = 3.94 as the distarce A C. The distance D C is now 5J£" — .694 = 4.806 inches. The length A C is 3.94 inches, as before explained ; we now have a right angled triangle, but we do not know what either of the acute angles are, and we can call DC or A C radius; one of these sides will be radius and the other tangent. Let us call D C radius, then A C is tangent to the angle ADC. We now divide the tangent A C (or 3.94) by the radius D C, or 4.806, which = .8198, or the tangent to the radius of 1 inch, and in the table of tangents we find this number to correspond with the angle 39° 21'. Having found the angle, we now find the secant of 39° 21', which is 1.2932, which, multiplied by the radius 4.806 = 6.214 + , the answer, or distance between centers D A. The following is another method, provided great accuracy is not required. In the same example, where two levers are to be keyed on a shaft, let Figure 38 represent the end of the shaft, which has a fine line lengthwise, say at B, and we want to make another line lengthwise at D, that will be just 80 degrees from B. The object now is to find the exact distance in a straight line from B to D in order to get the line lengthwise on the shaft at D. As the shaft is 2% 7/ in diameter, then in the angle A C B the radius is l%" ', and A B is the sine of the angle (40°.) The sine of 40° = .6428, and multiplied by the radius V/i = .8033, or the distance A B in a straight line; twice this = 1.60C6 inches, or the distance required D B. Having center lines on the two levers, and using a good new scale and a mag- nifying glass, very good results can be obtained in this man- ner. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 47 VD"> o y or un i- formly increasing in diameters, and it can easily be seen by the figure that if the belt is changed from the position shown to the center of the cones that it will be too slack. The dimensions of the cones are each 4, 6 and 8 inches for the diameters of their steps. We will now find the difference in the length of belt required for the center speed and the end of the cones. The difference between one half the diameters of the smallest step and one-half the diameter of the largest step is 2 inches, as shown; then 2'' -4- 16" = .125", or the tan- gent of the angle ABC, which corresponds to the angle 7° 8' ; the secant of this angle, as per table, is 1.0078, which, multiplied by the radius (16 r/ ) = 16.124'' for one side or A B, and twice this = 32.250 inches nearly, or about one- fourth inch too long fur the center speeds. On long belts this little difference would not make any trouble, but when they are very short they should be made so that the tension of the belt will be the same throughout. It is very easy to see that THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 53 1 6° 10 , i \ 1 • .s : *°"\ J 4^*9 36" | 3 i i j w _ . M".. i 54 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. in this case (Figure 44) that one-half of the circumference of both center steps ought to be exactly .125 of an inch longer, or .250'' in the whole circumference greater than shown in the figure. Thus, the circumference Of 6 /7 = 18.849, and adding .250 we have 19.1 nearly, and dividing by 3.1416 = 6.079 // , or about 6g 5 ? for the diameter. Figure 45 represents the same dimensions of cone and dis- tances as in Figure 41, but in the former we have a cross belt; it can readily be seen that wherever the belt is placed that the angles are the same; when the diameters are uniform as shown, the radius of the angles are 16 inches and the tangents are each 6 inches; then 6 -f- 16 = .375, or the tangent to radius of one inch, which in the table corresponds with the angle 20° 33 / , and the secant of this angle = 1.068 nearly, which, multiplied by the radius 16' f = 17.086 // for the length of one side, as A B, etc.; then twice this length, added to one-half the circumference of any pair of steps, as shown, will be the total length of belt, or about 53.02 7/ , the answer. This answer will not be strictly correct, for the reason (shown in diagram) that the belt will not be in a straight line to the center of the steps and, consequently, it will be a little longer than the figures given, but this need not be taken into consideration, except when the differences of their diameters are very great, which seldom happens in practice. It sometimes happens that in an example like Figure 44 that the middle steps are not of the same diameter; thus, sup- pose we have .250 // to be divided proportionately between the center steps of two cone pulleys, the diameter of which are 8 and 10 inches respectively; then 8 4- 10 = 18, and one should have 8/18 and the other 10/18, or, what is the same thing, 4/9 and 5/9 each of the .250 // . Generally it would not make any difference if the whole amount were to be added to either one of the pair. It will be seen from Figure 45 that in making cone pulleys for similar purposes shown that the diameters of their steps may increase uniformly if cross belts are to be used, but not for open belts, to secure good results. In making jigs and templates it is sometimes necessary (in order to obtain good results) that we know the distances in straight lines and in different directions between holes, etc. THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 00 A ^ / // ■** In Figure 46 I have shown a template in which three sets of holes, A B D, are to be drilled and reamed in a flange (nine holes in all); the flange has a hub 4 inches diameter, and 56 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. the template is bored to suit this hub and is fitted with a bind- ing screw, as shown, to hold it securely in any position. As there are to be three sets of holes, A, B, D, then after the first set is finished we will have to slack the binding screw and turn the template exactly 120°. But in order to know when it is exactly at that angle and at the same time to hold it in that position, we will make an extra hole (C); then, after the holes, A B D, are completed in the flange, a pin in the template at C is fitted to the hole in the flange at A, and it is then ready to drill the next three holes, etc. This is very easy to do when the template is at hand, but the first thing to do is to make the template and to know how to make it correctly ; it makes no difference what hole we start from to get the 120°. If we start from B we will have to make the template larger, and if we start from D we would have to build up on the template above A and to the left of it; by starting from A, as shown, we have less to add to the template than from any other place. Now, in boring holes in jigs and templates, if not too large, they should always be done in the lathe, if accuracy is required, because it is almost impossible in any drill press to bore a hole and find it true (at right angles to the surface of the plate being drilled or bored), and even when the drill press is in good condition when new, after using it for some time it will get out of shape. But, wherever done, in the lathe or drill press, a piece of cast iron (or something similar) should be turned four inches diameter to fit the template at F, and a plug to fit the spindle in place of the center, projecting outside of the face plate an inch or two, and turned true and of some exact size; for convenience only, suppose this to be one inch, then as the holes, A B G, are 8% inches from the center, our block at F, which sup- ports the template, can, by means of an end gauge 6.250 inches long placed between them, set the template in proper position 1o get this radius of S% // correct. This is the first operation; the template should in the meantime have these holes laid off by dividers, etc. roughly (so that we may know about where to start the first hole). We will now drill and bore the hole THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 57 at A. Never depend upon a drill to finish holes in tools of this kind, and if these holes are to be tapped, never depend on an ordinary tap to run straight, because they never will. Special taps with a pilot on the end that fits the hole closely should be used, or the thread should be finished or very nearly so with a tool in the tool post. If the thread is roughed out so that the tap will about enter, then it may be finished in this way, but the less you take out with the tap the better, and finishing with a tool is the best of all. The hole, A, is now supposed to be finished and, before moving the template, fit a plug neatly in it (A). The distance in a straight line, A B, or the chord A B of 40° is found by doubling the sine of half that angle ; thus, the sine of 20° = .342, and multiplied by the radius = .342 X 8.75 = 2.9925, and twice this = 5.985, or the distance between the centers, A B. If these two holes are to be the same size we will make an end gauge 5.985 inches long, and placing one end against the plug in A and the other end towards H., we will secure a block firmly on the faceplate of the lathe and, touching this end gauge near H, now take away the end gauge and swing the template so that the plug at A will just touch the block near H, and bore the hole B. If the template is %" thick, then the block will have to be raised high enough to let it pass under. The hole C can also be bored in a similar manner. The distance, C B, in a straight line is found in the following manner: As A C is 120° and A B 40°, then B C is the difference between these two angles, or 80°, and the sine of 40° = .643, which, multiplied by the radius (%%) = 5.626, and twice this = the chord of 80° = 11.252" for the distance B C, as shown. The hole D is 10° from C and is 10#" from the center. We will move the center piece upon which the tem- plate is held at F and set it by an end gauge as before. We will now find the distance in a straight line from the center of C to G, which is on the line D F; then, as C F is the long- est line of the angle G F C and is 8% inches long, it is the radius of this angle, andG C is the sine; as the sine of 10° = 58 THE MACHINIST AND TOOI, MAKER'S INSTRUCTOR. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 59 .1736, then multiplied by the radius {S^Q = 1.519 inches for the answer. A better way, perhaps, for locating the distances in this case would be to place the plug in the tail stock spindle (made to some standard size), as described, for the reason that it can always be moved out of the way, or brought up to the work just as close as occasion may require; the end gauge may then be used between this and a piece fitted in one of the holes already made, for the next operation, etc. It is well known that close, accurate work cannot be done by the most skillful hand without accurate measuring tools, as, for instance, in making the end gauges just mentioned I would recommend that they be made of ^g /7 or Jo" round steel. Drill rods are the most convenient for this purpose, and an instrument that will measure 6 inches will be suffi- cient for most purposes, as we can sometimes use two or three pieces together to get the necessary length required. In figure 46, B represents a bush, sometimes used for these templates. It is sometimes more convenient to have two or more of them fitted to the same hole, and by using threaded bushes (of standard sizes) we are enabled to use one with a plain hole, or we may want one with a tapped hole. The latter is far the better when tapping holes, and in this case the bushes should be made of extra length, and special taps would also wear better if made longer in the thread, and fluted for as short a distance as possible ; the extra length of the bush will guide the tap better and will also prevent its wearing away so rapidly; they may be used without temper- ing. Figure 47 represents a jig made for shaping the piece D, and also for drilling it, the surface shaded only being finished, including the two edges to form the angle of 37°. The plate A is held in the vise of the shaper, while B, on which D is clamped, is free to swivel on A. The three succes- sive positions, two to shape, and the third to drill (the last being the second operation after they are all shaped) are shown by the three holes I, I, I, in which a taper pin is used 60 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. to hold the plate B in the proper position on A; there is also a pin B, with a large head, as shown, on which B swivels, and this pin is forced down hard by the screw G, the point of which bears on one side only of a V groove; this binds the two plates B and A firmly together after the dowel-pin is in place; the manner of holding D is shown by the screw F pressing it against the two pins H H, and also the clamp, as shown. The plate C is raised high enough to let D pass under to be drilled. The work, D, as shown, is in the proper position for fin- ishing the edge, shown at J. In taking the finishing cut an end gauge, one inch long, is placed against the finished sur- face L, and the tool set so as to touch the other end of this gauge. After this operation the taper pin is withdrawn and B is swiveled to the next hole I; while in this position the shaded surface is finished as well as the edge, shown at O. Now, in boring these holes in the plate A, it should be swung from a pin on the face plate through the hole K, and the holes should be bored with a tool in the tool post, using the taper attachment; a taper plug should be used for sizing the holes. After both plates are bored in this manner, and put together, they may, by being careful, be slightly reamed to suit the taper pin. If we know the distance in a straight line between these holes, by means of gauges, as before stated, we can, with very little trouble, make the jig accurate enough for most any pur- pose. As the angle calls for 37°, we first find the sine of 18° 30', which is .317, then .317 X 4%" = 1.5057, and twice this = 3.0114 or the distance I' I"; and if the hole through C is parallel with E, as shown, and the hole through D is to be in the center, then F" C, should be 18° 30'. The drawing is wrong, for it shows 37°, but the figures are right, and the angle I" F" or the opening between these lines will be the differ- ence between 90° and 18° 30' which is 71° 30'; the sine of half this angle which is 35° 45' = .584, and multiplied by the radius (4%) = 2.775, then twice this sum = 5.550 for the answer, as shown. THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 61 Figure 48 shows a method of finding the size of a pulley (or anything similar) from a piece of the broken rim A B. Place the rim on a piece of paper and make the arc of a circle, as shown, A B, and in this arc make three or more circles, and at the points of intersection of these circles draw lines, and the point C, where these lines cross, is the center of the pulley, etc. 62 THE MACHINIST AND TOCXL MAKER'S INSTRUCTOR. Figure 49 shows a method of setting a machine at right angles to the main shaft. D 9/ 8 49 Drop a plumb line from the main shaft to the floor, and make a mark, D; from ten to twenty feet drop the line again and make another mark, K; with a good chalk line make the line D B on the floor; from two points on this line F G (any- where to suit) make the arcs H H, I I, and, where these lines cross each other draw a line, then this line will be at right angles to the line D E, as shown. Figure 50 represents a plate that is to have a number of holes bored through it, and two of these holes should be ex- actly six inches center to center, as shown, so that a pair of gears will run neatly on studs that are fitted to these holes. How shall we do this and know it to be correct? THE MACHINIST AND TOOL, MAKER' S INSTRUCTOR. <£»ye--ty <=>--t- _.x_. / 1 / ! 4 x I % CO 64 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. We have only to draw the line A B, and we have a right angled triangle CBD, the radins of which is six inches, and C D is the sine of the angle; dividing 3J4 by 6 we have the decimal .5416 which equals the sine to a radius of one inch. In the table of sines we fmd this number to correspond with the angle 32° 48', nearly; then as C D is the sine, E T> becomes cosine to this angle, and the cosine of the angle 32° 48', as per table, is .8406, and multiplied by the radius (6") = 5.0436 inches, as shown. Now, if we know these dimensions, by clamping parallels F G either on the drill press table or the face plate of a lathe, and then letting the plate rest on the parallel F, with an end gauge 5.043'' long placed between the end and the parallel G, it will be in the proper position for boring the hole C; then move the plate against the parallel G, and with two end gauges made of good sized wire (or another parallel) placed between the plate and the parallel F, just %}4? f l° n g> it will then be in the proper position for boring the hole, shown at E, etc. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 65 NATURAL, SINKS, TANGENTS, COSINES AND SECANTS. Deg. 1 DEG. MIS SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE . SEC. 2 4 .0 .00058 .00116 .0 .0C058 .00116 1. 1. 1. 1. 1. 1. 2 4 .01745 .01803 .01861 .01745 .01803 .01862 .99984 .99984 .99983 1.0001 10001 1.0002 6 8 10 .00174 .00233 .00291 .00174 .00233 .00291 .99999 .99999 .99999 1. 1. 1. 6 8 10 .01919 .01978 .02036 .01920 .01978 .02036 .99982 .999-50 .99979 1.0002 1.0002 1.0002 12 14 16 .00349 .00407 .00465 .00349 .00407 .00465 .99999 .99999 .99999 1. 1. 1. 12 14 16 .02094 .02152 .02210 .02095 .02163 .02211 .99978 .99977 .99975 1.0002 1.0002 1.0002 18 20 22 .00524 .00582 .00640 .00524 .00582 .00640 .99998 .99998 .99998 1. 1. 1. 18 20 22 .02269 .02327 .02385 .02269 .02327 .02386 .99974 .99973 .99971 1.0002 1.0003 1.0003 24 ! .00698 26 .00756 28 , .00814 .00698 .0U756 .00814 .99997 ,99997 .99997 1. 1. 1. 24 26 28 .02443 .02501 .02560 .02444 .02502 .02560 .99970 .99969 .99967 1.0003 1.0003 1.0003 30 32 34 .00872 .00931 .00989 .00872 .00931 .00989 .99996 .99996 .99995 1. 1. 1. 30 32 34 .02617 .02676 .02734 .02618 .02677 .02735 .99966 .99964 .99962 1.0003 1.0U03 1.0004 36 38 40 .01047 .01105 .01163 .01047 .01105 .01164 .99995 .99994 .99993 1.0001 1.0001 1.0001 36 38 40 .02792 .02850 .02908 .02793 .02851 .02909 .99961 .99959 .99958 1.0004 1.0004 1.0004 42 44 46 .01222 .01279 .01338 .01222 .01280 .01338 .99992 .99992 .99991 1.0001 1.0001 1.0001 42 44 46 .02966 .03025 .03083 .02968 .03026 .03084 .99956 .99954 .99952 1.0004 1.0004 1.0005 48 50 52 .01396 .01454 .01513 .01396 .01454 .U1513 .99990 .99989 .99988 1.0001 1.0001 1.0001 48 50 52 .03141 .03199 .032o7 .03142 .03201 .03259 .99951 .99949 .99947 1.0005 1.0005 1.0005 54 56 58 .01571 .01629 .01687 .01571 .01629 .01687 .99988 .99987 .99986 1.0001 1.0001 1.0001 54 56 58 .03315 .03374 .03132 .03317 .03375 .03434 .99945 .99943 .99941 1.0005 1.0006 1.0006 66 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 2 Deg. 3 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .08489 .03548 .03606 .03492 .03»50 .03608 .99939 .99937 .99935 1.0006 1.0006 1.0006 2 4 .05233 .05292 .05349 .05241 .05299 .05357 .99863 .99859 .99857 1.0014 1.0014 1.0014 6 8 10 .03664 03722 .03780 .03667 .03725 .03783 .99934 .99931 .99928 1.0007 1.0007 1.0007 6 8 10 .05408 .05466 .05524 .05416 .05474 .05532 .99853 .99850 .99847 1.0015 1.0015 1.0015 12 14 16 .03839 .03897 .03955 .03841 .03899 •03y58 .99926 .99924 .99922 1.0007 1.0008 1.0008 12 14 16 .05582 .05640 .05698 .05591 .05649 .05707 .99844 .99841 .99837 1.0016 1.0016 1.0016 18 20 22 .04013 .04071 .04129 .04016 .04074 .04133 .99919 .99917 .99915 1.0008 1.0008 1.0008 18 20 22 .05756 .05814 .05872 .05766 .05824 .05883 .99834 .99831 .99827 1.0017 1.0017 1.0017 24 26 28 .04187 .04245 .04304 .04191 .04249 .04308 .99912 .99909 .99907 1.0009 1.0009 1.0009 24 26 28 .05930 .05989 .06047 .05941 .05999 .06058 .99824 .99820 .99817 1.0018 1.0018 1.0018 30 32 34 .04362 .04420 .04478 .04366 .04424 .04482 .99905 .99902 .99900 1.0009 1.0010 1.0010 30 32 34 .06104 .06163 .06221 .06116 .06174 .06233 .99813 .99811* .99806 1.0019 1.0019 1.0019 36 38 40 .04536 .04594 .04652 .04541 .C4599 .04657 .99897 .99894 .99892 1.0010 1.0010 1.0011 36 38 40 .06279 .06337 .06395 .06291 .06349 .06408 .99803 .99799 .99795 1.0020 1.0020 1.0020 42 44 46 .04711 .04768 .04827 .04716 .04774 .04832 .99889 .99886 .99883 1.0011 1.0011 1.0012 42 44 48 .06453 .06511 .06569 .06467 .06525 .06583 .99791 .99788 .99784 1.0021 1.0021 1.0022 48 50 52 .04885 .04943 .05001 .04891 .04949 .05007 .99880 .99877 .99875 1.0012 1.0012 1.0012 48 50 52 .06627 .06685 .06743 .06642 .06700 .06759 .99780 .99776 .99772 1.0022 1.0022 1.0023 54 56 58 .05059 .05117 .05175 .05065 .05124 .05182 .99872 .99869 .99866 1.0013 1.0013 1.0013 54 56 58 .06801 .06859 .06917 .06817 .06876 .06934 .99768 .99764 .99760 1.0023 1.0024 1.0024 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 67 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 4 De G. 5 Deg. MIX SINE. TANG. COSINE. S^C. MIX SINE. TANG- COSINE- SEC. 2 4 .06975 .07034 .07091 .06992 .07051 .07109 .99756 .99752 .99748 1.0024 1.0025 1.0025 2 4 .08715 .08773 .08831 .08749 .08807 .08866 .99619 .99614 .99609 1.0038 1.0039 1.0039 6 8 10 .07150 .07208 .07266 .07168 .07226 .07285 .99744 .99740 .99736 1.0026 1.0026 1.0026 6 8 10 .08889 .08947 .09005 .08925 .08983 .09042 .99604 .99599 .99594 1.0040 1.0040 1.0041 12 14 16 .07324 .07382 ,07440 .07343 .07402 .07460 .99731 .99727 .99723 1.0027 1.0027 1.0027 12 14 16 .09063 .09121 .09179 .09100 .09159 .09218 .99588 .99583 .99578 1.0041 1.0042 1.0042 18 20 22 .07498 .07556 .07614 .07519 .07577 .07636 .99718 .99714 .99709 1.0028 1.0029 1.0029 18 20 22 .09237 .09295 .09353 .09276 .09335 .09394 .99572 .99567 .99562 1.0043 1.0043 1.0044 24 26 28 .07672 /TOO .07788 .07694 .07753 .07811 .99705 .99701 .99696 1.0029 1.0030 3.0030 24 26 28 .09411 .09468 .09526 .09453 .09511 .09570 .99556 .99551 .99545 1.0044 1.0045 1.0046 30 32 34 .07846 .07904 .07962 .07870 .07929 .07987 .99692 .99687 .99682 1.0031 1.0031 1.0032 30 32 34 .09584 .09642 .09700 .09629 .09687 .09746 .99539 .99534 .99528 1.0046 1.0047 1.0047 36 38 40 .08020 .08078 .08136 .03076 .08104 .08163 .99678 .99673 .99668 1.0032 1.0033 1.0033 36 38 40 .09758 .09816 .09874 .09805 .09863 .09922 .99523 .99517 .99511 1.0048 1.0048 1.0049 42 44 46 .08194 .08252 .08310 .08221 .08280 .08338 .99664 .996-^9 .99654 1.0034 1.0034 1.0035 42 44 46 .09932 .09990 .10047 .09931 .10040 .10099 .99505 .99499 .99494 1.0050 1.0050 1.0051 48 50 52 .08368 .08426 .08484 .08397 .08456 .08514 .99649 .99644 .99639 1.0035 1.0036 1.0036 48 50 52 .10105 .10163 .10221 .10157 .10216 .10275 ,99488 .99482 .99476 1.0051 1 .0052 1.0053 54 56 58 .08542 .08599 .08657 .08573 .08632 .08690 .99634 .99629 .99624 1.0037 1.0037 1.0038 54 56 58 .10279 .10337 .10395 .10334 .10393 .10451 .99470 .99464 .99458 1.0053 1.0054 1.0054 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 6 Deg. 7 Deg. MIN SINE. TANG- COSINE. SEC- MIN j SINE. TANG. COSINE. SEC 2 4 .10453 .10511 .10568 .10510 .10569 .10628 .99452 .99446 .99440 1.0055 1.0056 1.0056 2 4 .12187 .12245 .12302 .12278 .12337 .12396 .99254 .99247 .99240 1.0075 1.00', 6 1.0076 6 8 10 .10626 .10684 .10742 .10687 .10746 .10804 .99434 .99427 .99421 1 0057 1.0057 1.0058 6 8 10 .12360 .12418 .12475 .12455 .12515 .12574 .99233 .99226 .99219 1.0077 1.0078 1.0079 12 14 16 .10800 .10857 .10915 .10863 .10922 .10981 .99415 .99409 .99402 1.0059 1.0059 1.0060 12 14 16 .12533 .12591 .12648 .12633 .12692 .12751 .99211 .99204 .99197 1.0079 1.0080 1.0081 18 20 22 .10973 .11031 .11089 .11040 .11099 .11158 .99396 .99389 .99383 1.0060 1.0061 1.0062 18 20 22 .12706 .12764 .12822 .12810 .12869 .12928 .99189 .99182 .99174 1.0082 1.0082 1.0083 24 26 28 .11147 .11205 .11262 .11217 .11276 .11334 .99377 .99370 .9^363 1.0063 1.0063 1.0064 24 26 28 .12879 .12937 .12995 .12987 .13045 .13106 .99167 .99169 .99162 1.0084 1.0085 1.0085 30 32 34 .11320 .11378 .11436 .11393 .11452 .11511 .99357 .99350 .99344 1.0065 1.0065 1.0066 30 32 34 .13052 .13110 .13168 .13165 .13224 .13283 .99144 .99137 .99129 1.0086 1.0087 1.0088 36 38 40 .11494 .11551 .11609 .11570 .11629 .11688 .99337 .99331 .99324 1.0066 1.0067 1.0068 36 38 40 .13225 .13283 .13341 .13342 .13402 .13461 .99121 .99114 .99106 1.0089 1.0089 1.0090 42 44 46 .11667 .11725 .11782 .11747 .11806 .11865 .99317 .99310 .99303 1.0069 1.0069 10070 42 44 46 .13398 .13456 .13514 .13520 .13580 .13639 .99098 .93090 .99082 1.0091 1.0092 1.0092 48 50 52 .11840 .11898 .11956 .11924 .11983 .12042 .99296 .99289 .99283 1.0071 1.0071 1.0072 48 50 52 .13571 .13629 .13687 .13698 .13757 .13817 .99075 .99067 .99059 1.0093 1.0094 1.0095 54 56 58 .12014 .12071 .12129 .12101 .12160 .12219 .99276 .99269 .99262 1.0073 1.0074 1.0074 54 56 58 .13744 .13802 .13859 .13876 .13935 .13995 .99051 .99043 ,99035 1.0096 1.0097 1.0097 THE MACHINIST AXD TOOL MAKER'S INSTRUCTOR. 69 NATURAL, SINES, TANGENTS, COSINES AND SECANTS. 8 Deg. 9 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE- SEC. 2 4 .13917 .13975 .14032 .14054 .14113 .14172 .99027 .99018 .99010 1.0098 1.0099 1.0100 2 4 .15643 .15700 .15758 .15838 .15898 .15957 .98769 .98759 .98750 1.0125 1.0125 1.0126 6 8 10 .14090 .14148 .14205 .14232 .14291 .14351 .99002 .98994 .98986 1.0101 1.0102 1.0102 6 8 10 .15816 .15873 .15930 .16017 .16077 .16136 .98741 .98732 .98723 1.0127 1.0128 1.0129 12 14 16 .14263 .14320 ,14378 .14410 .14469 .14529 .98977 .98969 .98961 1.0103 1.U104 1.01U5 12 14 16 .15988 .16045 .16103 .16196 .16256 .16316 .98713 .98704 .98695 1.0130 1.0131 1.0132 18 20 22 .14435 .14493 .14551 .14588 .14648 .14707 .98952 .98944 .98935 1.0106 1.0107 1.0107 18 20 22 .16160 .16218 .16275 .16375 .16435 .16495 .98685 .98676 .98666 1.0133 1.0134 1.0135 24 26 28 .14608 .14666 .14723 .14766 .14826 .14885 .98927 .98918 .98910 1.0108 1.0109 1.0110 24 26 28 .16332 .16390 .16447 .16555 .16614 .16674 .98657 .98647 .98638 1.0136 1.0137 1.0138 30 32 34 .14781 .14838 .14896 .14945 .15004 .15064 .98901 .98893 .98884 1.0111 1.0112 1.0113 30 32 34 .16504 .16562 .16619 .16734 .16794 .16854 .98628 .98619 .98609 1.0139 1.0140 1.0141 36 38 40 .14953 .15011 .15068 .15123 .15183 .15242 .98875 .98867 .98858 1.0114 1.0115 1.0115 36 38 40 .16677 .16734 .16791 .16913 .16973 .17033 .98599 .98590 .98580 1.0142 1.0143 1.0144 42 44 46 .15126 .15183 .15241 .15302 .15362 .15421 .98849 .98840 .98832 1.0116 1.0117 1.0118 42 44 46 .16849 .16906 .16963 .17093 .17153 .17213 .98570 .98560 .9S551 1.0145 1.0146 1.0147 48 50 52 .15298 .15356 .15413 .15481 .15540 .15600 .98823 .98814 .98805 1.0119 1.0120 1.0121 48 50 52 .17021 .17078 .17135 .17273 .17333 .17393 ,98541 .98531 .98521 1.0148 1.0149 1.0150 54 56 58 .15471 .15528 .15586 .15659 .15719 .15779 .98796 .98787 .98778 1.0122 1.0123 1.0124 54 56 58 .17193 .17250 .17307 .17452 .17512 .17572 .98511 .98501 .98491 1.0151 1.0152 1.0153 70 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 10 Deg. II Deg. MIN SINE. TANG- COSINE. SEC- MIN SINE. TANG. COSINE. SEC 2 4 .17365 .17422 .17479 .17633 .17693 .17753 .98481 .98471 .98460 1.0154 1.0155 1.0156 2 4 .19081 .19138 .19195 .19438 .19498 .19559 .98162 .98151 .98140 1.0187 1.0188 1.0189 6 8 10 .17536 .17594 .17651 .17813 .17873 .17933 .98450 .98440 .98430 1.0157 1.0158 1.0159 6 8 10 .19252 .19309 .19366 .19619 .19679 .19740 .98129 .98118 .98107 1.0191 1.0192 1.0193 12 14 16 .17708 .17766 .17823 .17993 .18053 .18113 .98419 .98409 .98399 1.0160 1.0162 1.0163 12 14 16 .19423 .19480 .19537 .19800 .19861 .19921 .98095 .98084 .98073 1.0194 1.0195 1.0196 18 20 22 .17880 .17937 .17994 .18173 .18233 .18293 .98388 .98378 .98367 1.0164 1.0165 1.0166 18 20 22 .19594 .19652 .19709 .19982 .20042 .20103 .98061 .98050 .98038 1.0198 1.0199 1.0200 24 26 28 .18052 .18109 .18166 .18353 .18413 .18474 .98357 .98347 .98336 1.0167 1.0168 1.0169 24 26 28 .19766 .19823 .19880 .20163 .20224 .20285 .98027 .98015 .98004 1.0201 1.0202 1.0204 30 32 34 .18223 .18281 .18338 .18534 .18594 .18654 .98325 .98315 .98304 1.0170 1.0171 1.0172 30 32 34 .19937 .19994 .20051 .20345 .20406 .20466 .97992 .97981 .97969 1.0205 1.0206 1.0207 36 38 40 .18395 .18452 .18509 .18714 .18775 .18835 .98293 .98283 .98272 1.0174 1.0175 1.0176 36 38 40 .20108 .20165 .20222 .20527 .20587 .20648 .97957 .97946 .97934 1.0208 1.0210 1.0211 42 44 46 .18567 .18624 .18681 .18895 .18955 .19016 .98261 .98250 .98239 1.0177 1.0178 1 0179 42 44 46 .20279 .20336 .20393 .20709 .20769 .20830 .97922 .97910 .97898 1.0212 1.0213 1.0215 48 50 52 .18738 .18795 .18852 .19076 .19136 .19196 .98229 .98218 .98207 1.0180 1.0181 1.0182 48 50 52 .20449 .20506 .20563 .20891 .20952 .21012 .97887 .97875 .97863 1.0216 1.0217 1.0218 54 56 58 .18909 .18967 .19024 .19257 .19317 .19377 .98196 .98185 .98174 1.0184 1.0185 1.0186 54 56 58 .20P20 .20677 .20734 .21073 .21134 .21195 .97851 .97839 .97827 1.0220 1.0221 1.0222 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 71 NATURAL, SINES, TANGENTS, COSINES AND SECANTS. 12 Deg. 13 Deg. MIN SINK. TANG. COSINE. SEC MIN SINE. TANG. COSINE. SEC. 2 4 .20791 .20848 .20905 .21255 .21316 .21377 .97815 .97802 .97790 1.0223 1.0425 1.0226 2 4 .22495 .22552 .22608 .23087 .23148 .23209 .97437 .97424 .97411 1.0263 1.0264 1.0266 6 8 10 .20962 .21019 .21075 .21438 .21499 .21560 .97778 .97766 .97754 1.0227 1.0228 1.0230 6 8 10 .22663 .22722 .22778 .23271 .23332 .23393 .97397 .97384 .97371 1.0267 1.0268 1.0270 12 14 16 .21132 .21189 .21216 .21620 .21681 .21742 .97742 .97729 .97717 1.0231 1.0232 1.0234 12 14 16 .22835 .22892 .22948 .23455 .23516 .23577 .97358 .97345 .97331 1.0271 1.0273 1.0274 18 20 22 .21303 .21360 .21417 .21803 .21864 .21925 .97704 .97692 .97679 1.0235 1.0236 1.0237 18 20 22 .23005 .23062 .23118 .23639 .23700 .23762 .97318 .97304 .97291 1.0276 1.0277 1.0278 24 26 28 .21473 .21530 .21587 .21986 .22047 .22108 .97667 .97654 .97642 1.0239 1.0240 1.0241 24 26 28 .23175 .23231 .23288 .23823 .23885 .23946 .97277 .97264 .97250 1.0280 1.0281 1.0283 30 32 34 .21644 .21700 .21757 .22169 .22230 .22291 .97629 .97617 .97604 1.0243 1.0244 1.0245 30 32 34 .23344 .234U1 .23458 .24008 .24069 .24131 .97237 .97223 .97209 1.0284 1.0285 1.0287 36 38 40 .21814 .21871 .21928 .22352 .22114 .22475 .97592 .97579 .97566 1.0247 1.0248 1.0249 36 38 40 .23514 .23571 .23627 .24192 .24254 .24316 .97196 .97182 .97168 1.0288 1.0290 1.0291 42 44 46 .21984 .22041 .22098 .22536 .22597 .22658 .97553 .97540 .97528 1.0251 1.0252 1.0253 42 44 46 .23684 .23740 .23797 .24377 .24439 .24500 .97155 .97141 .97127 1.0293 1.0294 1.0296 48 50 52 .22155 .22211 .22268 .22719 .22780 .22842 .97515 .97502 .97489 1.0255 1.0256 1.0257 48 50 52 .23853 .23910 .23966 .24562 .24624 .24686 .97113 .97099 .97085 1.0297 1.0299 1.0300 54 56 58 .22325 .22382 .22438 .22903 .22964 .23025 .97476 .97463 .97450 1.0259 1.0260 1.0262 54 56 58 .24023 .24079 .24136 .24747 .24809 .24871 .97071 .97057 .97043 1.0302 1.0303 1.0305 THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. NATURAL, SINKS, TANGENTS, COSINES AND SECANTS. 14 DEG. 15 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC 2 4 .24192 .24248 .24305 .24933 .24994 .25056 .97029 .97015 .97001 1.0306 1.0308 1.0309 2 4 .25882 .25938 .2o994 .26795 .26857 .26919 .96593 .96577 .96562 1.0353 1.0354 1.0356 6 8 10 .24361 .24418 .24474 .25118 .25180 .25242 .96987 .96973 .96959 1.0311 1.0312 1.0314 6 8 10 .26050 .26106 .Z6163 .26982 .27044 .27107 .96547 .96.532 .96517 1.0358 1.0359 1.0361 12 14 16 .24531 .24587 .24643 .25304 .25366 .25427 .96944 .96930 .96916 1.0315 1.0317 1.0318 12 14 16 .26219 .26275 .26331 .27169 .27232 .27294 .96501 .96486 .96471 1.0362 1.03^4 1.0366 18 20 22 .24700 .24756 .24812 .25489 .25551 .25613 .96901 .968^7 .96873 1.0320 1.0321 1.0323 18 20 22 .26387 .26443 .26499 .27357 .27419 .27482 .96456 .96440 .96425 1.0367 1.0369 1.0371 24 26 28 .24869 .24925 .24982 .25675 .25737 .25800 .96858 .96844 .96829 1.0324 1.0326 1.0327 24 26 28 .26555 .26612 .26668 .27544 .27607 .27670 .96409 .96394 .96379 1.0372 1.0374 1.0376 84 .2*038 .25094 .25150 .25862 .25924 .25986 .96814 .96800 .96785 1.0329 1.0330 1.0332 30 32 34 36 38 40 .26724 .26780 .26836 .27732 .27795 .27858 .96363 .96347 .96332 1.0377 1.0379 1.0381 36 38 40 .25207 .25263 .25319 .26048 .26110 .26172 .96771 .96756 .96741 1.03a3 1.0335 1.0337 .26892 .2H948 .27004 .27920 .27983 .28046 .96316 .96300 .96285 1.0382 1.0384 1.0385 42 44 46 .25376 .25432 .25488 .26234 .26297 .26359 .96727 .96712 .96697 1.0338 1.0340 1.0341 42 44 46 .27060 .27116 .27172 .28108 .28171 .28204 .96269 .96253 .96237 1.0387 1.0389 1.0391 48 50 52 .25544 .25600 .25657 .26421 .26483 .26545 .96682 .96667 .96652 1.0343 1.0345 1.0346 48 50 52 .27228 .27284 .27340 .28297 .28360 .28423 .96222 .96206 .96190 1.0393 1.0394 1.0396 54 56 58 .25713 .25769 .25826 .26608 .26670 .26732 .96637 .96622 .96607 1.0348 1.0349 1.0351 54 56 58 .27396 .27452 .27508 .28486 .28548 .28611 .96174 .96)58 .96142 1.0398 1.0399 1.0401 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 73 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 16 Deg. 17 Deg. MIX SINE. TANG- COSINE. SEC- MIN SINE. TANG. COSINE. SEC 2 4 .27564 .27619 .27675 .28674 .28737 .28800 .96126 .96110 .96094 1.0403 1.0405 1.0406 2 4 .29237 .29293 .29348 .30573 .30637 .30710 .95630 .95613 .95596 1.0457 1.0459 1.0461 6 8 10 .27731 .27787 .27843 .28863 .28926 .28989 .96078 .96062 .96045 1.0408 1.0410 1.0412 6 8 10 .29404 .29459 .29515 .30764 .30827 .30891 .95579 .95562 .95545 1.0463 1.0464 1.0466 12 14 16 .27899 .27955 .28011 .29052 .29116 .29179 .96029 .96013 .95997 1.0413 1.0415 1.0417 12 14 16 .29571 .29626 .29682 .30955 .31019 .31082 .95528 .95510 .95493 1.0468 1.0470 1.0472 18 20 22 .28066 .28122 .28178 .29242 .29305 .29368 .95980 .95964 .95948 1.0419 1.0420 1.0422 18 20 22 .29737 .29793 .29848 .31146 .31210 .31274 .95476 .95459 .95441 1.0474 1.0476 1.0478 24 26 28 .28234 .28290 .28346 .29431 .29496 .29558 .95931 .95915 .95898 1.0424 1.0126 1.0428 24 26 28 .29904 .29959 .30015 .31338 .31402 .31466 .95424 .95406 .95389 1.0479 1.0481 1.0483 30 32 34 .28401 .28457 .28513 .29621 .29684 .29748 .95882 .95865 .95849 1.0429 1.0431 1.0433 30 32 34 .30070 .30126 .30181 .31530 .31594 .31658 .95372 .95354 .95336 1.0485 1.0487 1.0489 36 38 40 .28569 .28624 .28680 .29811 .29874 .29938 .95832 .9.5815 .95799 1.0435 1.0437 1.0438 ; 36 38 40 .30237 .30292 .30348 .31722 .31786 .31850 .95319 .95301 .95284 1.0491 1.0493 1.0495 42 44 46 .28736 .28792 .28847 .30001 .30065 .30128 .95782 .95765 .95749 1.0440 ! 1.0442 10444 42 44 46 .30403 .30458 .30514 .31914 .31978 .32012 .95266 .95248 .95231 1.0497 . 1.0499 1.0501 48 50 52 .28903 .28959 .29014 .30192 .30255 .30318 .95732 .95715 .95698 1.0446 1.0448 1.0449 48 50 52 .30569 .30625 .30680 .32106 .32170 .32235 .95213 .95195 .95177 1.0503 1.0505 1.0507 54 56 58 .29070 .29126 .29181 .30382 .30446 .30509 .95681 .95664 .95647 1.0451 1.0453 1.0455 54 56 58 .30735 .30791 .30846 .32299 .32363 .32427 .95159 .95141 .95123 1.0509 1.0511 1.0513 74 THE MACHINIST AND TOOL MAKERS INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 18 Deg. 19 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .30902 .30957 .31012 .32492 .32556 .32620 .95105 .95088 .95069 1.0515 1.0517 1.0519 2 4 .32557 .3261 2 .32667 .34433 .34498 .34563 .94552 .94533 .94514 1.0576 1.0578 1.0580 6 8 10 .31067 .31123 .31178 .32685 .32749 .32814 .95051 .95033 .95015 1.0521 1.0523 1.0525 6 8 10 .32722 .32777 .32832 .34628 .34693 .34758 .94495 .94476 .94457 1.0582 1.0585 1.0587 12 14 16 .31233 .31289 ,31344 .32878 .32943 .33007 .94997 .94979 .94961 1.0527 1.0529 1.0531 12 14 16 .32886 .32941 .32996 .34823 .31889 .34954 .94437 .94418 .94399 1.0589 1.0591 1.0593 18 20 22 .31399 .31454 .31509 .33072 .33136 .33200 .94942 .94924 .94906 1.0533 1.0535 1.0537 18 20 22 .33051 .33106 .33161 .35019 .35085 .35150 .94380 .94361 .94341 1.0595 1.0598 1.0600 24 26 28 .31565 .31620 .31675 .33265 .3334) .33395 .94887 .94869 .94851 1.0539 1.0541 1.0543 24 26 28 .33216 .33271 .33326 .35215 .35281 .35346 .94322 .94303 .94283 1.0602 1.0604 1.0606 30 32 34 .31730 .31785 .31841 .33459 .33524 .33589 .94832 .94814 .94795 1.0545 1.0547 1.0549 30 32 34 .33381 .33435 .33490 .35412 .35477 .35543 .94264 .94245 .94225 1.0608 1.0611 1.0613 36 38 40 .31896 .31951 .32006 .33654 .33718 .33783 .94777 .94758 .94739 1.0551 1.0553 1.0555 36 38 40 .33545 .33599 .33655 .35608 .35674 .35739 .94206 .9H86 .94166 1.0615 1.0617 1.0619 42 44 46 .32061 .32116 .32171 .33848 .33913 .33978 .94721 .94702 .94683 1.0557 1.0559 1.0561 42 44 46 .33709 .33764 .33819 .35805 .35871 .35936 .94147 .94127 .94108 1.0621 1.0624 1.0626 48 50 52 .32226 .32282 .32337 .34043 .34108 .34172 .94665 .94646 .94627 1.0563 1.0566 1.0568 48 50 52 .33874 .33928 .33983 .36002 .36068 .36134 ,940^8 .94068 .94048 1.0628 1.0630 1.0633 54 56 58 .32392 .32447 .32502 .34237 .34302 .34368 .94608 .94589 .94571 1.0570 1.0572 1.0574 54 56 58 .34038 .34092 .34147 .36199 .36265 .36331 .94029 .94009 .93989 1.0635 1.0637 1.0639 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 20 DEG. 21 Deg. min SINK. TANG. COSINE. SEC- MIN SINE. TANG. COSINE . SEC. 2 4 .34202 .34257 .34311 .36397 .36463 .36529 .93969 .93949 .93929 1.0642 1.0644 1.0646 ; 2 4 .35837 .35891 .35945 .38386 .38453 .38520 .93358 .93337 .93316 1.0711 10714 1.0716 6 8 10 .34366 .34420 .34475 .36595 .36661 .36727 .93909 .93889 .93869 1.0648 1.0651 1.0653 6 8 10 .36000 .36054 .36108 .38587 .386.53 .38720 .93295 .93274 .93253 1.0719 1.0721 1.0723 12 14 16 .34530 .34584 .34639 .36793 .36859 .36925 .93849 .93829 .93809 1.0655 1.0658 1 1.0660 12 14 16 .36162 .36217 .36271 .38787 .38854 .38921 .93232 .93211 .93190 1.0726 1.0728 1.07a! 18 20 22 .34693 .34748 .34803 .36991 .37057 .37123 .93789 .93769 .93748 1.0662 i 1.0664 1.0667 18 20 22 24 26 28 .36325 .36379 .36433 .38988 .39055 .39122 .93169 .93148 .93127 1.0733 1.0736 1.0738 24 26 28 .34857 .34912 .34966 .37189 .37256 .37322 .93728 .93708 .93687 1.0669 1.0672 1.0o74 .36488 .36542 .36596 .39189 .39257 .39324 .93105 .93084 .93063 1.0740 1.0743 1.07,15 30 32 34 .35021 .35075 .35129 .37388 .37455 .37521 .93667 .93647 .93626 1.0676 1.0678 1.0681 30 32 34 .36650 .36704 .36758 .39390 .39458 .39525 .93042 .93020 .92999 1.0748 1.075O 1.0753 36 38 40 .35184 .35238 .35293 .37587 .37654 .87720 .93606 .93585 .93565 1.0683 1.0685 1.0688 I 36 38 40 .36812 .36866 .36920 .39593 .39660 .39727 .92977 .92956 .929.35 1.0755 1.0758 1.0760 42 44 46 .35347 .35402 .35456 .37787 .37853 .379.0 .93544 .93524 .93503 1.0690 1.06y2 1.0695 42 44 46 .36975 .37029 .37o83 .39795 .39862 .39929 .92913 .92892 .92870 1.0762 1.0765 1.0768 48 50 52 .35511 .35565 .35619 .37986 .38053 .38119 .93483 .94462 .93441 1.0697 L.0699 j 1.0702 ; 48 50 52 .37137 .37191 .37245 .39997 .40064 .40132 .92848 .92827 .92805 1.0770 1.0773 1.0775 54 56 58 .35674 .35728 .35782 .38186 .38253 .383 J 9 .93420 .93399 .93379 1.0704 1.0707 1.0709 54 56 58 .37299 .37353 .37406 .40199 .40267 .40335 .92783 .92762 .92740 1.0778 1.0780 1.0783 76 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 22 Deg. 23 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TA*G. COSINE. SEC. 2 4 .37460 .37514 .37568 .40402 .40470 .40538 .92718 .92697 .92674 1.0785 1.0788 1.0790 2 4 .39073 .39126 .39180 .42447 .42516 .42585 .92050 .92028 .92005 1.0864 1.086^ 1.086a 6 8 10 .37622 .37676 .37730 .40606 .40673 .40741 .92653 .92631 .92619 1.0793 1.0795 1.0798 6 8 10 .39234 .39287 .39341 .42653 .42722 .42791 .91982 .91959 .91936 1.0872 1.0874 1.0877 12 14 16 .37784 .37838 .37892 .40809 .40877 .40945 .92587 .92565 .92543 1.0801 1.0803 1.0806 12 14 16 .39394 .39447 .39501 .42860 .4-929 .42998 .91913 .91890 .91867 1.0880 1.0882 1.0885 18 20 22 .37945 .37999 .38053 .41013 .41081 .41149 .92521 .92499 .92477 1.0808 1.0811 1.0813 18 20 22 .39554 .39608 .39661 .43067 .43136 .43205 .91845 .91821 .91798 1.0888 1.0891 1.0893 24 26 28 .38107 .38161 . .38214 .41217 .41285 .41353 .92454 .92432 .92410 1.0816 1.0819 1.0821 24 26 28 .39715 .39768 .39821 .43274 .43343 .43412 .91775 .917o2 .91729 1.0896 1.0899 1.0902 30 32 34 .38268 .38322 .38376 .41421 .41489 .41558 .92388 .92366 .92343 1.0824 1.0826 1.0829 30 32 34 .39875 .39928 .39981 .43481 .43550 .43619 .91706 .91683 .91659 1.0904 1.0907 1.0910 36 38 40 .38429 .3S483 .38537 .41626 .41694 .41762 .92321 .92298 .92276 1.0832 1.0834 1.0837 36 38 40 .40035 .40088 .40141 .43689 .43758 .43827 .91636 .91613 .91589 1.0913 1.0915 1.0918 42 44 46 .38590 .38*U4 .38698 .41831 .41899 .41967 .92254 .92231 .92208 1.0840 1.0842 1.0845 42 44 46 48 50 52 .40195 .40248 .40301 .43S97 .43966 .44036 .91566 .91543 .91519 1.0921 1.0924 1.0927 48 50 52 .38751 .38805 .38859 .42036 .42104 .42173 .92186 .92164 .92141 1.0847 1.0850 1.0853 .40354 .40407 .40461 .44105 .44175 .44244 .91496 .91472 .91449 1.0929 1.0932 1.0935 54 56 58 .38912 .38966 .39019 .42241 .42310 .42379 .92118 .92096 .92073 1.0855 1.0858 1.0861 54 56 58 .40514 .40567 .40620 .44314 .44383 .44453 .91425 .91402 .91378 1.0938 1.0941 1.0943 THE MACHINIST AND TOOX. MAKEE's INSTRUCTOR. 77 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 24 DEG. 25 DEG. MIN SINE. TANG- COSINE. sec. MIN SINE. TANG. COSINE. SEC 2 4 .40674 .40727 .40780 .44523 .44592 .44662 .91354 .91331 .91307 1.0946 1.0949 1.0952 2 4 .42262 .42314 .42367 .46631 .46701 .46772 .90631 .90606 .90581 1.1034 1.1037 1.1040 6 8 10 .40833 .40886 .40939 .44732 .44802 .44872 .91283 .91259 .91236 1.0955 1.0958 1.0961 6 8 10 .42420 .42472 .42525 .46843 .46914 .46985 .90557 .90532 .90507 1.1043 1.1046 1.1049 12 14 16 .40992 .41045 .41098 .44941 .45012 .45081 .91212 .91188 .91164 1.0963 1.0966 1.0969 12 14 16 .42578 .42630 .42683 .47056 .47127 .47198 .90482 .90458 .90433 1.1052 1.1055 1.1058 18 20 22 .41151 .41204 .41257 .45152 .45221 .45292 .91140 .91116 .91092 1.0972 1.0975 1.0978 18 20 22 .42736 .42788 .42841 .47270 .47341 .47412 .90408 .90383 .90358 1.1061 1.1064 1.1067 24 26 28 .41310 .41363 .41416 .45362 .45432 .45502 .91068 .91044 .91020 1.0981 1.0984 1.0987 24 26 28 .42893 .42946 .42999 .47483 .47555 .47626 .90333 .90308 .90283 1.1070 1.1073 1.1076 30 32 34 .41469 .41522 .41575 .45572 .45643 .45713 .90996 .90972 .90948 1.0989 1.0992 1.0995 30 32 34 .43051 .43103 .43156 .47697 .47769 .47840 .90258 .90233 .90208 1.1079 1.1082 1.1085 36 38 40 .41628 .41681 .41734 .45783 .45854 .45924 .90923 .90899 .90875 1.0998 1.1001 1.1004 36 38 40 .43209 .43261 .43313 .47912 .47983 .48055 .90183 .90158 .90133 1.1088 1.1092 1.1095 42 44 46 .41787 .41839 .41892 .45995 .46065 .46136 .90851 .90826 .90802 1.1007 1J010 11013 42 44 46 .43366 .43418 .43471 .48127 .48198 .48270 .90107 .90082 .90057 1.1098 1.1101 1.1104 48 50 52 .41945 .41998 .42051 .46206 .46277 .46348 .90778 .90753 .90729 1.1016 1.1019 1.1022 48 50 52 .43523 .43575 .43628 .48342 .48413 .48485 .9C032 .90006 .89981 1.1107 1.1110 1.1113 54 56 58 .42104 .42156 .42209 .46418 .46489 .46560 .90704 .90680 .90655 1.1025 1.1028 1.1031 54 56 58 .436S0 .43732 .43785 .48557 .48629 .48701 .89956 .89930 .89905 1.1116 1.1119 1.1123 78 THE MACHINIST AND TOOIi MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 26 Deg. 27 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .43837 .43889 .43941 .48773 .48845 .48917 .89879 .89854 .89828 1.1126 1.1129 1.1132 2 4 .45400 .45451 .45503 .50952 .51026 .51099 .89100 .89087 .89048 1.1223 1.1226 1.1229 6 8 10 .43994 .44046 .44098 .48989 .49062 .49134 .89803 .89777 .89751 1.1135 1.1138 1.1142 6 8 10 .45554 .45606 .45658 .51172 .51246 .51319 .89021 .88995 .88968 1.1233 1.1237 1.1240 12 14 16 .44150 .44203 ,44255 .49206 .49278 .49351 .89726 .89700 .89674 1.1145 1.1148 1.1151 12 14 16 .45710 .45761 .45813 .51393 .51466 .51540 .88941 .88915 .88888 1.1243 1.1247 1.1250 18 20 22 .44307 .44359 .44411 .49423 .49495 .49568 .89648 .89623 .89597 1.1155 1.1158 1.1161 18 20 22 .45865 .45916 .45968 .51614 .51687 .51761 .88862 .88835 .88808 1.1253 1.1257 1.1260 24 26 28 .44463 .44515 .44568 .49640 .49713 .49785 .89571 .89545 .89519 1.1164 1.1167 1.1171 24 26 28 .46020 .46071 .46123 .51835 .51909 .51983 .88781 .88755 .88728 1.1264 1.1267 1.1270 30 32 34 .44620 .44672 .44724 .49858 .49931 .50003 .89493 .89467 .89441 1.1174 1.1177 1.1180 30 32 34 36 38 40 o46175 .46226 .46278 .52057 .52130 .52204 .88701 .88674 .88647 1.1274 1.1277 1.1281 36 38 40 .44776 .44828 .44880 .50076 .50149 .50222 .89415 .89389 .89363 1.1184 1.1187 1.1190 .46329 .46381 .46433 .52279 .52353 .52427 .88620 .88593 .88566 1.1284 1.1287 1.1291 42 44 46 .44932 .14984 .45036 .50295 .50367 .50440 .89337 .89311 .89285 1.1193 1.1197 1.1200 42 44 46 .46484 .46536 .46587 .52501 .52575 .52649 .88539 .88512 .88485 1.1294 1.1298 1.1301 48 50 52 .45088 .45139 .45191 .50513 .50586 .50660 .89258 .89232 .89206 1.1203 1.1207 1.1210 48 50 52 54 56 58 .46639 .46690 .46741 .52724 .52798 .52873 ,88458 .88431 .88404 1.1305 1.1308 1.1312 54 56 58 .45243 .45295 .45347 .50733 .50806 .50879 .89179 .89153 .89127 - 1.1213 1.1217 1.1220 .46793 .46844 .46896 .52947 .53022 .53096 .88376 .88349 .88322 1.1315 1.1319 1.1322 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 79 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 28 Deg. 29 Deg. MIN SINE. TANG. COSINE. SEC- MIN SINE. TANG. COSINE. SEC. 2 4 .46947 .46998 .47050 .53171 .53245 .53320 .88295 .88267 .88240 1.1326 1.1329 1.1333 2 4 .48481 .48532 .48582 .55431 .55507 .55583 .87462 .87434 .87405 1.1433 11437 1.1441 6 8 10 .47101 .47152 .47204 .53395 .53470 .53544 .88213 .88185 .88158 1.1336 1.1339 1.1343 6 8 10 .48633 .48684 .48735 .55659 .55735 .55812 .87377 .87349 .87321 1.1445 1.1448 1.1452 12 14 16 .47255 .47306 .47357 .53619 .53694 .53769 .88130 .88103 .88075 1.1347 1.1350 1.1354 12 14 16 .48786 .48836 .48887 .55888 .55964 .56041 .87292 .87264 .87235 1.1456 1.1459 1.1463 18 20 22 .47409 .47460 .47511 .53844 .53919 .53994 .88047 .88020 .87992 1.1357 1.1361 1.1365 18 20 22 .48938 .48989 .49040 .56117 .56194 .56270 .87207 .87178 .87150 1.1467 1.1471 1.1474 24 26 28 .47562 .47613 .47664 .54070 .54145 .54220 .87965 .87937 .87909 1.1368 1.1372 1.1375 24 26 28 .49090 .49141 .49191 .56347 .56424 .56500 .87121 .87093 .87064 1.1478 1.1482 1.1486 30 32 34 .47716 .477*7 .47818 .54295 .54371 .54446 .87881 .87854 .87826 1.1379 1.1382 1.1386 30 32 34 .49242 .49293 .49343 .56577 .56654 .56731 .87035 .87007 .86978 1.1489 1.1493 1.1497 36 38 40 .47869 .47920 .47971 .54522 .54597 .54673 .87798 .87770 .87742 1.1390 1.1393 1.1397 36 38 40 .49394 .49445 .49495 .56808 .56885 .56962 .86949 .86920 .86892 1.1502 1.1505 1.1508 42 44 46 .48022 .48073 .48124 .54748 .54824 .54899 .87714 .87686 .87659 1.1401 1.1404 1.1408 42 44 46 .49546 .49596 .49647 .57039 .57116 .57193 .86863 .86834 .86805 1.1512 1.1516 1.1520 48 50 52 .48175 .48226 .48277 .54975 .55051 .55127 .87631 .87602 .87574 1.1411 L.1415 1.1419 48 50 52 .49697 .49748 .49798 .57270 .57348 .57425 .86776 .86747 .86718 1.1524 1.1528 1.1531 54 56. 58 .48328 .48379 .48430 .55203 .55279 .55355 .87546 .87518 .87490 1.1422 1.1426 1.1430 54 56 58 .49849 .49899 .49949 .57502 .57580 .57657 .86689 .86660 .86631 1.1535 1.1539 1.1543 80 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINKS, TANGENTS, COSINES AND SECANTS. 30 Deg. 31 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .50000 .50050 .50100 .57735 .57812 .57890 .86602 .86573 .86544 1.1547 1.1551 1.1555 2 4 .51504 .51554 .51603 .60086 .60165 .60244 .85717 .85687 .85657 1.1666 1.1670 1.1674 6 8 10 .50151 .50201 .50252 .57968 .58046 .58123 .86515 .86486 .86456 1.1559 1.1562 1.1566 6 8 10 .51653 .51703 .51753 .60324 .60403 .60482 .85627 .85596 .85566 1.1678 1.1683 1.1687 12 14 16 .50302 .50352 .50402 .58201 .58279 .58357 .86427 .86398 .86369 1.1570 1.1574 1.1578 12 14 16 .51803 .51852 .51902 .60562 .60642 .60721 .85536 .85506 .85476 1.1691 1.1695 1.1699 18 20 22 .50453 .50503 .50553 .58435 .58513 .58591 .86339 .86310 .86281 1.1582 1.1586 1.1590 18 20 22 .51952 .52001 .52051 .60801 .60880 .60960 .85446 .85415 .85385 1.1703 1.1707 1.1712 24 26 28 .50603 " .50653 .50703 .58669 .58748 .58826 .86251 .86222 .86192 1.1594 1.1598 1.1602 24 26 28 .52101 .52150 .52200 .61040 .61120 .61200 .85355 .85325 .85294 1.1716 1.1720 1.1724 30 32 34 .50754 .50804 .50854 .58904 .58983 .59061 .86163 .86133 .86104 1.1606 1.1610 1.1614 30 32 34 .52250 .52300 .52349 .61280 .61360 .61440 .85264 .85233 .85203 1.1728 1.1734 1.1737 36 38 40 .50904 .50954 .51004 .59140 .59218 .59297 .86074 .86044 .86015 1.1618 1.1622 1.1626 36 38 40 .52398 .52448 .52498 .61520 .61600 .6168J .85173 .85142 .85112 1.1741 1.1745 1.1749 42 44 46 .51054 .51104 .51154 .59375 .59454 .59533 .85985 .85955 .85925 1.1630 1.1634 1.1638 42 44 46 .52547 .52597 .52646 .61761 .61841 .61922 .85081 .85050 .85020 1.1753 1.1758 1.1762 48 50 52 .51204 .51254 .51304 .59612 .59691 .59770 .85896 .85866 .85836 1.1642 1.1646 1.1650 48 50 52 .52696 .52745 .52794 .62002 .6-083 .62164 .84989 .84958 .849>8 1.1766 1.1770 1.1775 54 56 58 .51354 .51404 .51454 .59848 .59928 .60007 .85806 .85776 .85746 1.1654 1.1658 1.1662 54 56 58 .52844 .52893 .52943 .62244 .62325 .62406 .84897 .84866 .84835 1.1779 1.1783 1.1787 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 81 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 82 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 34 DEG. 35 DEG. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC 2 4 .55919 .55967 .56016 .67451 .67535 .67620 .82903 .82871 .82838 1.2062 1.2067 1.2072 2 4 .57357 .57405 .57453 .70021 .70107 .70194 .81915 .81882 .81848 1.2208 1.2213 1.2218 6 8 10 .56064 .56112 .56160 .67705 .67790 .67875 .82806 .82773 .82741 1.2076 1.2081 1.2086 6 8 10 .57500 .57548 .57596 .70281 .70368 .70455 .81815 .81781 .81748 1.2223 1.2228 1.2233 12 14 16 .56208 .56256 ,56304 .67960 .68045 .68130 .82708 .82675 .82642 1.2091 1.2095 1.2100 12 14 16 .57643 .57691 .57738 .70542 .70629 .70716 .81714 .81681 .81647 1.2238 1.2243 1.2248 18 20 22 .56352 .56400 .56449 .68215 .68300 .68386 .82610 .82577 .82544 1.2105 1.2110 1.2115 18 20 22 .57786 .57833 .57881 .70804 .70891 .70979 .81614 .81580 .81546 1.2253 1.2258 1.2263 24 26 28 .56497 .56545 .56593 .68471 .68557 .68642 .82511 .82478 .82445 1.2119 1.2124 3.2129 24 26 28 .57928 .57975 .58023 .71066 .71154 .71241 .81513 .81479 .81445 1.2268 1.2273 1.2278 30 32 34 .56640 .56688 .56736 .68728 .68814 .68900 .82412 .82379 .82346 1.2134 1.2139 1.2144 30 32 34 36 38 40 .58070 .58117 .58165 .71329 .71417 .71505 .81411 .81378 .81344 1.2283 1.2288 1.2293 36 38 40 .56784 .56832 .56880 .68985 .69071 .69157 .82313 .82280 .82247 1.2149 1.2153 1.2158 .58212 .58260 .58307 .71593 .71681 .71769 .81310 .81276 .C1242 1.2298 1.2304 1.2309 42 44 46 .56928 .56976 .57023 .69243 .69329 .69415 .82214 .82181 .82148 1.2163 1.2168 1.2173 42 44 46 .58354 .58401 .58449 .71857 .71945 .72034 .81208 .81174 .81140 1.2314 1.2319 1.2324 48 50 52 .57071 .57119 .57167 .69502 .69588 .69674 .82115 .82082 .82048 1.2178 1.2183 1.2188 48 50 52 .58496 .58543 .58590 .72122 .72211 .72299 .81106 .81072 .81038 1.2329 1.2335 1.2340 54 56 58 .57214 .57262 .57310 .69761 .69847 .69934 .82015 .81982 .81948 1.2193 1.2198 1.2203 54 56 58 .58637 .58684 .58731 .72388 .72476 .72565 .81004 .80970 .80936 1.2345 1.2350 1.2355 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 83 NATURAL, SINES, TANGENTS, COSINES AND SECANTS. 1 36 Deg. 37 Deg. MIS SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .58778 .58825 ..58872 .72654 .72743 .72832 .80902 .80867 .80833 1.2361 1.2366 1.2371 2 4 .60181 .60228 .60274 .75355 .75446 .75538 .79863 .79828 .79793 1.2521 12527 1.2532 6 8 10 .58919 .58966 .59013 .72921 .73010 .73099 .80799 .80765 .80730 1.2376 1.2382 1.2387 6 8 10 .60321 .60367 .60413 .75629 .75721 .75812 .79758 .79723 .79688 1.2538 1.2543 1.2549 12 14 16 .59060 .59107 .59154 .73189 .73278 .73367 .80696 .80662 .80627 1.2392 1.2397 1.2403 12 14 16 .60460 .60506 .60552 .75904 .75996 .7o087 .79653 '79618 .79583 1.2554 1.2560 1.2565 18 20 22 .59201 .59248 .59295 .73157 .73547 .73636 .80593 .80558 .80524 1.2408 1.2413 1.2419 18 20 22 .60599 .60645 .60691 .76179 .76271 .7b363 .79547 .79512 .79477 1.2571 1.2577 1.2582 24 26 28 .59342 .59389 .594,55 .73726 .73816 .73906 .80489 .80455 .80420 1.2424 1.2429 1.2435 ! 24 26 28 .60738 .60784 .60830 .76456 .76548 .76640 .79441 .79406 .79371 1.2588 1.2593 1.2599 30 32 34 .59482 .59529 .59576 .73996 .74086 .74176 .80386 .80351 .80316 1.2440 1.2445 1.2451 30 32 34 .60876 .60922 .60968 .76733 .76825 .76918 .79335 .79300 .79264 1.2605 1.2610 1.2616 36 38 40 .59622 . 9669 .59716 .74266 .74357 .74447 .80282 .80247 .8U212 1.2456 1.2461 1.2466 36 38 40 .61014 .61060 .61107 .77010 .77103 .77196 .79229 .79193 .79158 1.2622 1.2627 1.2633 42 44 46 .59762 .59809 .59856 .74538 .74628 .74719 .80177 .80143 .80108 1.2472 1.2478 1.2483 42 44 46 .61153 .61198 .61245 .77289 .77382 .77475 .79122 .79087 .79051 1.2639 1.2644 1.2650 48 50 52 .59902 .59919 .69995 .74809 .7491-0 .74991 .80073 .80038 .80003 1.2488 1.2494 1.2499 48 50 52 .61291 .61337 .61382 .77568 .77661 .77754 .79015 .78980 .78944 1.2656 1.2661 1.2667 54 56 58 .60042 .60088 .60135 .75082 .75173 .75264 .79968 .79933 .79898 1.2505 1.2510 1.2516 54 56 58 .61428 .61474 .61520 .77848 .77941 .78035 .78908 .78872 .78837 1.2673 1.2679 1.2684 84 THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. NATURAL SINKS, TANGENTS, COSINES AND SECANTS. 38 Deg. 39 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .61566 .61612 .61658 .78128 .78222 .78316 .78801 .78765 .78729 1.2690 1.2696 1.2702 2 4 .62932 .62977 .63022 .80978 •81075 .81171 .77714 .77678 .77641 1.2867 1.2874 1.2880 6 8 10 .61703 .61749 ,61795 .78410 .78504 .78598 .78693 .78657 .78621 1.2707 1.2713 1.2719 6 8 10 .63067 .63113 .63158 .81268 .81364 .81461 .77604 .77568 .77531 1.2886 1.2892 1.2&8 12 14 16 .61841 .61886 .61932 .78692 .78786 .78881 .78586 .78550 .78514 1.2725 1.2731 1.2737 12 14 16 .63203 .63248 .63293 .81558 .81655 .81752 .77494 .77458 .77421 1.2904 1.2910 1.2916 18 20 22 .61978 .62023 .62069 .78975 .79070 .79164 .78477 .78441 .78405 1.2742 1.2748 1.2754 18 20 22 .63338 .63383 .63428 .81849 .81946 .82043 .77384 .77347 .77310 1.2922 1.2929 1.2935 24 26 28 .62115 .62160 .62206 .79259 .79354 .79448 .78369 .78333 .78297 1.2760 1.2766 1.2772 24 26 28 .63473 .63518 .63563 .82141 .82238 .82336 .77273 .77236 .77199 1.2941 1.2947 1.2953 30 32 34 .62251 .62297 .62342 .79543 .79638 .79734 .78261 .78224 .78188 1.2778 1.2784 1.2790 30 32 34 .63608 .63653 .63697 .82433 .82531 .82629 .77162 .77125 .77088 1.2960 1.296(5 1.2972 36 38 40 .62388 .62433 .62479 .79829 .79924 .80019 .78152 .78115 .78079 1.2796 1.2801 1.2807 36 38 40 .63742 .63787 .63832 .82727 .82825 .82923 .77051 .77014 .76977 1.2978 1.2985 1.2991 42 44 46 .62524 .62570 .62615 .80115 .80210 .80306 .78043 .78006 .77970 1.2813 1.2819 1.2825 42 44 46 .63877 .63921 .63966 .83021 .83120 .83218 .76940 .76903 .76865 1.2997 1.3003 1.3010 48 50 52 .62660 .62706 .62751 .80402 .80498 .80594 .77934 .77897 .77861 1.2831 1.2837 1.2843 48 50 52 .64011 .64055 .64100 .83317 .83415 .83514 .76828 .76791 .76754 1.3016 1.3022 1.3029 54 56 58 .62796 .62841 .62887 .80690 .80786 .80882 .77824 .77788 .77751 1.2849 1.2855 1.2861 54 56 58 .64145 .64190 .64234 .83613 .83712 .83811 .76716 .76679 .76642 1.3035 1.3041 1.3048 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 40 Deg. 41 Deg. MIS SINE. TANG. COSI.NE. SEC- j MIN SINE. TANG. COSINE. SEC. 2 4 .64279 .64323 .64368 .83910 .84009 .84108 .76604 .76567 .76529 1.3054 1.3060 1.3067 2 4 .65606 .65650 .6o694 .86928 .87u31 ,87133 .75471 .75433 .75394 1.3250 13257 1.3263 6 8 10 .64412 .64457 .64501 .84208 .84307 .84407 .76492 .76455 .76417 1.3073 1.3OS0 1.3086 6 8 10 .65737 .65781 .65825 .87235 .87338 .87440 .75356 .75318 .75280 1.3270 1.3277 1.3284 12 14 16 .64546 .64690 .64634 .84506 .84606 .84706 .76379 .76342 .76304 1.3092 1.3099 1.3105 12 14 16 .65869 .65913 .65956 .87543 .87646 .87749 .75241 '75203 .75165 1.3290 1.3297 1.3304 18 20 22 .64679 .64723 .61768 .84806 .84906 .85006 .76267 .76229 .76191 1.3112 1.3118 1.3125 18 20 22 24 26 28 .66000 .66044 .66087 .87852 .87955 .88058 .75126 .75088 .75049 1.3311 1.3318 1.3324 24 26 28 .64812 ,61856 .61900 .85106 .85207 .85307 .76154 .76116 .76078 1.3131 1.3138 1.3144 .66131 .66175 .66218 .88162 .88265 .88369 .75011 .74972 .74934 1.3331 1.3338 1.3345 30 32 34 .64945 .64989 .65033 .85408 .85509 .85609 .76040 •760U3 .75965 1.3151 1.3157 1.3164 30 32 34 .66262 .66305 .66348 .88472 .88576 .88680 .74895 .74857 .74818 1.3352 1.3359 1.3366 36 38 40 .65077 .65121 .65166 .85710 .86811 .85912 .75927 .75889 .75851 1.3170 1.3177 1.3184 36 38 40 .66392 .66436 .66479 .88784 .88888 .88992 .74780 .74741 .74702 1.3372 1.3379 1.3386 42 44 46 .65210 .65254 .65298 .86013 .86115 .86216 .75813 .75775 .75737 1.3190 1.3197 1.3202 42 44 46 .66523 .66566 .66610 .89097 .89201 .89305 .74664 .74625 .74586 1.3393 1.3400 1.3407 48 50 52 .65342 .65386 .65430 .86317 .86419 .86521 .75699 .75661 .75623 1.3210 1.3217 1.3223 48 50 52 .66653 .66696 .66740 .89410 .89515 .89620 .74547 .74509 .74470 1.3414 1.3421 1.3428 54 56 58 .65474 .65518 .65562 .86623 .86724 .86826 .75585 .75547 .75509 1.3230 1.3237 1.3243 54 56 58 .66783 .66826 .66870 .89725 .89830 .89935 .74431 .74392 •74353 1.3435 1.3442 1.3449 86 THE MACHINIST AND TOOI^ MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 42 DEG. i 43 Deg. MIN SINE. TANG. COSINE. SEC. MIN SIMS. TANG. COSINE. SEC. 2 4 .66913 .66956 .66999 .90040 .90146 .90251 .74314 .74275 .74236 1.3456 1.3463 1.3470 2 4 .68200 .68242 .68285 .93251 .93360 .93469 .73135 .73096 /J3056 1.3673 1.368 L 1.3688 6 8 10 .67043 .67086 .67129 .90357 .90462 .90568 .74197 .74158 .74119 1.3477 1.3485 1.3492 6 8 10 .68327 .68370 .68412 .93578 .93687 .93797 .73016 .72976 .72937 1.3695 1.3703 1.3710 12 14 16 .67172 .67215 .67258 .90674 .90780 .90887 .74080 .74041 .74002 1.3499 1.3506 1.3513 12 14 16 .68455 .68497 .68539 .93906 .94016 .94125 .72897 .72857 .72817 1.3718 1.3725 1.3733 18 20 22 .67301 .67344 .67387 .90993 .91099 .91206 .73963 .73924 .73885 1.3520 1.3527 1.3534 18 20 22 .68582 .68624 .68666 .94235 .94345 .94455 .72777 .72737 .72697 1.3740 1.3748 1.3756 24 26 28 .67430 .67473 .67516 .91312 .91419 .91526 .73845 .73806 .73767 1.3542 1.3549 1.3556 24 26 28 .68709 .68751 .68793 .94565 .94675 .94786 .72657 .72617 .72577 1.3763 1.3771 1.3778 30 32 34 .67559 .67602 .67645 .91633 .91740 .91847 .73728 .73688 .73649 1.3563 1.3571 1.3578 30 32 34 .68835 .68877 .68920 .94896 .95007 .95118 .72537 .72497 .72457 1.3786 1.3794 1.3801 36 38 40 .67687 .67730 .67773 .91955 .92062 .92169 .73609 .73570 .73531 1.3585 1.3592 1.3600 36 38 40 .68962 .69004 .69046 .95228 .95339 .95451 .72417 .72377 .72337 1.3809 1.3816 1.3824 42 44 46 .67816 .67859 .67901 .92277 .92385 .92493 .73491 .73452 .73412 1.3607 1.3614 1.3622 42 44 46 .69088 .69130 .69172 .95562 .9.673 .95785 .72297 .72256 .72216 1.3832 1.3839 1.3847 48 50 52 .67944 .67987 .68029 .92601 .92709 .92817 .73373 .73333 .73294 1.3629 1.3636 1.3644 48 50 52 .69214 .69256 .69298 .95896 .96008 .96120 .72176 .72136 .72095 1.3855 1.3863 1.3870 54 56 58 .68072 .68114 .68157 .92926 .93034 .93143 .73254 .73215 .73175 1.3651 1.3658 1.3666 54 56 58 .69340 .69382 .69424 .96232 .96344 .96456 .72055 .72015 .71974 1.3878 1.3886 1.3894 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 87 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 44 Deg. 45 Deg. MIN SINK. TANG. COSINE. SEC MIN bINE. TANG. COSINE. SEC. 2 4 .69466 .69507 .69549 .96569 .96681 .96794 .71934 .71893 .71853 1.3902 1.3909 1.3917 2 4 .70711 .70, o2 .70793 1. 1,0012 1.0023 .70711 .70669 .70628 1.4142 1.4150 1.4159 6 8 10 .69591 .69633 .69675 .96906 .97019 .97132 .71812 .71772 .71732 1.3925 1.3933 1.3941 6 8 10 .70834 .70875 .70916 1.0035 1.0047 1.0058 .70587 .70546 .70505 1.4167 1.41.5 1.4183 12 14 16 .69716 .69758 .69800 .97246 .97359 .97472 .71691 .71650 .71610 1.3949 1.3957 1.3964 12 14 16 .70957 .70998 .71039 1.0070 1.0082 1.0093 .70463 •70422 .70381 1.4192 1.4200 1.4208 18 20 22 .69841 .69883 .69925 .97586 .97699 .97813 .71569 .71528 .71488 1.3972 1.3980 1.3988 18 20 22 .71080 .71121 .71162 1.0105 1.0117 1.0129 .70339 .70298 .70257 1.4217 1.4225 1.4233 24 26 28 .69966 .70008 .70049 .97927 .9804 L .98155 .71447 .71406 .71366 1.3996 1.4004 1.4012 24 26 28 .71202 .71243 .71284 1.0141 1.0152 1.0164 .70215 .70174 .70132 1.4242 1.4250 1.4259 30 32 34 .70091 .70132 .70174 .98270 .98384 .98499 .71325 .71284 .71243 1.4020 1.4028 1.4036 30 32 34 .71325 .71366 .71406 1.0176 1.0188 1.0200 .70091 .70049 .70008 1.4267 1.4276 1.4284 36 38 40 .70215 .70257 .70298 .98613 .98728 .98843 .71202 .71162 .71121 1.4044 1.4052 1.4060 36 38 40 .71447 .71488 .71528 1.0212 1.0223 1.0235 .69666 .69925 .69883 1.4292 1.4301 1.4310 42 44 46 .70339 .70381 .70422 .98958 .99073 .99 J 89 .71080 .71039 .70998 1.4069 1.4077 1.4085 42 44 46 .71569 .71610 .71650 1.0247 1.0259 1.0271 .69841 .69800 .69758 1.4318 1.4327 1.43.5 48 50 52 .70463 .70505 .705,16 .99304 .99420 .99535 .70957 .70916 .70875 1.4093 L.4101 1.4109 48 50 52 .71691 .71732 .71772 1.0283 1.0295 1.0307 .69716 .69675 .69633 1.4344 1.4352 1.4361 54 56 58 .70587 .70628 .70669 .99651 .99767 .99884 .70834 .70793 .70752 1.4117 1.4126 1.4134 54 56 58 .71812 .71853 .71893 1.0319 1.0331 1.0343 .69591 .69549 .69507 1.4370 1.4378 1.4387 88 THE MACHINIST AND TOOI, MAKER'S INSTRUCTOR. NATURAL SINKS, TANGENTS, COSINES AND SECANTS. 46 Deg. i 47 DEG. MIN SINE. TANG. COSINE. SEC. MIN SIKE. TANG. COSINE. SEC. 2 4 .71934 .71974 .72015 1.0355 1.0367 1.0379 .69466 .69424 .69382 1.4395 1.4414 1.4413 2 4 .73135 .73175. .73215 . 1.0724 1.0736 1.0748 .68200 .68157 .68115 1.4663 1.4672 1.4681 6 8 10 .72055 .72095 72136 1.0391 1.0403 1.0416 .69310 .69298 .69256 1.4422 1.4430 1.4439 6 8 10 12 14 16 18 20 22 .73254 .73294 .73333 1.0761 1.0774 1.0786 .68072 .68029 .67987 1.4690 1.4699 l.47oy 12 14 16 .72176 .72216 .72256 1.0428 1.0440 1.0452 .69214 .69172 .69130 1.4448 1.4457 1.4465 .73373 .73412 .73462 1.0799 1.0812 1.0824 .67944 .67901 .67859 1.4718 1.4727 1.4736 18 20 22 .72297 .72337 .72377 1.0461 1.0476 1.048y .69088 .69046 .69004 1.4474 1.4483 1.4492 .73491 .73 )31 .73570 1.0837 1.0849 1.0862 .67816 .67773 .67730 1.4745 1.4755 1.4764 24 26 28 .72417 .72457 .72497 1.0501 1.0513 1.0525 .68962 .68920 .68878 1.4501 1.4510 1.4518 24 26 28 .73610 .73649 .73688 1.0875 1.0887 1.0900 .67687 .67645 .67602 1.4774 1.4783 1.4792 30 32 34 .72537 .72577 .72617 1.0538 1.0550 1.0562 .68835 .68793 .68751 1.4527 1.4536 1.4545 30 32 34 .73728 .73767 .73806 1.0913 1.0^26 1.0938 .67559 .67516 .67473 1.4802 1.4811 1.4821 36 38 40 .72657 .72697 .72737 1.0575 1.0587 1.0599 .68709 .68666 .68624 1.4554 1.4563 1.4572 36 38 40 .73845 .73885 .73924 1.0951 1.0964 1.0977 .67430 .67387 .67344 1.4830 1.4839 1.4849 42 44 46 .72777 .72817 .72857 1.0612 1.0624 1.0636 .68582 .68539 .68197 1.4581 1.4590 1.4599 42 44 46 .73963 .74002 .74041 1.0990 1.1003 1.1015 .67301 .67258 .67215 1.4858 1.4868 1.4877 48 50 52 .72897 .72937 .72976 1.0649 1,0661 1.0674 .68455 .68412 .68370 1.4608 1.4617 1.4626 48 50 52 .74080 .74119 .74158 1.1028 1.1041 1.1054 .67172 .67129 .67086 1.4887 1.4897 1.4906 54 56 (>8 .73016 .73056 .73096 1.0686 1.0699 1.0711 .68327 .68285 .68242 1.4635 1.4644 1.4654 54 56 58 .74197 .74236 .74275 1.1067 1.1080 1.1093 .67043 .66999 .66956 1.4916 1.4925 1.4935 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 48 DEG. I 49 DEG. MIS SINE. TANG. COSINE. SEC- MIN SINE. TANG. COSINE. SEC. 2 4 .74314 .74353 .74392 1.11061 1.11191 1.11321 .66913 .66870 .66827 1.4946 1.4954 1.4964 2 4 .75471 .75509 .75547 1.15037 1.15172 1.15308 .65606 .65562 .65518 1.5242 1 5253 1.5263 6 8 10 .74431 .74470 .74509 1.11452 1.11582 1.11713 .66783 .66740 .66697 1.4974 1.4983 1.49^3 6 8 10 .75585 .75623 .75661 1.15443 1J5579 1.15715 .65474 .65430 .65386 1.5273 1.5283 1.5294 12 14 16 .74548 .74586 .74625 1.11844 1.11975 1.12106 .66653 .66610 .66566 1.5003 1.5013 1.5022 12 14 16 .75700 .75738 .75775 1.15851 1.15987 1.16124 .65342 .65298 .65254 1.5304 1.5314 1.5325 18 20 22 .74664 .74703 .74741 1.12238 1.12369 1.12501 .66523 .66480 .66136 1.5032 1.5042 1.5052 18 20 22 .75813 .75851 .75889 1.16261 1.16398 1.16535 .65210 .65166 .65122 1.5335 1.5345 1.5356 24 26 28 .74780 .74818 .74857 1.12633 1.12765 1.12897 .66393 .66349 .66306 1.5062 1.5072 1.5082 24 26 28 .75927 .75965 .76003 1.16672 1.16809 1.16947 .65077 .65033 .64989 1.5366 1.5377 1.5387 30 32 34 .74896 .74934 .74973 1.13029 1.13162 3.13295 .66262 .66218 .66175 1.5092 1.5101 1.5111 30 32 34 .76041 .76078 .76116 1.17085 1.17223 1.17361 .64945 .61901 .64856 1.5398 1.5408 1.5419 36 38 40 .75011 .75050 .75088 1.L3428 1.13561 1.13694 .66131 .66088 .66044 1.5121 1.5131 1.5141 36 38 40 .76154 .76192 .76229 1.17500 1.17638 1.17776 .64812 .64768 .64723 1.5429 1.5440 1.5450 42 44 46 .75126 .75165 .75203 1.13828 1.13961 1.14095 .66000 .65956 .65913 1.5151 1.5161 1.5171 J 42 44 46 .76267 .76304 .76342 1.17916 1.18055 1.18194 .64679 .64635 .64590 1.5461 1.5471 1.5482 48 50 52 .75241 .75280 .75318 1.14229 1.14363 1.14498 .65869 .65825 .65781 1.5182 ! 1.5192 1.5202 ; 48 50 52 .76380 .76417 .76455 1.18334 1.18474 1.18614 .64546 .64501 .64457 1.5493 1.5503 1.5514 54 56 08 .75356 .75395 .75433 1.14632 1.14767 1.14902 .65738 .65694 .65650 1.5212 ! 1.5222 1.5232 54 56 5S .76492 .76530 .76567 1.18754 1.18894 1.19035 .64412 .64368 .64323 1.5525 1.5536 1.5546 90 THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 50 Deg. 51 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .76604 .76642 .76679 1.19175 1.19316 1.19457 .64279 .64234 .64 J 90 1.5557 1.5568 1 .5579 2 4 .77715 .77751 .77788 1.23490 1.23637 1.23784 .62932 .62887 .62842 1.5890 1.5901 1.5913 6 8 10 .76717 .76754 76791 1.19599 1.19740 1.19862 .64145 .64100 .64056 1.5590 1.560U 1.5611 6 8 10 12 14 16 .77824 .77861 .77897 1.23931 1.24079 1.24227 .62796 .62751 .62706 1.5924 1.5936 1.5947 12 14 16 .76828 .76866 .76903 1.20024 1.20166 1.20308 .64011 .63y66 .63922 1.5622 1.5633 1.5644 .77934 .77970 .78007 1.24375 1.24523 1.24672 .62660 .62615 .62570 1.5959 1.5971 1.5982 18 20 22 .76940 .76977 .77014 1.20451 1.20593 1.20736 .63877 .63832 .63787 1.5655 1.5666 1 .5677 18 20 22 .78043 .78079 .78116 1.24820 1.24969 1.25118 .62524 .62479 .62433 1.5994 1.6005 1.6017 24 26 28 .77051 .77088 .77125 1.20879 1.21023 1.21166 .63742 .63698 .63653 1.5688 1.5699 1.5710 24 26 28 .78152 .78188 .78225 1.25268 1.25417 1.25567 .62388 .62342 .62297 1.6029 1.6040 1.6052 30 32 34 .77162 .77199 .77236 1.21310 1.21454 1.21598 .63608 .63563 .63518 1.5721 1.5732 1.5743 30 32 34 .78261 .78296 .78333 1.25717 1.25867 1.26018 .62251 .62206 .62160 1.6064 1.6077 1.6088 36 38 40 .77273 .77310 .77347 1.21742 1.21885 1.22031 .63473 .63428 .63383 1.5755 1.5766 1.5777 36 38 40 .78369 .78405 .78442 1.26169 1.26319 1.26471 .62115 .62069 .6-024 1.6099 1.6111 1.6123 42 44 46 .77384 .77421 .77458 1.22176 1.22321 1.22467 .63338 .63293 .63248 1.5788 1.5799 1.5811 42 44 46 .78478 .78514 .78j50 1.26622 1.26775 1.26925 .61978 .61932 .61887 1.6135 1.6147 1.6159 48 50 52 .77494 .77531 .77568 1.22612 1.22758 1.22904 .63203 .63158 .63113 1.5822 1.5833 1.5845 48 50 52 .78586 .78622 .78658 1.27077 1.27230 1.27382 .61841 .61795 .61749 1.6170 1.6182 1.6194 54 56 58 .77605 .77641 .77677 1.23050 1.23196 1.23343 .63068 .63022 .62977 1.5856 1.5867 1.5879 54 56 58 .78694 .78729 .78765 1.27535 1.27688 1.27841 .61704 .61658 .61612 1.6206 1.6218 1.6231 THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 9L NATURAL SINES, TANGENTS, COSINES AND SECANTS. 52 Deg. 53 Deg. MIN SINK. TANG. COSINE. SEC- MIN SINE. TANG. COSINE. SEC. 2 4 .78801 .78837 .78873 1.27994 1.28148 1.28302 .61566 .61520 .61474 1.6243 1.6255 1.6267 2 4 .79864 .79899 .79934 1.32704 1.32865 1.33026 .60182 .60135 .60089 1.6618 16629 1.6642 6 8 10 .78908 .78944 .78980 1.28456 1.28610 1.28764 .61429 .61383 .61337 1.6279 1.6291 1.63U3 6 8 10 .79:68 .80003 .80038 1.33187 1.3334y 1.33511 .60042 .59994 .59949 1.6655 1.66t>8 1.6(581 12 14 16 .79016 .790ol .79087 1.28919 1.29074 1.29229 .61291 .61245 .61199 1.6316 1.6328 1.0340 12 14 16 .80073 .80108 .80143 1.33673 1.33835 1.33998 .59902 .59856 .59809 1.6694 1.6707 1.6720 18 20 22 .79122 .79158 .79193 1.29385 1.29541 1.29696 .61153 .61107 .61061 1.6352 1.6365 1.6577 18 20 22 .80178 .80212 .80247 1.34160 1.34323 1,34487 .59763 .59716 .59669 1.6733 1.6746 1.6759 24 26 28 .79229 .79264 .79300 1.29853 1.30009 1.30166 .61015 .60968 .60922 1.6389 1.6402 1.6414 24 26 28 .80282 .80316 .80351 1.34650 1.34814 1.34978 .59622 .59576 .59529 1.6772 1.6785 1.6798 30 32 34 .79335 .79371 .79406 1.30323 1.30480 1.30637 .60876 .60830 .60784 1.6427 1.6439 1.6452 30 32 34 .80386 .80420 .80455 1.35142 1.35307 1.35472 .59482 .59436 .59389 1.6812 1.6826 1.683d 36 38 40 .79441 .79477 .79512 1.30795 1.30952 1.31110 .60738 .60691 .60645 1.6464 1.6477 1.6489 36 38 40 .80489 .80524 .80558 1.35637 1.35802 1.35968 .59342 .59295 .59248 1.6S51 1.6865 1.6878 42 44 46 .79547 .79583 .79618 1.31269 1.31427 1.31586 .60599 .60553 .60506 1.6502 1.6514 1.6527 42 44 46 .80593 .80627 .80662 1.36134 1.36300 1.36466 .59201 .59154 .59108 1.6891 1.6905 1.6918 48 50 52 .79653 .79688 .79723 1.31745 1.319U4 1.32064 .60460 .60414 .60367 1.6540 1.6552 1.6565 48 50 52 .80696 .80730 .80765 1.36633 1.36800 1.36967 .59061 .59014 .58967 1.6932 1.6945 1.6959 54 56 58 .79758 .79793 .79829 1.32224 1.32384 1.32544 .60321 .60274 .60228 1.6578 1.6591 1.6603 54 56 58 .80799 .80833 .80867 1.37134 1.37302 1.37470 .58920 .58873 .58826 1.6972 1.6986 1.6999 92 THE MACHINIST AND TOOL, MAKEE's INSTRUCTOR. NATURAL SINKS, TANGENTS, COSINES AND SECANTS. 54 DEG. 55 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .80902 .80936 .80970 1.37638 1.37807 1.37976 .58779 .58731 .58684 1.7013 1.7027 1.7040 2 4 .81915 .81949 .81982 1.42815 1.42992 1.43169 .57358 .57310 .57262 1.7434 1.7449 1.7463 6 8 10 .81004 .81038 .81072 1.38145 1.38314 1.38484 .58637 .58590 .58543 1.7054 1.7068 1.7081 6 8 10 12 14 16 18 20 22 .82015 .82048 .82082 1.43347 1.43525 1.43703 .57215 .57167 .57119 1.7478 1.7493 1.7507 12 14 16 .81106 .81140 .81174 1.38653 1.38824 1.36994 .58496 .58449 .58401 1.7095 1.7109 1.7123 .82115 .82148 .82181 1.43881 1.44060 1.44239 .57071 .57024 .56976 1.7522 1.7537 1.7551 18 20 22 .81208 .81242 .81^76 1.39165 1.39336 1.39507 .58354 .58^07 .58260 1.7137 1.7151 1.7164 .82214 .82248 .82281 1.44418 1.44598 1.44778 .56928 .56880 .56832 1.7566 1.7581 1.7596 24 26 28 .81310 .81344 .81378 1.39679 1.39850 1.40022 .58212 .58165 .58118 1.7178 1.7192 1.7206 24 26 28 .82314 .82347 .82380 1.44958 1.45139 1.45320 .56784 .56736 .56689 1.7610 1.7625 1.7640 30 32 34 .81412 .81445 .81479 1.40195 1.40367 ] .40540 .58070 .58023 .57976 1.7220 1.7234 1.7249 30 32 34 .82413 .82446 .82478 1.45501 1.45682 1.45864 .56641 .56593 .56545 1.7655 1.7670 1.7685 36 38 40 .81513 .81546 .81580 1.40714 1.40887 1.41061 .57928 .57881 .57833 1.7263 1.7277 1.7291 36 38 40 .82511 .82544 .82577 1.46046 1.4622y 1.46411 .56497 .56449 .56401 1.7700 1.7715 1.7730 42 44 46 .81614 .81647 .81681 1.41235 1.41409 1.41584 .57786 .57738 .57691 1.7305 1.7319 1.7334 42 44 46 48 50 52 .82610 .82643 .82675 1.46595 1.46778 1.46962 .56353 .56305 ,56256 1.7745 1.7760 1.7775 48 50 52 .8171.4 .81748 .81782 1.41759 1,41934 1.42110 .57643 .57596 .57548 1.7348 1.7362 1.7377 .82708 .82741 .82773 1.47146 1.47330 1.47514 .56208 .56160 .56112 1.7791 1.7806 1.7821 54 56 58 .81815 .81848 .81882 1.42286 1.42462 1.42638 .57501 .57453 .57405 1.7391 1.7405 1.7420 54 56 58 .82806 .82839 .82871 1.47699 1.47885 1.48070 .56064 .56016 .55968 1.7837 1.7852 1.7867 THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR^ 9S NATURAL SINES, TANGENTS AND SECANTS. , COSINES 56 Deg. 57 Deg. MIK SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .32904 .8293t .82969 1.48256 1.48442 1.48629 .55919 .55871 .55823 1.7883 1 1.7898 1.7914 2 4 .83867 .83899 .83930 1.53986 1,54183 1.54379 .54464 .54415 .54366 1.8361 18377 1.8394 6 8 10 .83001 .83034 .83066 1.48816 1=49003 1.49190 .55775 .55726 .55678 1.7929 1.7945 1.7960 6 8 10 .83962 .83994 .84025 1.54576 1.54774 1.54972 .54317 .54269 .54220 1.8410 1.8427 1.8443 12 14 16 .83098 .83131 c83163 1.49378 1.49566 1.49755 .55630 .55581 .55533 1.7976 1.7992 1.8007 12 14 16 .84057 .84088 .84120 1.55170 1.55368 1.55567 .54171 .54122 .54073 1.8460 1.8477 1.8493 18 20 22 .83195 .832z8 .83260 1.49944 1.50133 1.50322 .55484 .55436 .55388 1.8023 1.8039 1.8054 18 20 22 .84151 .84182 .84214 1.55766 1.55966 1.56165 .54024 .53975 .53926 1.8510 1.8527 1.8544 24 26 28 .83292 .83324 .83356 1.50512 1.50702 1.50893 .55339 .55291 .55242 1.8070 1.8086 1.8102 24 26 28 .84245 .8427 7 .84308 1.56366 1.56566 1.56767 .53877 .53828 .53779 1.8561 1.8578 1.8595 30 32 34 .83389 .83421 .83453 1.51084 1.51275 1.51466 .55194 .55145 .55097 1.8118 1.8134 1.8150 30 32 34 .84339 .84370 .84402 1.56969 1.57170 1.57372 .53730 .53681 .53632 1.8611 1.8629 1.8646 36 38 40 .83485 .83517 .83^49 1.51658 1.51850 1.52043 .55048 .54999 .54951 1.8166 1.8182 1.8198 36 38 40 .84433 .84464 .84495 1.57575 1.57778 1.57981 .53583 .53534 .53484 1.8663 1.8680 1.8697 42 44 40 .83581 .83633 .83645 1.52235 1.52429 1.52622 .54902 .54854 .54805 1.8214 ' 1.8230 1 1.8246 42 44 46 .84526 .84557 .8M5SS 1,58184 1.58388 1.58593 .53435 .53386 .53337 1.8714 1.8731 1.8749 48 50 52 .83676 .83708 .83740 1.52816 1.53010 1.53205 .54756 .54708 .54659 1.8263 1.8279 1.8295 48 50 52 .84619 .84650 .84681 1.58797 1.59002 1.59208 .53288 .53238 .53189 1.8766 1.8783 1.6801 54 56 58 .83772 .83804 .83835 1.53400 1.53595 1.53791 .54610 .54561 .54513 1.8311 1.8328 1.8344 54 56 58 .84712 .84743 .84774 1.59414 1.59620 1.59826 .53140 .53091 =53041 1.8818 1.8836 1.8853 94 THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 58 Deg. 59 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .84805 .84836 .84866 1.60033 1.60241 1.60449 .52992 .52943 .52893 1.8871 1.8888 1.8906 2 4 .85717 .85747 .85777 1.66428 1.66647 1.66867 .51504 .51454 .51404 1.9416 1.9435 1.9454 6 8 10 .84897 .84928 .84959 1.60657 1.60865 1.61074 .52844 .52794 .52745 1.8924 1.8941 1.8959 6 8 10 .85806 .85836 .85866 1.67088 1.67309 1.67530 .51354 .51304 .51254 1.9473 1.9491 1.9510 12 14 16 .84989 .8o020 .850ol 1.61283 1.61493 1.61703 .52696 .52646 .52597 1.8977 1.8995 1.9013 12 14 16 .85896 .85926 .85956 1.67752 1.67974 1.68196 .51204 .51154 .51104 1.9530 1.9549 1.9568 18 20 22 .85081 .85112 .85142 1.61914 1.62125 1.62336 .52547 .52498 .52448 1.9030 1.9048 1.9066 18 20 22 .85985 .86015 .86045 1.68419 1.68643 1.68866 .51054 .51004 .50954 1.9587 1.9606 1.9625 24 26 28 .85173 .85203 .85254 1.62548 1.62760 1.62972 .52399 .52349 .52299 1.9084 1.9102 1.9121 24 26 28 .86074 .86104 .86133 1.69091 1.6W316 1.69541 .50904 .50854 .50804 1.9645 1.9664 1.9683 30 32 34 .85264 .85294 .85325 1.63185 1.63398 1.63612 .52250 .52200 .52151 1.9139 1.9157 1.9175 30 32 34 .86163 .86192 .86222 1.69766 1.69992 1.70219 .50754 .50704 .50654 1.9703 1.9722 1.9742 36 38 40 .85355 .85385 .85416 1.63826 1.64041 1.64256 .52101 .52051 .52002 1.9193 1.9212 1.9230 36 38 40 .86251 .86281 .86310 1.70446 1.70673 1.70901 .50603 .50553 .50503 1.9761 1.9781 1.9801 42 44 46 .85446 .8S476 .85506 1.64471 1.64687 1.64903 .51952 .51902 .51852 1.9248 1.9267 1.9285 42 44 46 .86340 .86369 .86398 1.71129 1.71358 1.71588 .50453 .50403 .50352 1.9820 1.9840 1.9860 48 50 52 .85536 .85567 .85597 1.65120 1.65337 1.65554 .51803 .51753 .51703 1.9304 1.9322 1.9341 48 50 52 .86427 .86457 .86186 1.71817 1.72047 1.72278 .50302 .50252 .50201 1.9880 1.9900 1.9920 54 56 58 .85627 .85657 .85487 1.65772 1.65990 1.66209 .51653 .51604 .51554 1.9360 1.9378 1.9397 54 56 58 ,86515 .86544 .86573 1.72509 1.72741 1.72973 .50151 .50101 .50050 1.9940 1.9960 1.9980 THE MACHINIST AND TOOL MAKERS INSTRUCTOR. NATURAL, SINES, TANGENTS, COSINES AND SECANTS. 60 Deg. 61 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .86603 .86632 .86661 1.73205 1.73438 1.73671 .5 .49950 .49899 2. 2.002 2.004 2 4 .87462 .87490 .87518 1.80405 1,80653 1.80901 .48481 .48430 .48379 2.063 2 065 2.067 6 8 10 .86690 .86719 .86748 1.73905 1.74140 1.74375 .49849 .49798 .49748 2.006 2.008 2.010 6 8 10 .87546 .87575 .87603 1.81150 1.81399 1.81649 .48328 .48277 .48226 2.069 2.071 2.073 12 14 16 .86777 .86805 .86834 1.74610 1.74846 1.75082 .49697 .49647 .49596 2.012 2.014 2.016 12 14 16 .87631 .87659 .87b87 1.81899 1.82150 1.82402 .48175 .48124 .48073 2.076 2.078 2.080 18 20 22 .86863 .86892 .86921 1.75319 1.75556 1.75794 .49546 .49495 .49445 2.018 2.020 2.022 18 20 22 .87715 .87743 .87770 1.82654 1.82906 L 83159 .48022 .47971 .47920 2.082 2.085 2.087 24 26 28 .86949 .86978 .87007 1.76032 1.76271 1.76510 .49394 .49344 .49293 2.024 2.026 2.029 24 26 28 .87798 .87826 .87854 1.83413 1.83667 1.83922 .47869 .47818 .47767 2.089 2.091 2.093 30 32 34 .87036 .87064 .87093 1.76749 1.76990 1.77230 .49242 .49192 .49141 2.031 2.033 2.035 30 32 34 .87882 .87909 .87937 1.84177 1.84433 1.84689 .47716 .47665 .47614 2.096 2.098 2.100 36 38 40 .87121 .87150 .87178 1.77471 1.77713 1.77955 .49090 .49040 .48999 2.037 2.039 2.041 36 38 40 .87965 .87993 .88020 1.84946 1.85204 1.85462 .47562 .47511 .47460 2.102 2.105 2.107 42 44 46 .87207 .87235 .87264 1.78198 1.78441 1.78685 .48938 .48888 .48837 2.043 2.045 2.047 42 44 46 .88048 .88075 .88103 1.85720 1.85979 1.86239 .47409 .47358 .47306 2.109 2.112 2.114 48 50 52 .87292 .87321 .87349 1.78929 1.79174 1.79419 .48786 .48735 .48684 2.050 2.052 2.054 48 50 52 .88130 .88158 .88185 1.86499 1.86760 1.87021 .47255 .47204 .47153 2.116 2.1)8 2,121 54 56 58 .87377 .87406 .87434 1.79665 1.79911 1.80158 .48634 .48583 .48532 2.056 2.058 2.060 54 56 58 .88213 .88240 .88267 1.87283 1.87546 1.87809 .47101 .47050 .46999 2.123 2.125 2.128 96 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 62 Deg. 63 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .88295 .88322 .88349 1.88073 1.88337 1.88602 .46947 .46896 .46844 2.130 2.132 2.135 2 4 .89101 .89127 .89153 1.96261 1.96544 1.96827 .45399 .45347 .45^95 2.203 2.205 2.208 6 8 10 .88377 88404 .88431 1.88867 1.89133 1.89400 .46793 .46742 .46690 2.137 2.139 2.142 6 8 10 .89180 .89206 .89232 1.97111 1.97395 1.97681 .45243 .45192 .45140 2.210 2.213 2.215 12 14 16 .88458 .88485 .88512 1.89667 1.89935 1.90203 .46639 .46587 .46536 2.144 2.146 2.149 12 14 16 .89259 .89285 .89311 1.97966 1.98253 1.98540 .45088 .45036 .44984 2.218 2.220 2.223 18 20 22 .88539 .88566 .88593 1.90472 1.90741 1.91012 .46484 .46433 .46381 2.151 2.154 2.156 18 20 22 .89337 .89363 .89389 1.98828 1.99116 1.99406 .44932 .44880 .44828 2.226 2.228 2.231 24 26 28 .88620 .88647 .88674 1.91282 1.91554 1.91826 .46330 .46278 .46226 2.158 2.161 2.163 24 26 28 .89415 .89441 .89467 1.99695 1.99986 2.00277 .44776 .44724 .44672 2.233 2.236 2.238 30 32 34 .88701 .88728 .88755 1.92098 1.92371 1.92645 .46175 .46123 .46072 2.166 2.168 2.170 30 32 34 .89493 .89519 .89545 2.00569 2.00862 2.01155 .44620 .44568 .44516 2.241 2.244 2.246 36 38 40 .88782 .88808 .88835 1.92920 1.93195 1.93470 .46020 .45968 .45917 2.173 2.175 2.178 36 38 40 .89571 .89597 .89623 2.01449 2.01743 2.02039 .44464 .44411 .44359 2.249 2.252 2.254 42 44 46 .88862 .88888 .88915 1.93746 1.94023 1.94301 .45865 .45813 .45762 2.180 2.183 2.185 42 44 46 .89649 .89674 .89700 2.02335 2.02631 2.02929 .44307 .44255 .44203 2.257 2.260 2.262 48 50 52 .88942 .88968 .88995 1.94579 1.94858 1.95137 .45710 .45658 .45606 2.188 2.190 2.193 48 50 52 .89726 .89752 .89777 2.03227 2.03526 2.03825 .44151 .44098 .44046 2.265 2.268 2.27$ 54 56 58 .89021 .89048 .89074 1.95417 1.95698 1.95979 .45554 .45503 .45451 2.195 2.198 2.200 54 56 58 .89803 .89828 .89854 2.01125 2.04426 2.04728 .43994 .43942 .43889 2.273 2.2,6 2.2^8 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 97 NATURAL, SINES, TANGENTS, COSINES AND SECANTS. 64 DEG. 65 Deg. MIX SINE. TAXG- COSINE. SEC- MIX SINE. TANG. COSINE. SEC 2 4 .89879 .89905 .89930 2.0503 2.0533 2.0564 .43837 .43785 .43733 2.281 2.284 2.287 2 4 .90631 .90655 .90680 .21445 .21478 .21510 .42262 .42209 .42156 2.366 2.369 2.372 6 8 10 .89956 .899*1 .90007 2.0594 2.0625 2.0655 .43680 .43628 .43575 2 289 2.292 2.295 6 8 10 .90704 .90729 .90753 .21543 .21576 .21609 .42104 .42051 .41998 2.375 2.378 2.381 12 14 16 .90032 .90057 .90082 2.0686 2.0717 2.0747 .43523 .43471 .43418 2.297 2.300 2.303 12 14 16 .90778 .90802 .90826 .21642 .21675 .21708 .41945 .41892 .41840 2.384 2.387 2.390 18 20 22 .90108 .90133 .90158 2.0778 2.0809 2.0840 .43366 .43313 .43261 2.306 2.309 : 2.311 18 20 22 .90851 .90875 .90899 .21742 .21775 .21808 .41787 .41734 .41681 2.393 2.396 2.399 24 26 28 .90183 .90208 .90233 2.0872 2.0903 2.0934 .43209 .43156 .43104 2.314 2.317 2.320 24 26 28 .90924 .90948 .90972 .21842 .21875 .21909 .41628 .41575 .41522 2.402 2.405 2.408 30 32 34 .90259 .90281 .90309 2.0965 2.0997 2.1028 .43051 .42999 .42946 2.323 2.326 2.328 30 32 34 .90996 .91020 .91044 .21943 .21977 .22011 .41469 .41416 .41363 2.411 2.414 2.417 36 38 40 .90334 .90358 .90383 2.1060 2.1092 2.1123 .42894 .42841 .42788 2.331 2.334 2.337 36 38 40 .91068 .91092 .91116 .22045 .22079 .22113 .418x0 .41257 .41204 2.421 2.424 2.427 42 44 46 .90408 .90433 .90458 2.1155 2.1187 2.1219 .42736 .42683 .42631 2.340 2.343 2 346 42 44 46 .91140 .91164 .91188 .22147 .22182 .22216 .41151 .41098 .41045 2.430 2.433 2.436 48 50 52 .90483 .90507 .90532 2.1251 2.1283 2.1315 .42578 .42525 .42473 2.349 2.351 2.a54 48 50 52 .91212 .91236 .91260 .22251 .22286 .22320 .40992 .40939 .40886 2.439 2.443 2.446 54 56 58 .90557 .90582 .90606 2.1348 2.1380 2.1412 .42420 .42367 .42315 2.357 2.360 2.363 54 56 58 .91283 .91307 .91331 .22355 .22390 .22425 .40833 .40780 .40727 2.449 2.452 2.455 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 68 Deg. 67 DEG. MIN SINE. TANG. COSINE. SEC. MIH SINE. TANG. COSINE. SEC. 2 4 ,91355 .91378 .91402 2.2460 2.2495 2.2531 .40674 .40621 .40567 2.458 2.462 2.465 2 4 .92050 .92073 .92096 2.3558 2.3597 2.3635 .39073 .39020 .38966 2.559 2.563 2.566 6 8 10 .91425 .91449 .91472 2.2566 2.2602 2.2637 .40514 .40461 .40408 2.468 2.471 2.475 6 8 10 .92119 .92141 .92164 2.3673 2.3712 2.3750 .38912 .38859 .38805 2.570 2.573 2.577 12 14 16 .91496 .91519 ,91543 2.2673 2.2709 2.2745 .40355 .40301 .40248 2.478 2.481 2.485 12 14 16 .92186 .92209 .92231 2.3789 2.3828 2.3867 .38752 .38698 .38644 2.580 2.584 2.588 18 20 22 .91566 .91590 .91613 2.2781 2.2816 2.2853 .40195 .40141 .40088 2.488 2.491 2.494 18 20 22 .92254 .92276 .92299 2.3906 2.3945 2.3984 .38591 .38537 .38483 2.591 2.595 2.598 24 26 28 .91636 .91660 .91683 2.2889 2.2925 2.2962 .40035 .39982 .39928 2.498 2.501 2.504 24 26 28 .92321 .92343 .92366 2.4023 2.4063 2.4102 .38430 .38376 .38322 2.602 2.606 2.609 30 32 34 .91706 .91729 .91752 2.2998 2.3035 2.3072 .39875 .39822 .39768 2.508 2.511 2.515 30 32 34 .92388 .92410 .92432 2.4142 2.4182 2.4222 .38268 .38215 .38161 2.613 2.617 2.620 36 38 40 .91775 .91799 .91822 2.3108 2.3145 2.3183 .39715 .39661 .39608 2.518 2.521 2.525 36 38 40 .92455 .92477 .92499 2.4262 2.4302 2.4.342 .38107 .38053 .37^99 2.624 2.628 2.632 42 44 46 .91845 .91868 .91891 2.3219 2.3257 2.3294 .39555 .39501 .39448 2.528 2.532 2.535 42 44 46 .92521 .92543 .92565 2.4382 2.4423 2.4464 .37946 .37892 .37838 2.635 2.639 2.643 48 50 52 .91914 .91936 .91959 2.3332 2.3369 2.3407 .39394 .39341 .39287 2.538 2.542 2.545 48 50 52 .92587 .92609 .92631 2.4504 2.4545 2.4586 .37784 .37730 .37676 2.647 2.650 2.654 54 56 58 .91982 .92005 .92028 2.3445 2.3482 2.3520 .39234 .39180 .39127 2.549 2.552 2.556 54 56 58 .92653 .92675 .92697 2.4627 2.4668 2.4709 .37622 .37569 .37515 2.658 2.662 2.666 THE MACHINIST AND TOOD MAKER'S INSTRUCTOR. 99 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 68 Dec. 69 Deg. MIS SINK. TANG. COSINE. SEC MIN SINE. TANG. COSINE . SBC. 2 4 .92718 .92740 .92762 2.4751 2.4792 2.4834 .37461 .37407 .37353 2.669 2.673 2.677 2 4 .93358 .93379 .93400 2.6051 2.6096 2.6142 .35837 .35782 .35728 2.790 2 795 2.799 6 8 10 .92784 .92805 .92827 2.4876 2.4918 2.4960 .37299 .37245 .37191 2.681 2.685 2.689 6 8 10 .93420 .93441 .93462 2.6187 2.6233 2.6279 .35674 .35619 .35565 2.803 2.807 2.812 12 14 16 .92849 .92870 .92892 2.5002 2.5044 2.5086 .37137 .37083 .37029 2.693 2.697 2.700 12 14 16 .93483 .93503 .93524 2.6325 2.6371 2.6418 .35511 .35456 .35402 2.816 2.820 2.825 18 20 22 .92913 .92935 .92956 2.5129 2.5171 2.5214 .36975 .36921 .36867 2.704 2.708 2.712 18 20 22 .93544 .93565 .93585 2.6464 2.6511 2 6557 .35347 .35293 .35239 2.829 2.833 2.838 24 26 28 .92978 .92999 .93020 2.5257 2.5300 2.5343 .36812 .36758 .36704 2.716 2.720 2.724 24 26 28 .93606 .93626 .93647 2.6604 2.6651 2.6699 .35184 .35130 .35075 2.842 2.847 2.851 30 32 34 .93042 .93063 .93084 2.5386 2.5430 2.5473 .36650 .36596 .36542 2.728 2.732 2.737 30 32 34 .93667 .93688 .93708 2.6746 2.6794 2.6841 .35021 .34966 .34912 2.855 2.860 2,864 36 38 40 .93106 .93127 .93148 2.5517 2.5561 2.5604 .36488 .36434 .36379 2.741 2.745 2.749 36 38 40 .93728 .93748 .93769 2.6889 2.6937 2.6985 .34857 .34803 .34748 2.869 2.873 2.878 42 44 46 .93169 .93190 .93211 2.5649 2.5693 2.5737 .36325 .36271 .36217 2.753 2.757 2.761 42 44 46 .93789 .93809 .93829 2.7033 2.7082 2.7130 .34694 .34639 .34584 2.882 2.887 2.891 48 50 52 .93232 .93253 .93274 2.5781 2.5826 2.5871 .36162 .36108 .36054 2.765 2.769 2.773 48 50 52 .93849 .93869 .93889 2.7179 2.7228 2.7277 .34530 .34475 .34421 2.896 2.900 2.905 54 56 58 .93295 .93316 .93337 2.5915 2.5960 2.6005 .36000 .35945 .35891 2.778 2.782 2.786 54 56 58 .93909 .93929 .93949 2.7326 2.7375 2.7425 3.4366 3.4311 3.4257 2.910 2.914 2.919 100 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 70 DEG. 71 DEG. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .93969 .93989 .94009 2.7475 2.7524 2.7575 .34202 .34147 .34093 2.924 2.928 2.933 2 4 .94552 .94571 .94590 2.9042 2.9097 2.9152 .32557 .32502 .32447 3.071 3.077 3.082 6 8 10 M029- .94049 .94068 2.7625 2.7675 2.7725 .34038 .33983 .33929 2.938 2.943 2.947 6 8 10 .94609 .94627 .94646 2.9207 2.9263 2.9319 .32392 .32337 .32282 3.087 3.092 3.098 12 14 16 .94088 .94108 .94127 2.7776 2.7827 2.7878 .33874 .33819 .33764 2.952 2.957 2.962 12 14 16 .94665 .94684 .94702 2.9375 2 9431 2.9487 .32227 .32171 .32116 3.103 3 108 3.1U 18 20 22 .94147 .94167 .94186 2.7929 2.7980 2.8032 33710 .33655 .33600 2.966 2.971 2.976 18 20 22 .94721 .94740 .94758 2.9544 2.9600 2.9657 .32061 .32006 .31951 3.119 3.124 3.130 24 26 28 .94206 .94225 .94245 2.8083 2.8135 2.8187 .33545 .33490 .33436 2.981 2.986 2.991 24 26 28 .94777 .94795 .94814 2.9714 2.9772 2.9829 .31896 .31841 .31786 3.135 3.141 3.146 30 32 34 .94264 .94284 .94303 2.8239 2.8291 2.8344 .33381 .33326 .33271 2.996 3. 3.005 30 32 34 .94832 .94851 .94869 2.9887 2.9945 3.0003 .31730 .31675 .31620 3.151 3.157 3.162 36 38 40 .94322 .94342 .94361 2.8396 2.8449 2.8502 .33216 .33161 .33106 3 011 3.016 3.021 36 38 40 .94888 .94906 .94924 3.0061 3.0119 3.0178 .31565 .31510 .31454 3.168 3.174 3.179 42 44 46 .94380 .94399 .94418 2.8555 2.8609 2.8662 .33051 .32997 .32942 3.026 3.031 3.036 42 44 46 .94943 .94961 .94979 3.0237 3.0296 3.0356 .31399 .31344 .31289 3.185 3.190 3.196 48 50 52 .94438 .94457 .94476 2.8716 2.8770 2.8824 .32887 .32832 .32777 3.041 3.046 3.051 48 50 52 .94997 .95015 .95033 3.0415 3.0475 3.0535 .31233 .31178 .31123 3.202 3.207 3.213 54 56 58 .94495 .94514 .94533 2.8878 2.8933 2.8987 .32722 .32667 .32612 3.056 3.061 3.066 54 56 58 .95052 .95070 .95088 3.0595 3.0655 3.0716 .31068 .31012 .30957 3.219 3.224 3.230 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 101 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 72 Deg. 73 Deg. MIN SINE. TANG- COSINE. SEC. MIN SINE. TANG. COSINE. SEC 2 4 .95106 .95124 .95142 3.0777 3.0838 3.0899 .30902 .30846 .30791 3.236 3.242 3.248 2 4 .95630 .95647 .95664 .32708 .32777 .32845 .29237 .29182 .29126 3.420 3.427 3.433 6 8 10 .95159 .95177 .95195 3.0961 3.1022 3.1084 .30736 .30680 .30625 3.253 3.259 3.265 6 8 10 .95681 .95698 .95715 .32914 .32983 .33052 .29070 .29015 .28959 3.440 3.446 3.453 12 14 16 .95213 .9523 L .95248 3.1146 3.1209 3.1271 .30570 .30514 .30459 3.271 3.277 3.283 12 14 16 .95732 .95749 .95766 .33122 .33191 .33261 .28903 .28847 .28792 3.460 3.466 3.473 18 20 22 .95266 .95284 .95301 3.1334 3.1397 3.1460 .30403 .30348 .30292 3.289 3.295 3.301 18 20 22 .95782 .95799 .95816 .33332 .33402 .33473 .28736 .28680 .28625 3.480 3.487 3.493 24 26 28 .95319 .95337 .95354 3.1524 3.1588 3.1652 .30237 .30182 .30126 3.307 3.313 3.319 24 26 28 .95832 .95849 .95865 .33544 .33616 .33687 .28569 .28513 .28457 3.500 3.507 3.514 30 32 34 .95372 .95389 .95407 3.1716 3.1780 3.1845 .30071 .30015 .29960 3.325 3.332 3.338 30 32 34 .95882 .95898 .95915 .33759 .33832 .33904 .28402 .28346 .28290 3.521 3.528 3.535 36 38 40 .95424 .95441 .9M59 3.1910 3.1975 3.2041 .29904 .29849 .29793 3.344 3.350 3.356 36 38 40 .95931 .95948 .95964 .33977 .34050 .34124 .28234 .28178 .28123 3.542 3.549 3.556 42 44 46 .95476 .95493 .95511 3.2106 3.2172 3.2238 .29737 .296*2 .29626 3.363 3.369 3 375 42 44 46 .95981 .95997 .96013 .34197 .34271 .34346 .28067 .28011 .27955 3.563 3.570 3.577 48 50 52 .95528 .95545 .95562 3.2305 3.2371 3.2438 .29571 .29515 .29460 3.382 3.388 3.394 48 50 52 .96029 .96046 .96062 .34420 .34495 .34570 .27899 .27843 .27787 3.584 3.591 3.599 54 56 58 .95579 .95596 .95613 3.2505 3.2573 3.2641 .29404 .29348 .29293 3.401 3.407 3.414 54 56 58 .96078 .96094 .96110 .34646 .3^722 .34798 .27731 .27676 .27620 3.606 3.613 3.621 102 TKE MACHINIST AND TOOIi MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 74 DEG. 75 Deg. MIN SINE. TANG. COSINE SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .96126 .96142 .96158 3.4874 8.4951 3.5028 .27564 .27508 .27452 3.628 3.635 3.643 2 4 .96593 .96608 .96623 3.7320 3.7407 3.7495 .25882 .25826 .25769 3.864 3.872 3.880 6 8 10 .96174 .96190 .96206 3.5105 3.5183 3.5261 .27396 .27340 .27284 3.650 3.658 3.665 6 8 10 .96638 .96653 .96667 3.7583 3.7671 3.7759 .25713 .25657 .25601 3.899 3.898 3.906 12 14 16 .96222 .96238 ,96253 3.5339 3.5418 3.5497 .27228 .27172 .27116 3.673 3.680 3.688 12 14 16 .96682 .96697 .96712 3.7848 3.7938 3.8028 .25545 .25488 .25432 3.915 3.923 3.932 18 20 22 .96269 .96285 .96301 3.5576 3.5656 3.5736 .27060 .27004 .26948 3.695 3.703 3.711 18 20 22 .96727 .96742 .96756 3.8118 3.8208 3.8299 .25376 .25320 .25263 3.941 3.949 3.958 24 26 28 .96316 .96332 .96347 3.5816 3.5897 3.5977 .26892 .26836 .26780 3.719 3.726 3.734 24 26 28 .96771 .96786 .96800 3.8391 3.8482 3.8574 .25207 .25151 .25094 8.967 3.976 3.985 30 32 34 .96363 .96379 .96394 3.6059 3.6140 3.6222 .26724 .26668 .26612 3.742 3.750 3.758 30 32 34 36 38 40 .96815 .96829 .96844 3.8667 3.8760 3.8854 .25038 .24982 .24925 3.994 4.003 4.012 36 38 40 .96410 .96425 .96440 3.6305 3.6387 3.6470 .26556 .26500 .26443 3.766 3.774 3.782 .96858 .96873 .96887 8.8947 3.9042 3.9136 .24869 .24813 .24756 4.021 4.030 4.039 42 44 46 .96456 .96471 .96486 3.6554 3.6638 3.6722 .26387 .26331 .26275 3.790 3.798 3.806 42 44 46 .96902 .96916 .96930 3.9232 3.9327 3.9423 .24700 .24644 .24587 4.049 4.058 4.067 48 50 52 .96502 .96517 .96532 3.6806 3.6891 3.6976 .26219 .26163 .26107 3.814 3.822 3.830 48 50 52 .96945 .96959 .96973 3.9520 3.9616 3.9714 .24531 .24474 .24418 4.076 4.086 4.095 54 56 58 .96547 .96562 .96578 3.7062 3.7148 3.7234 .26050 .25994 .25938 3.839 3.847 3.855 54 56 58 .96987 .97001 .97015 3.9812 3.9910 4.0009 .24362 .24305 .24249 4.105 4.114 4.124 THE MACHINIST AND TOOIi MAKER y S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 103 76 Deg. 77 Deg. MIN SINK. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .97030 .97044 .97058 4.0108 4.0207 4.0307 .24192 .24136 .24079 4.134 4.143 4.153 2 4 .97437 .97450 .97463 4.3315 4.3430 4.3546 .22495 .22438 .22382 4.445 4 457 4.468 6 8 10 .97072 .97086 .97100 4.0408 4.0509 4.0611 .24023 .23966 .23910 4.163 4.172 4.182 6 8 10 .97476 .97489 .97502 4.3662 4.3779 4.3897 .22325 .22268 .22212 4.479 4.491 4.502 12 14 16 .97113 .97127 .97141 4.0713 4.08.15 4.0918 .23853 .23797 .23740 4.192 4.202 4.212 12 14 16 .97515 .97528 .97541 4.4015 4.4134 4.4253 .22155 .22098 .22041 4.514 4.525 4.537 18 20 22 .97155 .97169 .97182 4.1022 4.1126 4.1230 .23684 .23627 .23571 4.222 4.232 4.242 18 20 22 .97553 .97566 .97579 4.4373 4.4494 4.4615 .21985 .21928 .21871 4.549 4.560 4.572 24 26 28 .97196 .97210 .97223 4.1335 4.1440 4.1546 .23514 .23458 .23401 4.253 4.263 4.273 24 26 28 .97592 .97604 .97617 4.4737 4.4860 4.4983 .21814 .21758 .21701 4.584 4.596 4.608 30 32 34 .97237 .97251 .97264 4.1653 4.1760 4.1867 .23345 .23288 .23231 4.284 4.294 4.304 30 32 34 .97630 .97642 .97655 4.5107 4.5232 4.5357 .21644 .21587 .21530 4.620 4.632 4.645 36 38 40 .97278 .97291 .97304 4.1976 4.2084 4.2193 .23175 .23118 .23062 4.315 4.326 4.336 36 38 40 .97667 .97680 .97692 4.5483 4.5609 4.5736 .21474 .21417 .21360 4.657 4.669 4.682 42 44 46 .97318 .97331 .97345 4.2303 4.2413 4.2524 .23005 .22948 .22892 4.347 4.358 4.368 42 44 46 .97705 .97717 .97729 4.5864 4.5993 4.6122 .21303 .21246 .21189 4.694 4.707 4.719 48 50 52 .97358 .97371 .97384 4.2635 4.2747 4.2859 .22835 .22778 .22722 4.379 4.390 4.40 L 48 50 52 .97742 .97754 .97766 4.6252 4.6382 4.6514 .21132 .21076 .21019 4.732 4.745 4.758 54 56 6b .97398 .97411 .97424 4.2972 4.3086 4.3200 .22665 .22608 .22552 4.412 4.423 4.434 54 56 58 .97778 .97791 .97803 4.6646 4.6779 4.6912 .20962 .20905 .20848 4.771 4.783 4.797 104 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 78 Deg. 79 Deg. WIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .97815 .97827 .97839 4.7046 4.7181 4.7317 .20791 .20734 .20677 4.810 4.823 4.836 2 4 .98163 .98174 .98185 5.1445 5.1606 5.1767 .19081 .19024 .19067 5.241 5.257 5.272 6 S3 1J .97851 97863 .97875 4.7453 4.7591 4.7729 .20620 .20563 .20506 4.850 4.863 4.876 6 8 10 .96196 .98207 .98218 5.1929 5.2092 5.2257 .18909 .18852 .18795 5.288 5.304 5.320 12 14 16 .97887 .97899 .97910 4.7867 4.8007 4.8147 .20449 .20393 .20336 4.890 4.904 4.917 12 14 16 .98229 .98240 .98250 5.2422 5.2588 5.2755 .18738 .18681 .18624 5.337 5.353 5.369 38 20 L2 .97922 .97934 .97916 4.8288 4.8430 4.8573 .20279 .20222 .20165 4.931 4.945 4.959 18 20 22 .98261 .98272 .98283 5.2923 5.3093 5.3263 .18567 .18509 .18452 5.386 5.403 5.419 24 26 !28 .97958 .97969 .97981 4.8716 4.8860 4.9006 .20108 .20051 .19994 4.973 4.987 5.001 24 26 28 .98294 .98304 .98315 5.3434 5.3607 5.3780 .18395 .18338 .18281 5.436 5.453 5.470 30 32 34 .97992 .98004 .98016 4.9152 4.9298 4.9446 .19937 .19880 .19823 5.016 5.030 5.045 30 32 34 .98325 .98336 .98347 5.3955 5.4131 5.4308 .18223 .18166 .18109 5.487 5.505 5.522 36 38 40 .98027 .98039 .98050 4.9594 4.9744 4.9894 .19766 .19709 .19652 5.059 5.074 5.089 36 38 40 .98357 .98368 .98378 5.4486 5.4665 5.4845 .18052 .17994 17937 5.540 5.557 5.575 42 44 46 .98061 .98073 .98084 5.0045 5.0197 5.0350 .19594 .19537 .19480 5.103 5.118 5.133 42 44 46 .98389 .98399 .98409 5.5026 5.5209 5.5393 .17880 .17823 .17766 5.593 5.611 5.629 49 6. J 52 .98096 .98107 .98118 5.0504 5.0658 5.0814 .19423 .19366 .19309 5.148 5.164 5.179 48 50 52 .98420 .98430 .98440 5.5578 5.5764 5.5951 .17708 .17651 .17594 5.647 5.665 5,684 51 56 58 .98129 .98140 .98152 5.0970 5.1128 5.1286 .19252 .19195 .19138 5.194 5.210 5.225 54 56 58 .98450 .98461 .98471 5.6139 5.6329 5.6520 .17536 .17479 .17422 5.702 5.721 5,740 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 105 NATURAL SINES, TANGENTS, COSINES AND SECANTS. 80 Deg. 81 Deg. MIN SINE. TANG. COSINE. SEC- MIN SINE. TANG. COSINE. SEC. 2 4 .98481 .98491 .98501 5.6713 5.6906 5.7101 .17365 .17307 .17250 5.759 5.778 5.797 2 4 .98769 .98778 .98787 .63137 .63376 .63616 .15643 .15586 .15528 6.392 6.416 6.440 6 8 10 .9*511 .98521 .98531 5.7297 5.7495 5.7694 .17193 .17135 .17078 5.816 5.836 5.855 6 8 10 .98796 .98805 .98814 .63859 .64103 .64348 .15471 .15+13 .15356 6.464 6.488 6.512 12 14 16 ■9854 L .985^1 .98560 5.7894 5.8095 5.8298 .17021 .16963 .16906 5.875 5.895 5.915 12 14 16 .98823 .9*832 .98840 .64596 .64846 .65097 .15298 .15241 .15183 6.536 6.561 6.586 18 20 22 .98570 .98580 .98590 5.8502 5.8708 5.8915 .16849 .16791 .16734 5.935 5.955 5.976 18 20 22 .98849 .98858 .98867 .65350 .65605 .65863 .15126 .15068 .15011 6.611 6.636 6.662 24 26 28 .98599 .98609 .98619 5.91-24 5.9333 5.9545 .16677 .16619 .16562 5.996 6.017 6.038 24 26 28 .98875 .98884 .98893 .66122 .66383 .66646 .14953 .14896 .14838 6.687 6.713 6.739 30 32 34 .98628 .98638 .98647 5.9758 5.9972 6X188 .16504 .16447 .16390 6.059 6.080 6.101 30 32 34 .98901 .98910 .98918 .66912 .67179 .67449 .14781 .14723 .14666 6.765 6.792 6.818- 36 38 40 .98657 .98666 .98676 6.0405 6.0624 6.0844 .16332 .16275 .16218 6.123 6.144 6.166 36 38 40 .98927 .98935 .98944 .67720 .67994 .68269 .14608 .14551 .14493 6.845 6.872 - 6.900 42 44 46 .98685 .98695 .98704 6.1066 6.1290 6.1515 .16160 .16103 .16045 6.188 6.210 6 232 42 44 46 .98952 .98961 .98969 .68547 .68828 .69110 .14435 .14378 .14320 6.927 6.955 6.983 48 50 52 .98713 .98723 .98732 6.1742 6.1970 6.2200 .15988 .15930 .15873 6.255 6.277 6.300 48 50 52 .98977 .98986 .98994 .69395 .69682 .69972 .14263 .14205 .14148 7.011 7.040 7.068 54 56 58 .98741 .98750 .98759 6.2432 6.2665 6.2901 .15816 .15758 .15700 6.323 6.346 6.369 54 56 58 .99002 .99010 .99018 .70264 .70558 .70855 .14090 .14032 .13975 7.097 7.126 7.156 106 THE MACHINIST AND TOOL MAKERS' INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 82 DEG. 83 Deg. MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .99027 .99035 .99043 7.1154 7.1455 7.1759 .13917 .13859 .13802 7.185 7.215 7.245 2 4 .99255 .99262 .99269 8.1443 8.1837 8.2234 .12187 .12129 .12071 8.205 8.245 8.284 6 8 10 .99051 .99059 .99067 7.2066 7.2375 7.2687 .13744 .13687 .13629 7.276 7.306 7.337 6 8 10 .99276 .99283 .99289 8.2635 8.3041 8.3450 .12014 .11956 .11898 8.324 8.364 8.405 12 14 16 .99075 .99082 ,99090 7.3002 7.3319 7.3639 .13571 .13514 .13456 7.368 7.400 7.431 12 14 16 .99296 .99303 .99310 8.3862 8.4279 8.4701 .11840 .11782 .11725 8.446 8.487 8.529 18 20 22 .99098 .99106 .99114 7.3962 7.4287 7.4615 .13398 .13341 .13283 7.463 7.496 7.528 18 20 22 .99317 .99324 .99331 8.5126 8.5555 8.5989 .11667 .11609 .11551 8.571 8.614 8.657 24 26 28 .99121 .99129 .99137 7.4946 7.5281 7.5618 .13225 .13168 .13110 7.561 7.594 7.628 24 26 28 .99337 .99344 .99350 8.6427 8.6870 8.7317 .11494 .11436 .11378 8.700 8.744 8.789 30 32 34 .99144 .99152 .99159 7.5957 7.6300 7.6647 .13052 .12995 .12937 7.661 7.695 7.730 30 32 34 .99357 .99363 .99370 8.7769 8.8225 8.8686 .11320 .11262 .11205 8.834 8.879 8.925 36 38 40 .99167 .99174 .99182 7.6996 7.7348 7.7703 .12879 .12822 .12764 7.764 7.799 7.834 36 38 40 .99377 .99383 .99389 8.9152 8.9623 9.0098 .11147 .11089 .11031 8.971 9.018 9.065 42 44 46 .99189 .99197 .99204 7.8062 7.8424 7.8789 .12706 .12648 .12591 7.870 7.906 7.942 42 44 46 .99396 .99402 .99409 9.0579 9.1065 9.1555 .10973 .10915 .10857 9.113 9.161 9.210 48 50 52 .99211 .99219 .99226 7.9158 7.9530 7.9906 .12533 .12475 .12418 7.979 8.016 8.053 48 50 52 .99415 .99421 .99427 9.2052 9.2553 9.3060 .10800 .10742 .10684 9.259 9.309 9.360 54 56 58 .99233 .99240 .99247 8.0285 8.0667 8.1054 .12360 .12302 .12245 8.090 8.128 8.167 54 56 58 .99434 .99440 .99416 9.3572 9.4090 9.4614 .10626 .10568 .10511 9.410 9.462 9.5U THE MACHINIST AND TOOL MAKER'S INSTRUCTOR, 107 NATURAL, SINES, TANGENTS AND SECANTS. , COSINES 84 Deg. 85 Deg. KIN SINK. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .99452 .99458 .99464 9.5144 9.5679 9.6220 .10453 .10395 .10337 9.567 9.620 9.674 2 4 .99619 .99624 .99629 11.4301 11.5072 11.5853 .08716 .08657 .08599 11.474 11550 11.628 6 8 10 .99470 .99476 .99482 9.6768 9.7322 9.7882 o!0279 .10221 .10163 9.728 9.783 9.839 6 8 10 .99634 .99639 .99644 11.6645 11.7448 11.8262 .08542 .08484 .08426 11.707 11.787 11.868 12 14 16 .99458 .99494 .99499 9.8448 9.9021 9.9601 .10105 .10047 .09990 9.895 9.952 10.010 12 14 16 .99649 .99654 .99659 11.9087 11.9923 12.0772 .08368 .08310 .08252 11.950 12.034 12.118 18 20 22 .99505 .9951 L .99517 10.0187 10.0780 10.1381 .09932 .09874 .09816 10.068 10.127 10.187 18 20 22 .99664 .99668 .99673 12.1632 12.2505 12.3390 .08194 .08136 .08078 12.204 12.291 12.379 24 26 28 .99523 .99528 .99534 10.1988 10.2602 10.3224 .09758 .09700 .09642 10.248 10.309 10.371 24 26 28 .99678 .99682 .99687 12.4288 12.5199 12.6124 .08020 .07962 .07904 12.469 12.560 12.652 30 32 34 .99539 .99545 .99551 10.3854 10.4491 10.5136 .09584 .09526 o09468 10.433 10.497 10.561 30 32 34 .99692 .99696 .99701 12.7062 12.8014 12.8961 .07846 .07788 .07730 12.745 12.840 12.937 36 38 40 .99556 .99562 .99567 10.5789 10.6450 10.7119 .09411 .09353 .09295 10.626 10.692 10.758 36 38 40 .99705 .99709 .99714 12.9962 13.0958 13.1969 .07672 .07614 .07556 13.034 13.134 13.235 42 44 46 .99572 .99578 .99583 10.7797 10.8483 10.9178 .09237 .09179 .09121 10.826 10.894 10.963 42 44 46 .99718 .99723 .99727 13.2996 13.4039 13.5098 .07498 .07440 .07382 13.337 13.441 13.547 48 50 52 .99588 .99594 .99599 10.9882 11.0594 11.1316 .09063 .09005 .08947 11.033 11.105 11.176 48 50 52 .99731 .99736 .99740 13.6174 13.7267 13.8378 .07324 .07266 .07208 13.654 13.763 13.874 54 56 58 .99604 .99609 .99614 11.2048 11.2789 11.3540 .08889 .08831 .08773 11.249 11.323 11.398 54 56 58 .99744 .99748 .99752 13.9507 14.0655 14.1821 .07150 .07091 .07034 13.986 14.100 14.217 108 THE MACHINIST AND TOOIi MAKER'S INSTRUCTOR. NATURAL SINES, TANGENTS, COSINES AND SECANTS. 86 DEG. 87 DEG. .MIN SINE. TANG. COSINE. SEC. MIN SINE. TANG. COSINE. SEC. 2 4 .99756 .99760 .99764 14.3007 14.4212 14.5438 .06976 .06917 .06859 14.335 14.456 14.578 2 4 .99863 .99866 .99869 19.0811 19.2959 19.5156 .05234 .05175 .05117 19.108 19.322 19.541 6 8 10 .99768 .99772 .99776 14.6685 14.7954 14.9244 .06801 .06743 .06685 14.702 14.829 14.958 6 8 10 .99872 .99875 .99877 19.7403 19.9702 20.2056 .05059 .05001 .04943 19.766 19.995 20.230 12 14 16 .99780 .99784 .99788 15.0557 15.1893 15.3254 .06627 .06569 .06511 15.089 15.222 15.35y 12 14 16 .99880 .99883 .99886 20.4465 20.6932 20.9460 .04885 .04827 .04768 20.471 20.717 20.970 18 20 22 .99791 .99795 .99799 15.4638 15.6048 15.7483 .06453 .06395 .06337 15.496 15.637 15.780 18 20 22 .99889 .99892 .99894 21.2049 21.4704 21.7426 .04711 .04652 .04594 21.228 21.494 21.765 24 26 28 .99803 .99806 .99810 15.8945 16.0435 16.1952 .06279 .06221 .06163 15.926 16.075 16.226 24 26 28 .99897 .99900 .99902 22.0217 22.3081 22.6020 .04536 .04478 .04420 22.044 22.330 22.624 30 32 34 .99813 .99817 .99820 16.3499 16.5075 16.6681 .06104 .06047 .05989 16.380 16.538 16.698 30 32 34 .99905 .99907 .99909 22.9038 23.2137 23.5321 .04362 .04304 .04245 22.925 23.235 23.553 36 38 40 .99824 .99827 .99831 16.8319 16.9990 17.1693 .05930 .05872 .05814 16.861 17.028 17.198 36 38 40 .99912 .99915 .99917 23.8593 24.1957 24.5418 .04187 .04129 .04071 23.880 24.216 24.562 42 44 46 .99834 .99837 .99841 17.3432 17.5205 17.7035 .05756 .05698 .05640 17.372 17.549 17.730 42 44 46 .99919 .99922 .99924 24.8978 25.2644 25.6418 .04013 .03955 .03897 24.918 25.284 25.661 48 50 52 .99844 .99847 .99850 17.8863 18.0750 18.2677 .05582 .05524 .05466 17.914 18.103 18.295 48 50 52 .99926 .99928 .99931 26.0307 26.4316 26.8450 .03839 .03780 .03722 26.050 26.450 26.864 54 56 58 .99853 .99857 .99859 18.4645 18.6656 18.8711 .05408 .05349 .05292 18.491 18.692 18.897 54 56 58 .99934 .99^35 .99937 27.2715 27.7117 28.1664 .03664 .03606 .03548 27.290 27.730 28.184 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 109 NATURAL, SINES, TANGENTS, COSINES AND SECANTS. 88 Deg. 89 Deg. MIS SINE. TANG. COSINE. SEC* MIN SINE. TANG. COSINE. SEC. 2 4 .99939 .99941 .99943 28.636 29.122 29.624 .03490 .03432 .03374 28.654 29.139 29.641 2 4 .99985 •99986 .99987 57.290 59.266 61.383 .01745 .01687 .01629 57.298 59 274 61.390 6 8 10 .99945 .99947 .99949 30.145 30.683 31.242 .03315 .03257 .03199 30.161 30.698 31.257 6 8 10 .99988 .99988 .99989 63.657 66.106 68.750 .01571 .01513 .01454 63.664 66.112 68.757 12 14 16 .99951 .99952 .99954 31.820 32.421 33.045 .03141 .03083 .03025 31.836 32.437 33.060 12 14 16 .99990 .99991 .99992 71.615 74.729 78.126 .01396 .01338 .01279 71.622 74.736 78.133 18 20 22 .99956 .99958 .99959 33.694 34.368 35.069 .02966 .02908 .02850 33.708 34.382 35.084 18 20 22 .99992 .99993 .99994 81.847 85.940 90.463 .01222 .01163 .01105 81.833 85.984 92.469 24 26 28 .99961 .99962 .99964 35.801 36.563 37.358 .02792 .02734 .02676 35.814 36.576 37.371 24 26 28 .99995 .99995 .99996 95.489 101.107 107.43 .01047 .00989 .00931 95.495 101.112 107.411 30 32 34 .99966 .99967 .99969 38.188 39.057 39.965 .02617 .02560 .02501 38.201 39.069 39.978 30 32 34 .99996 .99997 .99997 114.59 122.78 132.22 .00872 .00814 .00756 114.548 122.850 132.275 36 38 40 .99970 .99971 .99973 40.917 41.916 42.964 .02443 .02385 .02327 40.931 40.928 42.976 36 38 40 .99997 .99998 .99998 143.24 156.26 171.89 .00698 .00640 .00582 143.266 156.250 171.821 42 44 46 .99974 .99975 .99977 44.066 45.226 46.449 .02269 .02210 .02152 44.077 45.237 46.460 42 44 46 .99998 .99999 .99999 190.98 214.86 245.55 .00524 .00465 .00407 190.840 215.05 245.70 48 50 52 .99978 .99979 .99980 47.739 49.104 50.548 .02094 .02036 .01978 47.750 49.114 50.559 48 50 52 .99999 .99999 .99999 286.48 343.78 429.72 .00349 .00291 .00233 286.53 343.65 429.20 54 56 58 .99982 .99983 .99984 52.081 53.709 55.441 .01919 .01861 .01803 52.090 53.718 55.450 54 56 58 .99999 1. 1. 572.96 859.44 1718.87 .00174 .00116 .00058 574.71 862.0 1718.8 90 Deg. Sine — 1. Tangent — Infinite. Cosine — 0. Secant = Infinite. The preceding Table of Sines, Tangents, Cosines and Secants is given for every two minutes of the Quadrant to a Radius of 1. 110 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. CHAPTER II. GEARING. In the transfer of motion from one axis to another gear- ing is the most important system that can be introduced, whether the shafts are to be parallel or at an angle to each other. For this purpose when the shafts are parallel, we use spur gears, so called from their teeth being parallel with their axis. *■■ When the gears are not of the same size, the smaller one is usually called a pinion, although the name u spur gear " is also correct. Bevel gears are used for transmitting motion between a pair of shafts that are at right angles to each other, and one wheel is always larger than the other. Mitre gears are so called because the pitch line of their teeth are always at an angle of 45° from their axis. When we say a pair of mitre gears, we know that they are both alike, that the same cutter will cut both gears, and also that their shafts are to be at right angles to each other. Thus it can be seen that a mitre is a bevel wheel (although we do not usually call it such), but a bevel wheel is never a mitre, except as stated that the pitch line is at an angle of 45° to its axis (or bore. ) Angle gears are also bevel gears, but they are so called from the fact that they are usually employed for transmitting motion between shafting that is less or greater than a right angle or 90°. Internal gears also have their teeth, like spur gears, par- allel to its axis, but the teeth of the internal gear are inside of the rim, while those of the spur gear are outside. Spiral gears are so called from having their teeth formed like a screw of a very coarse pitch, and for this reason are THE MACHINI8T AND TOOL MAKER'S INSTRUCTOR. Ill sometimes called screw gears. A spiral gear, like a spiral milling cutter, runs smoother than with straight teeth, and sometimes can be used to greater advantage than a worm and worm wheel. When a pair of spiral gears are required to connect a pair of shafts that are parallel to each other, one should be right hand and the other left hand; they may be one degree or twenty degrees, to suit, and whatever angle is given to one must also be given to the other, regardless of their size, but the pitch of the spiral will be in proportion to the number of teeth; thus in a spiral gear of twenty teeth 10 degrees angle and a certain pitch, another gear to mesh with it of 40 or 60 teeth, should also be cut at 10 degrees, but the pitch would be twice or three times as great, or in proportion to the number of their teeth. If a pair of spiral gears are to connect at right angles to each other, then both wheels will be right hand or left hand, according to the direction they are to drive. In spur gears when we say a 4-inch, an 8-inch or a 20-inch gear, it is always understood to be the pitch diameter, and if these gears were of 8 pitch then their diameters would be 4|, 8f and 20f respectively; but if they were to be 10 pitch, then instead of adding eighths, you would add tenths, or if they were to be 2 pitch you would add J^ // > or if 1 pitch you would then add inches. Thus a gear of 1 pitch, 20 teeth, would be 20 inches diameter on the pitch line and 22 inches outside diameter, or if you want a pair of gears, say 24 teeth and 30 teeth of six pitch, then the pair would be 2 / and 3 g° re- spectively, or 4 inches and 5 inches diameter respectively on the pitch line or 4f and 5f inches outside diameter. In other words, the gear will measure on the pitch line as many 8ths, lOths or 32ds as there are teeth; thus, in a wheel of 20 teeth, 8 pitch 20/8ths ; 20 teeth 10 pitch, 20/10ths; 20 teeth 32 pitch, 20/32ds, etc. of inches, and two more eighths should be added if of 8 pitch, or 2/32 more if of 32 pitch, etc. This is called diametral pitch and means so many teeth per inch in diam- eter, measured on the pitch line. 112 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. Nearly all gears are now made on this system, for the reason that in building machinery the distances between two or more shafts that are to be driven by gears will always be so many 8ths apart if 8 pitch, or so many lOths, if 10 pitch, etc., while in the old system of circular pitch it is common to have the distances so many inches and a sixty-fourth full or a thirty-second scant, etc. The clearance at the bottoms of teeth should always be one-tenth (1/10) of the thickness of the tooth on the pitch line. To find the thickness of a tooth on the pitch line, divide the circumference of the wheel on the pitch line by the num- ber of teeth and spaces, or by twice the number of teeth, and one-tenth of this will be the depth of clearance at the bottom of the teeth. Thus, to find the thickness of an 8 pitch tooth, as there are eight teeth to every inch in the diameter measured on the pitch line, then a wheel of one inch diameter = 3.1416" in circumference, and divided by 16 = .196" for the thickness, and one-tenth of this for clearance = .020 nearly, for any 8 pitch gear. If this was a 6 pitch gear, we would divide by 12 ; thus, 3.1416 -T- 12 = .262" nearly, for the thickness of any 6 pitch tooth, and this divided by 10 = .026" for the depth of the clearance of any 6 pitch gear. If it was a 30 pitch gear, then 3.1416 -~ 60 = .052" for the thickness of the tooth on the pitch line, and one-tenth of this = .005" for the depth of the clearance, etc. In any gear of 8 pitch the depth of tooth is always one- eighth above the pitch line and one-eighth below the pitch line, and also the clearance ; then 2/8 = .250 and added to .020 = .270" for the total depth of any 8 pitch gear. In a 6 pitch gear the depth of tooth is one-sixth of an inch above the pitch line and one-sixth of an inch below the pitch line, plus the clearance ; thus we have explained that the clearance for a 6 pitch gear was .026", then added to 2/6 of an THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 113 inch = .359" for the depth of the teeth for any 6 pitch gear, etc. It must be remembered that in all cut gearing the width of space on the pitch line should always be the same as the thickness of the teeth, and consequently, the circular pitch of any gear is equal to twice the thickness of the tooth on the pitch line. Or, dividing the circumference of the gear on the pitch line by the number of teeth will be the same thing. Thus, a wheel one inch diameter on the pitch line would be 3.1416 // in circumference, and if it was to be of 8 pitch, then there would be 8 teeth on the wheel; then 3.141 6 -r- 8 = .393" nearly, for the circular pitch, or, if it was 10 pitch, then 3.1416 -r- 10 = .314" for the circular pitch, etc. Remember, that in calculating the dimensions of gear teeth, as in the two examples just shown, as for instance dividing 3.1416 by 10, it is not necessary to go higher than thousandths, or three decimals, and we always take for the last figure the one nearest to a whole number. Or we can find the diametral pitch in the following man- ner when the circular pitch is known : The circular pitch of a gear is %"; what is the diametral pitch? y s " = .875" and 3.1416 -r- .875 = 3.59, the answer. In other words 3.1416 divided by the circular pitch reduced to a decimal number, will be the diametral pitch. If we have a pair of gears that are 12" and 8" respect- ively, pitch diameters, we know that the distance between centers will be equal to one-half of the diameters of the two gears; or, 12 + 8 = 20, and 20 -f- 2 = 10". Or we may have the two gears like the following example : Find the distance be- tween centers of two gears in mesh of 28 and 24 teeth respect- ively and 32 pitch. 28 + 24 = 52, and 52 -r- 32 = 1% or 1.625; this divided by 2 = .8125, the answer. Suppose we have a pair of gears of 26 and 32 teeth respect- ively and 8 pitch, what is the distance between centers of these wheels when in mesh ? 26 + 32 = 58, and 58 -=- 8 pitch = 7.25", and this again divided by 2 = 3.625", the answer. Now let us examine a pair of gears in circular pitch. We 114 THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. GEARING.— TOOTH DIMENSIONS. Thickness Diametral Circular of Tooth on Depth to Working Clearance. Pitch. Pitch. the Pitch Cut. Depth. Iyine. 1 3.1416 1.5708 2.157 2.000 .157 VA 2.5132 1.2566 1.725 1.600 .125 1J* 2.0944 1.0472 1.437 1.333 .104 1% 1.7952 .8976 1.232 1.143 .089 2 1.5708 .7854 1.078 1.0 .078 2M 1.3962 .6981 .958 .888 .070 2^ 1.2566 .6283 .863 .800 .063 2% 1.1423 .5711 .784 .727 .057 3 1.0470 .5235 .719 M6 .053 3^ .9666 .4833 .664 .616 .048 sy 2 .8976 .4488 .616 .571 .045 4 .7854 .3927 .539 .500 .039 5 .6283 .3141 .431 .400 .031 6 .5236 .2618 .359 .333 .026 7 .4488 .2244 .308 .286 .022 8 .3927 .1964 .270 .250 .020 9 .3490 .1745 .240 .222 .018 10 .3141 .1570 .216 .200 .016 11 .2856 .1428 .196 .182 .014 12 .2618 .1309 .180 .167 .013 14 .2244 .1122 .154 .143 .011 16 .1963 .0982 .135 .125 .010 18 .1745 .0872 .120 .111 .009 20 .1570 .0785 .108 .100 .008 22 .1428 .0714 .098 .001 .007 24 .1309 .0654 .090 .083 .007 26 .1208 .0604 .083 .077 .006 28 .1122 .0561 .077 .071 .006 30 .1047 .0523 .072 .066 .006 32 .0982 .0491 .067 .062 .005 34 .0924 .0462 .063 .059 .004 36 .0872 .0436 .060 .055 .004 40 .0785 .0392 .054 .050 .004 42 .0748 .0374 .051 .048 .003 46 .0683 .0341 .047 .043 .004 50 .0628 .0314 .043 .040 .003 54 .0581 .0290 .040 .037 .003 60 .0523 .0261 .036 .033 .003 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 115 have a pair of gears of 24 and 30 teeth and % pitch ; what is the distance between the centers of these two wheels when in mesh? 24 X %" (or .375") = 9.000", the circumference on the pitch line; 9-^-3.1416 = 2.864+, the diameter on the pitch line. 30 X Y&" (or .375") = 11.250", or the circumference on the pitch line, and 11.250 -r- 3.1416 =3.581, nearly, for the diameter on the pitch line ; and 2.864" + 3.581" = 6.445", then divided by 2 = 3.2225 inches, or the answer. This can also be found in a more simple manner as fol- lows : Multiply the constant number .3183 by the circular pitch, and the result by one-half the number of teeth in both wheels; thus, in the same example, .3183 X .375 = .11936+, and one-half the number of teeth in both wheels = 27; then .11936 X 27 = 3.222", the answer, or the distance between the centers of the two wheels. Any example in circular pitch can be done in the same manner. The decimal .3183 is found by dividing one by 3.1416. The depth of teeth in circular pitch is found in the fol- lowing manner: Thus, for a gear of J^ 7/ circular pitch 3.1416 divided by .5 {%) = 6.283, for the diametral pitch, and 1 divided by 6.283 = .159", for one -half of the running depth of the tooth, and twice this, or, .318" = the running depth. Now if the circular pitch is 3^3", then the tooth is exactly J4" thick on the pitch line, and the clearance we said was one- tenth (1/10) of the thickness of the tooth on the pitch line; then 1/10 of J£, (or .250") = .025" and added to .318" = .343" for the depth to cut any gear of 3^ 7/ circular pitch. We can also find the depth of the teeth in a more simple manner, thus : Multiply the number (constant) .3183 by the circular pitch and this will give one-half of the running depth of the tooth; thus, in y 2 " circular pitch 3^" = .5"; .3183 X .5 = .159, and doubled equals .318, and one-tenth of the thickness of the tooth = .343", the answer, or total depth as before. It can readily be seen that if we want to have a pair of gears in mesh, and the holes are bored to the proper distance 116 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. for standard gears, and we then take a pair of gears that are 8 or 10 thousands too large or too small in the diameter, it will be too tight or too loose, and if the gear has 15 teeth and should be ten thousandths of an inch too small in the diam- eter, then it would be about 30 thousandths of an inch too small or to be divided into the 15 teeth, or about two thou- sandths of an inch for each tooth, etc. In a great many instances this would never be known, and especially in a rough class of work, such as rolling mill ma- chinery, etc. But in a fine class of work the greatest care should be taken to have the blanks turned to an exact size by means of the micrometer caliper or the vernier, as the case may be, and then if the gear cutter or milling machine is graduated correctly (some are not) we will have no trouble in getting good work. In designing the shapes of teeth, it is well to remember that the teeth of a pair of wheels should always be in contact at least equal to the pitch. Or, in other words, if we have a pair of wheels in mesh with each other of 3 7/ pitch, the teeth should not separate from each other until the next pair of teeth are well in contact, otherwise the friction from the slid- ing motion will be very great and a uniform motion cannot be maintained. It has also been demonstrated by practice that any describing curve rolled on the outside of one pitch line and on the inside of another will work correctly together. An epicycloid is somewhat similar, except that it is traced by a point on the circumference of a circle, as shown in figure 7, rolling upon the pitch line of another, and, when rolling on the inside it is called a hypocycloid. The circle C is called the generating circle, because it generates (or forms) the curves for the shapes of the teeth. This generating circle is usually made equal to the radius (half the diameter) of the smallest wheel in a train of gearing. Suppose this smallest wheel to be 3 pitch, 15 teeth or 5 inches in diameter, then the generating circle would be V/%" diameter. This same circle would be used for all the wheels con- THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 117 necting in a train, rolled on the outside of the pitch line (fig- ure 7) for generating the faces of the teeth, and on the inside (Figure 8) for the flanks of the teeth. Now if we roll this ^Yz' generating circle on the inside of a 5" pitch circle, we will have a perfectly straight line from the pitch line through the center of the gear, or, as we call them, radial lines, for the flanks of the teeth of this 5 /7 gear. It is not necessary that this size of generating circle should be used for this 5" gear, but it is the size best adapted for general purposes. If we use a generating circle that is less than one-half the diameter of the pitch circle, the flanks of the teeth will be more nearly parallel with each other, while the faces of the teeth will be more pointed than before, and the latter will naturally make the teeth crowd harder on each other, and, at the same time, if the flanks of the teeth are parallel, even fo r a short distance, the cutters will work hard, as there can be no clearance on the teeth of cutters when parallel. (This re- fers to formed cutters only.) In the epicycloidal form of gear, one of the most simple, is shown at Figure 1, and is usually called pin gearing, so called because one of the gears is made of two discs a little larger than the outside of the pins which are inserted for the teeth, as shown at C, this wheel is made somewhat wider than the large wheel, otherwise they could not run together. It must not be forgotten that the pitch lines G G, of both wheels must be in proportion to the numbers of teeth in the two wheels; thus, if the large wheel has 14 teeth, and the small one 7 teeth (or pins), the pitch diameter of one wheel must be just twice as great as the other. In designing a pair of wheels of this kind we will suppose the pins C to be % /f diameter ; the distance between two holes on the pitch line should be % /7 or % circular pitch, and, if there were 7 pins in the small wheel, then the circumference of the pitch circle G would be 7 times %" around it ; this will, of course, make the teeth on the large wheel %" thick on the pitch line also. 118 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. To draw the teeth on the large wheel, first lay them out just %f f between centers on the pitch circle as shown, and draw the circles equal to the size of the pins for the bottom of the teeth, and, with the dividers set in one of these circles, H, and the other leg of the dividers touching the edge of the next circle, draw the addendum (or that part of the tooth outside the pitch circle) shown at I, etc. » JF - ^*-/5» av 1 1/ »i; 73 i i mt l/B THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 119 Figure 2 represents the single curve gear, and is of 4 pitch (diametral) 13 teeth, and consequently is 13-4ths or 3 \^ ,f diameter on the pitch line A A, which should be drawn first, and as any 4 pitch gear is .393 7/ thick on the pitch line, lay out with the dividers several points equal to this distance on the line A A ; then with bevel pro- Pi > tractor, or what is better, a piece of thin sheet steel cut to an angle of 75° 30', and placing the corner of this template on the pitch line at one of these points as shown, and with one edge in line with the center, measure off on the other side from the corner of the template a distance equal to one-fourth (34) of 120 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. the radius, or from the pitch line to the center of the gear, which in this case is about .406 7/ , as shown at B, and, with the dividers set in the center of this gear describe at B a circle, which is called the base circle, shown dotted in the figure. Having drawn the base circle, we do not have to use this tem- plate again on this gear for the reason that having spaced off on the pitch circle (not the base circle) as directed, and with the dividers set to .406 // as nearly as possible, we place one leg of the dividers on the base circle and the other on the pitch circle at one of these points stepped off and draw the curves for the faces of the teeth to the base circle only. Now it has been found that for 12 and 13 teeth that paral- lel lines should be used for that portion of the teeth inside of the base circle, and drawing a circle at the center of the gear, as shown, for convenience, equal to the width of space at the base circle, draw lines C C, and for the fillets at the bottom of the teeth, the radius should be about one-sixth (1/6) of the distance B B. With gears having 20 teeth the curves may ex- tend from the addendum to the fillets at the bottom (or two pitches deep.) In all gears from 13 to 20 teeth, there should be parallel lines inside of the base circle for a portion of the distance. With a little practice in drawing the teeth of a pair of wheels in mesh with each other, and in different positions, making the lines as fine and accurate as possible, and also practicing on a wheel of very coarse pitch, you will soon learn how to form the inside or flanks of the teeth. Remember the outside should be made as described, and the inside, or that portion inside of the base circle to suit the outside or addendum of the teeth. If you want to make a drawing at any time, say of a pair of gears of any pitch (in order to find the shapes of the teeth), draw the teeth as stated, in different positions, and make them five or ten times (for convenience) as large as required; this will be on the principle of a magnifying glass, and will show the errors very plainly; this refers to gears of any pitch, style, etc. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 121 Remember that a gear of 60 pitch and any number of teeth is of the same shape as a gear of 2 pitch (or any other pitch) with the same number of teeth, and, of course, using the same Style of a tooth. That is, an involute tooth, of 32 pitch 15 teeth, is of the same shape as an involute tooth of 1 pitch 15 teeth; the only difference is, that one is much smaller or finer than the other. I have had a great many cutters made by making the drawings of three or four teeth of a gear on smooth stiff paper and then cutting out with a sharp knife ; but when regular standard cutters are wanted, it is cheaper and better to get them from parties who make a business of this class of work. In Figure 3, I have shown a double curve gear of Y^ n pitch, 40 teeth. There are no fillets shown at the bottom of these teeth, because they are strong enough without them ; in gears with a small number of teeth, however, it will be neces- sary to have fillets the same as in a single curve. This style of gear teeth was invented by Professor Willis, after a great deal of experimenting, and very nearly approach- es a perfect epicycloid. It is drawn in the following manner The example is 40 teeth }/%" pitch ; therefore, 40 times y%' around on the pitch line, or 20 //f circumference, and divided by 3.1416= 6.366 inches in diameter, on the pitch line A A, which should be drawn first ; as the gear is J£ inch pitch, the tooth should be exactly \^ ,r thick on this line. Now having made several points on the pitch circle A A, y±" from each other (for }/%" pitch, or 5/16" for %" pitch), with a bevel protractor, or what is better, a piece of thin sheet steel for a template, as shown at B C D of 75° placed at one of these points; we find in the table (that follows) for 40 teeth, J^ /7 pitch, the number 4 ; this means 4/20 of an inch, and is for the face of the teeth; then with the dividers set to 4/20 to the right of the corner of the template, make the point F, and from this point lay off the face of the tooth as shown by the small circle. 122 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. ■<- \JE/ 75 For the flank, or inside of the same tooth, we move the instrument one pitch, whatever it may be; in this case it is y%" to the left, as shown, and in the table for the flanks of teeth we find the number 7 for 40 teeth y% f pitch, and we now measure off 7/20" to the left of the corner, and from this point describe the curve for the flank, as shown by the large circle. It must be remembered that the numbers in the table are so many 20ths of inches. If you want to draw several of these TABLE FOR DOUBLE CURVE GEAR TEETH. (wims SYSTEM.) CENTERS FOR THE FLANKS OF THE TEETH. PITCH IN INCHES AND PARTS. NUMBER TEETH. Y2 % % 1 im m i 2 2M 2^ 3 \m 13 64 80 96 129 160 193 225 257 289 321 3a6 450 14 35 43 52 69 87 104 121 139 156 173 208 242 15 25 31 37 49 62 74 86 99 111 123 148 17a 16 20 25 30 40 50 59 69 79 89 99 191 138 17 17 21 25 34 42 50 59 67 75 84 101 117 18 15 19 22 30 37 45 52 59 67 74 89 104 19 13 17 20 27 35 40 47 54 60 67 80 94 20 12 16 19 25 31 37 43 49 56 62 74 86 22 11 14 16 22 27 33 39 43 49 54 65 76 24 10 12 15 20 25 30 35 40 45 49 59 69 26 9 11 14 18 23 27 32 37 41 46 55 64 28 13 22 26 30 35 40 43 52 60 30 8 10 12 17 21 25 29 33 37 41 49 58 35 9 11 16 19 23 26 30 34 38 45 53 40 7 15 18 21 25 28 32 35 42 49 60 6 8 9 13 15 19 22 25 28 31 37 43 80 7 12 17 20 23 26 29 35 41 100 8 11 14 22 25 28 34 3& 150 5 13 16 19 21 24 27 32 38 Rack, 6 7 10 12 15 17 20 22 25 30 34 CENTERS FOR THE FACES OF THE TEETH. PITCH IN INCHES AND PARTS. NUMBERS OF TEETH. H % % 1 iM IJfi 1% 2 m 2^ 3 3^ 12 2 3 4 5 6 7 9 10 11 12 15 17 15 3 7 8 10 11 12 14 17 19 20 4 5 6 8 9 11 12 14 15 18 21 30 4 7 9 10 12 14 16 18 21 25 40 6 8 11 13 15 17 19 23 2a 60 5 10 12 14 16 18 20 25 29 80 9 11 13 15 17 19 21 26 30 100 7 18 20 22 31 150 5 6 14 16 19 21 23 27 32 Rack. 10 12 15 17 20 22 25 30 34 For 4" Pitch, double, 2"; For 5" Pitch, double, 2J£", etc. 124 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. teeth, then with one leg of the dividers set to the center of the gear at I, draw arcs through the centers of these two cir- cles shown in the figure, and you will not have to use the template again on this gear. Figure 4 represents a perfect epicycloidal tooth, both on the pinion and the rack; they are of 4 pitch, and the pinion has 15 teeth, and were drawn in the following manner: For the pinion, turn up a block of wood with a radius equal to the radius of the pinion on the pitch line, also another piece (concave) to match the other. Now take the dividers and lay off the pitch circle on paper, and place one of these wood blocks (made as thin as possible) and with another round block turned up just one-half the diameter of the pitch circle of the pinion, and, holding a lead pencil close to the periphery, and rolling it on this circle, describe the curves, first on one side and then the other (inside and outside of the pitch circle); these curves will form a perfect epicycloid. One straight block of wood will answer for the rack, plac- ing it on the pitch line G D, and rolling the same block first on one side and then the other as before. Two sizes of rolls can be used on a pair of gears, but what- ever size is used for the outside, or face, of one gear must, in all cases, be used for the inside of its mate. This rack and pinion can also be done in the same man- ner as described in the double curve gears; thus, for the rack, place the instrument with the corner touching the pitch line as shown to the left of the figure, and, as there is no center to a rack as in the pinion, the end A will have to be placed at 90° to the pitch line ; in the table find the number correspond- ing to the pitch required, and describe the curve for the face, and then move one pitch to the left, and describe the flank as already explained. Figure 5 also shows a pinion of 15 teeth in mesh with an internal gear of 30 teeth, and was drawn in the same manner as in Figure 4. It will be seen that the teeth of the internal wheel are similar in shape to that of the rack, except that in THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 125 126 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 12? the internal wheel the teeth are thicker at the root than in the rack, this being due to the curve of the wheel. An internal wheel can be drawn in the same manner as any other gear, the difference being that the teeth as drawn becomes the space and the space the teeth. In figure 6 I have shown the manner of constructing the perfect involute shape of gear teeth, which is as follows : From the center D, describe the pitch circle A (a small por- tion only being seen) and on this circle step off points for the teeth as already explained ; we then place our template or pro- tractor of 75° 30 / with the corner on the pitch line A; and the edge in line with the center of the gear, measuring off one- fourth of the radius of the gear, which, in this example, is ^ 7/ (the radius being 3 // ). We now have a point from which with the dividers set (the other point in the center at D) we describe the arc C B, which is called the base circle. Remember that the involute always extends from the base circle, and not the pitch circle, to the top of the tooth, while that portion of the tooth inside the base circle is the same as explained in single curve gears. We then lay out a number of radial lines, extending from the base line to the center ; it is not necessary to start from any particular place in the figure. I have started nearly %" from the side of the tooth. I could have made it %"; the more lines and the closer they are, if properly drawn, the more correct will be the tooth, but it can easily be seen that I have made them close enough for all purposes ; but whatever distance you make 1 and 2, make the rest the same, or in other words, space them evenly. Next from these radial lines draw the lines marked V 2 / 3', etc., exactly at 90° from the radial lines, and, with the dividers set with one point at 1 on the base line, and the other point at F, also on the base line, draw that portion of the tooth between F and 2', then open the dividers, and placing one leg at 2 on the base circle, and with the pencil at 2 / extend the tooth to 3 7 , then open again, and placing one point of the dividers at 3 on the base circle, and the pencil at 3' extend the tooth to 4', etc. Continue in this w r ay until you 128 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. THE MACHINIST AND TOOL. MAKEK'S INSTRUCTOR. 129 finish the tooth as shown. It can easily be seen by the draw- ing that a single arc struck from any position will not be an involute. The true involute tooth is the best by far for the shapes of gear teeth for most purposes, for the reason that when the distance between centers are not correct it will make no dif- ference in their uniformity of speed, etc* Figure 7 illustrates a method of drawing the perfect epicycloid, as follows : B is the center of the gear B A, the pitch circle, and C the generating circle, which in this case I have made a little less in diameter than the radius of the gear, in order to show a curve in the next figure, as both these figures 7 and 8 show but one side of a tooth ; figure 7 the face of a tooth, and figure 8 the flank of the same tooth. In Figure 7 draw radial lines from the center B, through the pitch circle A B, as shown; they must be produced fa enough to reach the top of the tooth. Make these lines uni- form in their distance one from the other; it is not important what the distance is, but do not get them too far apart. Sup- pose them to be one-eighth of an inch starting from A, toward B; then on the generating circle C, also commencing at A, step off points exactly the same distance J£ 7/ , as shown at 1, 2, 3, etc., and with the dividers set to the center of the circle at B, draw arcs from these points on the generating circle, as shown at 1, 2, 3, etc. Remember to draw the arc from the line F B to 1 on the generating circle ; next draw the arc from the line F B to 2 on the generating circle ; in the same man- ner draw all the arcs, and as there are 13 points on the gener- ating circle, then the 13th arc will be continued through to the 13th radial line. Now to find the shape of the face of the tooth A D above the pitch line, place one leg of the dividers on the 13 point on the generating circle near I and the other leg at G, then re- move the dividers to H, and mark the point I; next with one leg of the dividers in the 12 space on the generating circle and the other leg at J, remove as before to K, and describe the point shown at L. Continue in this way until you pro- 130 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR • I \ \ \ \ i J -i j-\^r * j * j j / / / / h't 'ii' m m\\\\\\ii!u llil.l'll ////// \ i 1 I I * l ^9? ^<> III''/////// \ u Villi//////// into' i .' ^ THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 131 duce all the points shown on the arc A D, and a curve through these points will be an epicycloid. Figure 8 shows the same system for drawing the flanks of the same tooth. B is the center of the gear, A E is the pitch circle, and C the generating circle. Draw radial lines from the center B to the pitch circle, as shown; as before stated the distance between points is not particular, except that what- J 5' *•. 3 ever they may be in the circle K A, they must be stepped off the same on the generating circle C, and all must start from the same point A. Now with the dividers set at the center B describe arcs through the points 1, 2,3, 4, etc., on the gener- ating circle, the 10th arc passing through the 10 point on the generating circle to the 10th radial line ; the 9th arc passing through the 9th point on the generating circle to the 9th radial line, etc. Then with one point of the dividers set at F, the 10th point on the circle, and the other at G, the 10th radial 132 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. line, transfer to the line A B at H and produce I ; and from J, the 9th point on the circle, to K, the 9th radial line, transfer to L, and produce M; continue in this manner as shown, and a curve drawn through these points, if not too far apart, will be an epicycloid for the flank of this gear. As already explained, in a combination of gearing in which they are all to interchange, the generating circle is usually made y% the diameter of the smallest gear in the set. But if you want a pair 24' ' diameter, or one that is 24" and the other 30 /r diameter, on the pitch line, then I would use for either case a generating circle that was 12 r/ diameter for general purposes, but, if you want to increase the strength of the teeth, in these two latter cases I should make the generating circle a little less than one-half the diameter of the smallest of the pair, on the pitch circle. The teeth of racks are sometimes made with straight faces; some parties make them at an angle of 15° on each side, and some make them 14^° on each side. I prefer the latter, as that is the angle (29°) of a worm thread. BEVEL, WHEELS. I will now show you how to draw a pair of mitre wheels in gear. Figure 9 represents a pair of 6 pitch 18 teeth; 18 -T- 6 = 3 inches for the largest pitch diameter shown at A A; draw these two lines first as near as possible to the proper length and exactly at right angles (90°) to each other ; next draw lines through the center of both these lines rep- resenting the axis of the gears and meeting at B, as shown. From the extreme ends of the largest pitch diameters draw the lines C, meeting at B. These lines (C) are the pitch lines of the teeth ; at right angles to the lines C B, draw E F, the length of which depends on the pitch of the gear; if 4 pitch, draw one-fourth of an inch on the outside and one-fourth plus the clearance on the inside. The clearance on bevel gears of all kinds is the same depth on the large end of the gear as in spur gears; as the example is 6 pitch, then draw E F one-sixth of an inch, or .166" on the outside of the line C shown at E THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 133 te ^ 134 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. y=o THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 135 and one-sixth plus the clearance or .192 shown at F; next draw all these lines B F, according to the length of the face of the gear wanted, toward the center B; this length and also the general outlines of hub, etc., depend on circumstances. The center angle of a pair of mitre gears is always 45°. as shown. It sometimes happens that we have to drive two shafts by gearing that are less or greater than a right angle (90°). In Figure 10 is an example of this kind, in which the shafts are at an angle of 100° from each other, as shown. The pinion has 16 teeth and the wheel 72, and are 8 pitch. First draw the lines A A and B B at an angle of 100°; at right angles to B B draw the line D, exactly 2 inches long (8 pitch 16 teeth) repre- senting the largest pitch diameter of the pinion. The end of this line at B should be just 4^2 inches from A A, because the large wheel has 72 teeth, and consequently is just 9 inches in diameter at the largest part (or 4% // radius). At the inter- section of the lines A A and B B shown at C, draw E C and G C; as the gears are 8 pitch, then the teeth will extend J^ // above the pitch line and also y^' plus the clearance below the pitch line at the large end only, and at right angles to the £itch lines B C and G C ; all these lines are drawn to the cen- ter at C. Draw the shape of the gears to suit circumstances. Figure 11 shows the manner of constructing the teeth of bevel gears. At right angles to the pitch line and about one- third of the length of the tooth from the large end, draw the line B B extending to the axis of the gear B B, and, with the dividers set to this length, transfer to one side and draw the circle C C. As this gear is 4 pitch, then the tooth and space are each .393 // wide on the pitch line ; now space off three or four points on the line C, and with our template or protractor of 75° 3(K, draw the shapes in the same manner as explained in spur gears, using for the circle A a radius equal to one- fourth of the line BB. Remember that the shape and dimensions of these teeth on the circle C C are just the same as on a spur gear of the same diameter; this refers only to the extreme end of the bevel gear tooth at the largest pitch diameter, and from this point 136 THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 137 the teeth gradually get smaller in all dimensions, depending on the length of face, etc. Now, we have found the shape of the teeth, but in cutting bevel gears the cutters will have to be made thin enough to cut through at the small end of the teeth, and the longer the face of the tooth, the thinner we will have to make our cutter. I have shown the shape of the cutter for this gear, and the arcs D D represent the pitch lines of the gear at the two ex- treme ends of the teeth. This will be more fully explained hereafter. I will now explain the method of finding the angles of bevel gears. This is a very important matter for the reason that if we do not know the angles, we can neither turn nor cut them accurately. Figure 12 represents a pair of bevel wheels of 4 pitch ; O G is a line through the center of the large wheel which has 30 teeth, and O E, the center line through the pinion which has 12 teeth, a portion only is shown ; the wheels are supposed to be in mesh, O I representing the pitch line between the two gears. Now, as the large gear has 30 teeth, we know the radius (or one-half the diameter) is just &%" which is radius to the angle BOE; we also know the radius of the pinion (or the length of B E), because it is one-half the diameter of the pinion on the pitch line, and, as the pinion has 12 teeth, dividing 12 by 4=3 inches diameter, or lj^ 7 ' radius ; this 1J^ /7 becomes tangent to the angle BOE, then 1.5" divided by %-%" = .400", which is tangent to the radius of one inch. In the table of tangents we find the angle 21° 48' to correspond with the decimal .400. 21° 48' is then the center angle of the pinion on the pitch line as shown, and the difference between this and 90° or 68° 12' is also the center angle of the large gear, also shown. It must be remembered that the lines B O and I O are the pitch lines of the teeth, the latter for both the pinion and the large wheel as they are supposed to be in gear. Now, the center angle of the gear being found on the 138 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. a ♦ P l r .^, .-- ''^ h / ^1 ~*~\ * ,-"** o et im*~j^' ^ t MOf/V/d v^ \ • Mi- * - n HI33J. ZlH^'ft £ \ \ Hfc "* 1 :' ! '\§§§§>.,^ ' / iSR^ °° ; \ \? t 55 IV * 1 *-$\ \ x j£^>^ ' \ \ vV \ \ * ^ f \ v\cf)»\ V 0D| >* f* ta Y\\ « « 5: \ \ " ^. ° ^ a ^ *-*.« Ui \ \ \\ O 'J? \ir^ 5 a0 i c> \ \ w \ \ \^ ' CO \\V'^ \V\ ; A* i ^ : «; A\ i \\ 1 5 o _t„ 4 -i Q" THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 139 pitch line, then the complement of this angle, or 68° 12 7 , will be the proper angle to finish the teeth A C, measured from the back of the gear, because A 3? should always be made 90 a from the pitch line B O as shown, and not from the outside or top of the teeth. The outside diameter of bevel gears is found in a differ- ent manner from spur gears, because in bevel gears, if of 4 pitch, as in the example, we measure J4" (B A) at right angles to the pitch line B O. For convenience I have shaded a small angle A B D at the top of this pinion, and, in order to show it more clearly have removed it to one side and have also enlarged it. This angle is for the purpose of finding the exact diameter of this pinion at the highest point shown at A. In the angle A B D, the longest side A B is known, be- cause it is always equal to the pitch. Thus in 10 pitch, it would be .100"; in 8 pitch, .125 // , and in 4 pitch, .250". As the example is 4 pitch, then A B is .250". We also know the angle, because it is always the same as the center angle on the pitch line; thus, in the figure the angle is 21° 48', and A D is sine and D B cosine to this angle. Now, in the table we find the cosine of 21° 48' = .928, and, multiplied by the radius (.250) = .232", and this is the distance D B to be added to the radius of the pinion; that is, the radius is 1.5"; then 1.5" + .232=1.732" for one side, and doubled = 3.4642" for the largest diameter of the pinion. Let us now find the angle to turn the face of this pinion. We have found the center angle at the pitch line, and for con- venience we now want to know the length of this pitch line, (B O). As O E is radius and B K tangent, then B O is the secant of the angle BOB. In the table of secants we find opposite 21° 48' the decimal 1.077, which, multiplied by the radius 3% = 4.0387, or say 4.039" for the length O B, as shown. Now as A C is always at right angles to B O, then we have a right angled triangle A O B, figures 12 and 13. In the latter figure I have separated that portion of the tooth above the pitch line, and also that portion including the clearance 140 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 1 ^ I ->«-^o, / / s^L*-* .11 * v** +■» i^ « THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 141 below the pitch line, forming two separate right angled trian- gles : the one above the pitch line for finding the angle to turn the pinion, and the one below the pitch line for finding the angle to cut the teeth. As A C is 90° from O B and the length is also known to be 4.039", also that A O is longer than B O, then A O is the secant to the angle A O B. We know A B is .250", and divided by the radius 4.039 = .0618+, or the tangent to the radius 1, and in the table of tan- gents this corresponds to the angle 3° 32', which added to 21° 48' = 25° 20' for center angle; or 90° — 25° 20' = 64° 40' measured from the back as shown, for turning the face of the pinion. For cutting the teeth the angle B O C is shown, and we have to add the clearance, which is always on the large end of bevel wheels the same as in spur wheels; in 4 pitch the clear- ance is .039", and added to .250" = .289", which divided by 4.039 = .0715 4- = tangent to radius 1, which corresponds to the angle 4° 6', and 21° 48' — 4° 6' = 17° 42', for the center angle; or 90° — 17° 42' = 72° 18', the angle to cut this gear, measured from the back as shown in Figure 12. It is also necessary in making cutters for bevel gears to know the smallest pitch diameter of the gears in order that the cutters may be made of the proper thickness on the pitch Line for cutting them. Figure 14 shows the method of finding this diameter [figures 12, 13 and 14 representing the same gear.) I have shown that the line B O is 4.039", and in figure 12 he face is marked 1^"; then 4.039 — 1.25" = 2.790" nearly. In figure 14 I have shaded an angle A O C for the purpose >f showing the method of finding the smallest pitch diameter >f any mitre or bevel gear. We have found the center angle on the cone pitch line to >e 21° 48', and as the longest line (A O) is radius, then A C becomes sine, and the sine of 21° 48' = .371 + , which multi- plied by the radius (2.790") = 1.036" for the length of A C, 142 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. V-prrtiy' ryy& CO * , )),«#«*& 7tf0 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 143 and twice this = 2.072" for the smallest pitch diameter of the pinion. 2.072" X 3.1416 = 6.509" for the circumference, and, divided by twice the number of teeth (24) = .2712", for the thickness of the cutter or tooth on the pitch line. It must not be forgotten that the cutter is made of the proper thickness to suit the teeth on the smallest pitch diame- ter, and, in making the cutter, it is also necessary to know the depth from the pitch line to the bottom of the tooth on the small end. Referring back to Figure 14, I show a method of finding this depth by the angle EOF. I have shown A O to be 2.790" long, and this line is represented by E O. Now this angle is already explained in Figure 13, which is the same thing, and has been found to be 4° 8 7 ; then as E O represents the pitch line, and E F at right angles to it the tangent, we multiply the tangent of 4° 6' by the radius 2.79; thus tangent 4° 6' = .0717 X 2.79 = .200" for the depth from the pitch line to the bottom of the teeth on the small end of the gear. Now, as the teeth of bevel gears are of the same depth, thickness, etc., at the largest pitch diameter as those of spur gears, then this gear of 4 pitch will be .289" from the pitch line to the bottom of the teeth on the large end, and only .200" on the small end, a difference of .089" in the depth from the pitch line, and of course, the cutter will be too thin for the large ends of the teeth. To finish the teeth of bevel wheels correctly (or I might say approximately correct, for bevel gears cannot be made correct by rotary cutters) we will have to pass the cutter through each space at least twice. There are two ways of cutting these teeth to get them of the proper thickness on the large end of the teeth, and, if pos- sible, the wheels should always be on an arbor that is fitted in the taper hole of the index head spindle, no matter how slight the angle of the teeth may be, as otherwise the teeth may be thicker on one side than the other (or rather not spaced uni- ormly). 144 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. One way of cutting the teeth of bevel wheels to get them the right thickness on the large end of the teeth is to place the cutter central with the headstock spindle, and, having the wheel elevated to the proper angle, pass the cutter through, then rotate the wheel by the index crank, say two or three holes on the index plate, depending altogether on the gear you are cutting, that is, whether the cutter is much thinner than standard size. Now, if you revolve the wheel as stated, but one hole, you will have to move the table slightly in or out depending upon which way you revolve it, so that the cutter will pass through the small end of the gear without touching the sides of the tooth, and, as the large end of the wheel will revolve through a greater space than the small end, so will the cutter passing through take out more and more as it reaches the large end of the teeth. Remember that whatever the number of holes you turn one way, you must also turn the same number in the opposite direction, otherwise you will get crooked teeth. It can easily be seen that the teeth will be rounded off a little too much unless the cutters have been made for cutting in this way. The other way for cutting the teeth of bevel wheels is to set the Universal Head at the proper angle to the table, and cutting all the teeth in this position, then swivel the head to the same angle in the opposite , direction and pass the cutter through the second time. The greatest care should be used to have the cutter central with the work. I will now show you how to get the exact angle for swiv- eling the Universal Head to cut the teeth of a bevel wheel. In figure 15, which also corresponds with Figure II, but greatly enlarged, we have found the thickness of the tooth, or space at the pitch line on the small end of the gear to be .247 "\ we will also have to know the thickness of the cutter that touches the pitch line as it passes through the large end. In figure 11 I have shown the position of the template of 75° 30' on the pitch line, with the circle A equal to one-fourth THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 145 146 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 04) the radius, thus 39/16" -r- 4 = .890". Now in Figure 15 the pitch line would touch at A 1 , which corresponds with C C, Figure 11. In Figure 15 H H represents the template of 75° 30' in line with the center of the gear and a right angled tri- angle (A 1 O representing the top of the template) with the ver- tex at O as shown, will be the complement of 75° 30' or 14° 30' as shown in the figure. We have found the depth from the pitch line to the bottom of the tooth on the small end of the gear, which is .182", and we know that the depth from the pitch line to the bottom of the tooth on the large end is the difference between the whole depth .539" and .250", or .289" (for 4 pitch); then draw two lines D D on this cutter as shown, which represents the height of the pitch lines at the two ends of the teeth in the bevel gear. We know how wide the space should be on the pitch line at the large end of the tooth, because it is the same as any tooth of a 4 pitch spur gear or .393"; as the cutter was made thin enough for the small end, we will have to find the width across the cutter that touches at the largest pitch diameter by the angles shown. In other words, the object now is to find the distance between 1 1 and H H. As A 1 O is .890" we call it radius, A 1 B the sine and B O cosine of the angle A 1 O B. The cosine of 14° 30' = .968 + and multiplied by the radius (.890") = .8626" for the length of B O. Now we have found the depth from the pitch line to the bottom of the small end of the gear (or the cutter) to be .182" as stated, and the depth on the large end .289"; then the difference between these pitch lines, as represented by the lines D D, Figure 15 = .107"; we next find the distance A 1 B, or the sine of the angle A 1 O B. The sine of 14° 30' = .2503 and multiplied by the radius (.890) = .223"; then subtracting the distance between D D which we said was .107" from .223", we have .116" for the sine C G of the second angle (the line C O being dotted). Now to find what this angle is we divide .116" by .890 and we have .1301 for the sine to a radius of 1 inch, which in the table corresponds to the angle 7° 29 / . We have now found the degrees and minutes of these two THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 147 right angled triangles, and we now want to know the cosines of these two angles in order to know the distance between the two lines 1 1 and H H. The cosine of 14° 30' = .968, and multi- plied by the radius (.890) = .861". The cosine of 7° 29 / = .9915, and multiplied by the radius (.890) = .883. Now the difference between .861" and .883" = .022", and this is the distance between 1 1 and H H. Now draw an angle as shown dotted at the bottom of fig. ure 15, representing the length of the teeth (which in this case is one inch), one end equal in width to the thickness of the space between two teeth at the small end on the pitch line, the other end equal to the width of the space be- tween two teeth on the large end on the pitch line, minus twice the width of space between the lines I I and H H ; this will be the correct angle for cutting the teeth of this bevel gear. The angle for cutting any bevel gear or mitre wheel can also be found in a similar manner. SPIRAL GEARS. Figure 16 shows the manner of finding the correct diame- ter of gears when they are to be cut spiral and with regular standard cutters. Find the diameter oi the gear on the pitch line in the usual manner and multiply it by the secant of the angle you wish to cut it, then add two pitches in the usual way. Thus, suppose you want a spiral gear of 8 pitch 24 teeth, or 10 pitch 30 teeth. The diameters for regular spur gears would be 3" diameter on the pitch line. But if you want them cut at an angle of 10°, then multiply the secant of 10° = 1.0154 by 3", and the result, 3.046", will be the correct diameter on the pitch line; if 8 pitch, add two 8ths to this, making 3.296" for the diameter of the outside. If the same gear was to be 10 pitch, then add two lOths, making it 3.246" outside diameter. Now, if the same gear was to be cut at an angle of 23° 10', then the secant of 23° 10' = 1.0877, multiplied by 3" =3.263" for the diameter on the pitch line. 148 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 149 If they were to be cut at an angle of 45°, then the secant of 45° = 1.414 X 3" = 4.242" for the diameter at the pitch line. Suppose you want a spiral gear of 4 pitch, 36 teeth and 10° angle. 36 -H 4= 9, this multiplied by the secant of 10° = 9.1386" for the pitch diameter and 2/4ths added = 9.6386 r/ for the outside diameter of the gear. It sometimes happens, however, that we have a standard size blank, say 2%" diameter on the pitch line, and we want it to have 22 teeth ; this would be right for an ordinary spur gear of 8 pitch. But if we should want to cut this gear at an angle of 45° (spiral), then we would have to make a special cutter, <6* A3 / 2,2.751 -y 150 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. unless it would suit some ether pitch. To find the thick- ness of the cutter on the pitch line when the diameter of the gear (on the pitch line) and the number of teeth is given, multiply the cosine of the angle you wish to cut it by the diameter of the gear on the pitch line, and this again by 3.1416, then divide the product by twice the number of teeth, and the result will be the thickness of the tooth on the pitch line, thus : In the example, figure 17, suppose we have a gear blank 2%f / diameter on the pitch line, and we want the teeth to be cut spiral at an angle of 45°, 22 teeth. The cosine of 45° = .707, and .707 X 2.75" = 1.9442", and this again multiplied by 3.1416 = 6.105"; then 6.105" -r- 44 = .138 + " for the thick- ness of the cutter on the pitch line. This would be nearly right for a cutter of 11 pitch, and, when the difference is so slight, we can sometimes make the gear a little larger or smaller to suit circumstances. Thus, suppose instead of mak- ing it 11 pitch, which is a little thicker than in the example, we make the blank to suit a 12 pitch cutter, which is about .007" thinner, then, as there are 22 teeth and 22 spaces, a 12 pitch cutter being .131" thick on the pitch line, we multiply .131 by 44 = 5.764"; divide this by 3.1416 = 1.834" which multiplied by the secant of 45° (1.414) = 2.593", or the diame- ter. Suppose a party came into the shop and ordered a pair of spiral gears, one of the gears to be twice as large as the other to connect or drive two spindles that were 6" apart, and parallel, and the teeth to be cut at an angle of 10° from their axis. What will be the thickness of the cutters at the pitch line? If the spindles are 6" apart, and one twice as large as the other, then the radius of one gear will be 2" and the other 4", or the diameters 4" and 8". The cosine of 10° = .9848, and multiplied by 8" (the large wheel) = 7.878", and this again multiplied by 3.1416 = 24.738"; this divided by twice the number of teeth in the wheel will be the thickness of the cutter on i he pitch line. Thus: suppose therj are 48 teeth THE MACHINIST AND TOOT, MAKEE's INSTRUCTOR. 151 on the wheel, then 24.738 -7- 96 = .257 + ", the answer; and this, of course, will be the thickness of cutter for the small wheel also. Now as these shafts are to run parallel, then one of the wheels will be cut right hand and the other left hand. They , of course, will be of the same angle, 10°, but the pitch of the large wheel will be twice as great as the small one. In cutting the teeth of spiral gears the angle is usually given, but sometimes, instead of the angle, the number of inches to one turn is given ; thus, suppose you have a spindle 2J£" diameter, and you want to mill grooves on it that will have, say one turn in 30". In doing work like this, it is nec- essary that we know the proper angle to swing the table, which is done as follows : Multiply the diameter by 3.1416 and divide the product by the number of inches to one turn of the groove, which will give the tangent. In the table of tangents the angle can be found. Thus, 2.5X3.1416 = 7.850; 7.850 -r- 30 = .2617, the tangent, which, in the table corresponds to the angle 14° 40', the answer. In cutting spiral gears, standard cutters (as regards shape) can generally be used, excepting when the angle is very great and the wheel is very small. In the latter case the teeth will be rounded off too much. If the cutter is large in the diame- ter at the same time, it will also increase the trouble. Special cutters should be made for this class of work if accuracy is required. It must be remembered that in milling most kinds of work when the angle is very great, that the shape of the groove will not be the same as the cutter that produced it, except when using end mills. In this case the table is not set over, but end mills, if small, are too slow and consequently expensive. WORM GEARING. In making worms and worm wheels it is more convenient to do so by circular pitch than diametral pitch, for the reason that a worm is nothing more or less than a screw with a special 152 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. shape of thread, (29° angle, 14^° each side). As all lathes are geared to cut so many threads per inch, for instance 4 threads per inch, which means X 7/ pitch, then the worm wheel must also be cut to suit it, and for this reason we use circular pitch for this class of work. If you want to cut a worm and worm wheel with 9 threads per inch, dividing 1 by 9, the result .111 + " would be the pitch of both the worm and worm wheel, and dividing by 2, the re- sult, .0555 // , will be the thickness of the thread and also the tooth on the pitch line. Most parties in ordering worms and worm wheels will say 3 pitch, 4 pitch, 5 pitch, etc., but they simply mean so many threads per inch (called linear pitch), and means so many threads measured in a straight line. Figure 18 represents a worm and worm wheel of %" cir- cular pitch; the wheel has 84 teeth. When the distance be- tween centers should be some even figure, say 6", 8", etc., we make the worm wheel the right diameter to suit the pitch and number of teeth required, and then the worm is made of the proper size to " fill out," as you would call it. In the figure the distance between centers is not import- ant, and I have shown the worm to be 4" outside diameter. As the wheel has 84 teeth and is %" pitch, we multiply % or .375 by 84, and we have 31.5 inches for the circumference, which, divided by 3.1416 = 10.0267 inches for the diameter on the pitch line, as shown at K ; we find the working depth for %" pitch to be . 238", and one-half of this or .119" added to each side will make the worm wheel 10.265" diameter at the throat, as shown on the line H. This is the only thing about worm wheels of much importance, that is, to get the correct diameter at the throat. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 153 154 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. WORMS AND WORM WHEELS. a W THICK- WIDTH CIRCU « z DIAME- NESS OF WORKING WHOLE OF WIDTH OF LAK a S TRAL TOOTH ON DEPTH OF DEPTH OF THREAD THREAD PITCH. PITCH. PITCH LINE. TOOTH. TOOTH. TOOL AT END. AT TOP. 8" J6 1.0472 1.5000 1.9098 2.0598 .9285 1.0061 m 4/11 1.1423 1.375 1.7506 1.8881 .8511 .9223 w* 2/5 1.2566 1.250 1.5915 1.7165 .7737 .8384 2M 4/9 1.3962 1.125 1.4324 1.5449 .6963 .7545 2 K 1.5708 1.0 1.2732 1.3732 .6190 .6707. m 8/15 1.6755 .9375 1.1936 1.2874 .5803 .6288 i% 4/7 1.7952 .875 1.1141 1.2016 .5504 .5869 m 8/13 1.9332 .8125 1.0345 1.1158 .5029 .5449 iy 2 % 2.0944 .750 .9549 1.0299 .4642 .5030 Ws 8/11 2.2847 .6875 .8754 .9441 .4255 .4611 m 4/5 2.5132 .625 .7958 .8583 .3870 .4192 V/8 8/9 2.7925 .5625 .7162 .7724 .3482 .3770 1 1 3.1416 .500 .6366 .6866 .3095 .3353 15/16 11/15 3.3510 .4687 .5968 .6437 .2902 .3144 % 11/7 3.5903 .4375 .5570 .6008 .2710 .2934 13/16 13/13 3.8664 .4062 .5173 .5579 .2514 .2724 3/4 i>6 4.1888 .375 .4774 .5149 .2321 .2515 11/16 15/11 4.5694 .3437 .4377 .4720 .2130 .2305 X 13/5 5.0264 .3125 o3978 .4290 .1935 .2096 9/16 17/9 5.5850 .2812 ,3581 .3862 .1741 .1886 % 2 6.2831 .250 .3183 .3433 .1550 .1676 7/16 2 2/7 7.1806 .2187 .2788 .3006 .1353 .1466 2/5 2^ 7.8540 .200 .2546 .2746 .1242 .1342 % 2% 8.3776 .1875 .2387 .2575 .1160 .1257 Ks 3 9.4248 .1666 .2122 .2289 .1032 .1116 5/16 3 1/5 10.0528 .1562 .1989 .2145 .0967 .1048 2/7 3^ 10.9955 .1429 .1819 .1962 .0885 .0956 M 4 12.5663 .125 .1591 .1716 .0799 .0838 2/9 43^ 14.1371 .1111 .1415 .1526 .0686 .0741 1/5 5 15.7079 .100 .1273 .1372 .0620 .0670 3/16 5% 16.7552 .0937 .1193 .1287 .0580 .0628 1/6 6 18.8496 .0833 .1061 .1144 .0515 .0556 1/7 7 21.9912 .0714 .0910 .0981 .0442 .0478 1/8 8 25.1327 .0625 .0795 .0858 .0386 .0419 1/16 16 50.2654 .0312 .0398 ,0429 .0193 .0209 This table refers to single threads only. If double threads are required divide by 2. If triple by 3, and if quadruple by 4, etc. THE MACHINIST AND TOOE MAKER' S INSTRUCTOR. 155 Now, if it was necessary that the distance between centers should be say 6", then we would take one-half of the diameter or the radius of the worm wheel, as shown on the pitch line at G = 5.0133", and subtracting it from 6" we would have .9867" for one-half the diameter of the worm on the pitch line, or 1.973" for the whole diameter on the pitch line, and adding .238" we would have 2.211" for the outside diameter of the worm. The exact diameter of any worm wheel at the largest part as shown on the line I, although not very important, can easily be found if we know the angle of the sides of the teeth; in the figure the angle is 30°, as shown. For convenience I have shown a right angled triangle at the center of the worm A O B; the length O A is known, be- cause it is one-half of the diameter of the worm, less the work- ing depth (.238") of the tooth ; then 2" — .238 + " = 1.761" for the radius A O ; the cosine of 30° = .866, which multiplied by the radius 1.761 = 1.525", for the distance O B. Now we know that from the center of the worm to the throat of the worm wheel it is 2" — .238", or 1.761"; then subtracting 1.525" from 1.761" = .236" in a vertical direction from B to the throat of the wheel at the center ; then twice this, or .472", added to 10.265" = 10.737" for the largest diameter of the worm wheel, as shown on the line I. The dimensions of worm threads and the width of tools for cutting them are found in the following manner: In the example the thread is %" circular pitch, and divid- ing 3.1416 by %, or .375, we have 8.378 for the diametral pitch; then dividing 1 by 8.378 = .1193" for one half of the working depth, or .238", for the working depth of the tooth or thread ; as the pitch is %" ', then the tooth and also the space is one-half or 3/16" thick on the pitch line ; one-tenth of 3/16", or .187" = about .019", and this added to twice .119" = .257", for the whole depth of any worm, worm wheel, or spur wheel of %" circular pitch. The width of the top of thread and also the bottom of the space is found in the following manner : 156 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. Figure 18% represents the side of this %" pitch tooth greatly enlarged, shown at A A. The angle of worm threads, as explained, is 14%° on each side, and drawing a line perpen- dicular to the pitch line, as shown, we produce a right angled triangle on each side of the pitch line, with the radius given. Then, as the tangent of 14%° = .2586, we multiply this by .119", and the re- sult, .0307", is the distance B B, and twice this, or .0615, subtracted from the thickness of the tooth on the pitch line (.1875)" = .125" for the width of thread on top. The tangent, .2586, multiplied by .138", the distance from the pitch line to the bottom of the tooth, thus, .2586 X .138" = .0356 for the distance C C; twice this, or .0713", subtracted from the thickness of the tooth on the pitch line = .116", nearly, for the width of space on the bottom, or the width of the tool for cutting the thread, etc., for a worm of %" circular pitch, The dimensions of any worm thread can be found in the same manner. In cutting the teeth of worm wheels they should first be driven on an arbor, and the table set over to the proper angle (corresponding to the angle of the worm thread), then select an ordinary spur gear cutter, not much, if any, larger in the diameter than the worm, and a little thinner on the tooth than THE MACHINIST AND TOOl, MAKER' S INSTRUCTOR. 157 the thread of the worm, and placing the worm wheel directly under the center of the cutter, raise the wheel up into the cutter sufficiently to cut it two-thirds or more of the whole depth, depending upon the size and shape of cutter, and also whether the angle is correct. The teeth are roughed out only in this way by indexing; after which a hob, made like the worm, and with a diameter equal to that of the worm, plus twice the clearance added, is placed on the arbor of the milling machine, with the table set back to zero (right angled to the cutter arbor) ; the dog is then taken off the worm wheel arbor, and the wheel is then raised up into the hob as before. The hob will revolve the worm wheel while it is gradually raised to the proper depth. Now let us see what the angle should be in this worm wheel. As the worm is 3.761" diameter on the pitch line, then the circumference would be about 11.809", and dividing the pitch by the circumference, thus, %" = .375 -r- 11.809" = .0317" for the tangent (this will be more fully explained here- after). This tangent corresponds to the angle 1° 49", which is the correct angle to swivel the table for roughing out the worm wheel. Remember to take the circumference on the pitch diame- ter of the worm, and not the outside diameter. The angle for roughing out any worm wheel is found in a similar manner. It sometimes happens that when worm wheels are very small, or of a very coarse pitch, that it is necessary when com- mencing to hob the wheel (after it has been roughed out) to help revolve it by hand until it has made two or three turns, after which it will take care of itself. In figure 19 I have shown an attachment to be 'used on the Universal Milling Machine for cutting the teeth of worm wheels complete, by hobbing, when a great many of one size are wanted, the details of which are as follows: A A is a spindle fitted at one end (shown at the left) to the milling machine spindle, the outer end projecting through the bear- ing jB, which is held sidewise by the* bevel wheel C on one side and a tight fitting nut and collars on the other, as shown. 158 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 159 The outer end of this spindle is also bored taper to receive the shanks of small hobs, the outer end of the hob being supported by the center in the overhanging arm. The bevel wheel C meshes with D, which is screwed on the shaft B. The bear- ing F has a hollow shaft or sleeve passing through it, and is bored to slip over the worm shaft on the Universal Head, which it is to drive. The mitre wheel G is supposed to work on either side of the bearing F, in order to cut both right hand and left hand worm wheels as desired. In the figure I have shown 40 teeth on the driving wheel C, and 30 teeth on the driven D. The pair at the other end are mitres, and can be made any size to suit circumstances (say 30 teeth each). As it takes 40 turns of the worm shaft to make one revolution of the spindle in the Universal Head, then, if we want to cut a worm wheel of 30 teeth, a pair of bevel wheels with 40 teeth for the driver, and 30 for the driven, as shown, will give the required number of teeth. If 32 teeth are wanted on the worm wheel, then a bevel wheel with 32 teeth should be placed at D, and if 50 teeth are wanted, then the wheel at D should have 50 teeth also. Of course, the other wheels need not be changed as regards the number of teeth. The worm wheel to be cut is placed on an arbor between the centers in the usual way, and driven by a dog, the attach- ment being made long enough to suit the work. As the shaft H telescopes the shaft E, we are not limited to an exact length of the arbor on which the work is placed. Both bear- ings F andB are of the same design and better understood by the side elevation shown at I, partly broken off at one of the bearings for want of space. 160 THE MACHINIST AND TOQIL MAKER'S INSTRUCTOR. CHAPTER III. Extracts from Brown & Sharps Manufacturing Co.'s Treatise on Milling Machines. (published by permission.) MHJJNG MACHINES AND SKIU,. No one who has had sufficient experience with the Milling Machine in its various forms to acquire a reasonably clear idea of its capabilities, and who has an opportunity to see the ma- chine in use in the various shops, can fail to see that in many of them it is very imperfectly understood, and that, as a con- sequence, comparatively poor results are obtained from its use — results, we mean, which are very poor compared with those which should be obtained, and are obtained in every case where the legitimate functions of the machine are clearly recognized, and the conditions necessary to its successful operation secured. The Milling Machine intelligently selected or constructed, with reference to the work it is expected to do, provided with well designed and well made special fixtures, where the nature of the work calls for them, and then skillfully handled, is a surprisingly efficient tool, but used as it is being used in many shops today, it is a delusion, a failure and an injury alike to the users, to the builders and to the good name of the mill- ing machines generally. While it is true that there is scarcely a machine tool in use which will yield more satisfactory returns for a given out- lay, when pains are taken to use it in the best possible man- ner, it is also true, we think, that there is no tool in common use, the efficiency of which is so much reduced by careless or ignorant handling and abuse. Considerable intelligence and skill, as well as constant attention, must be bestowed upon the milling machines in order to secure anything like a satis- factory performance from them, either in the quality of the work done or in its quantity. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 161 In some cases this skill and intelligence must be pos- sessed and exercised by the man who actually handles the machine; in other cases by some one who, though he does not actually operate it, supervises its operation, and is responsible for the work done by it. But in any case, the skill, the intelli- gence and the careful attention must be exercised, or the re- sults will be anything but satisfactory. We hear a great deal about the comparatively cheap labor required to do milling machine work, and it is evident that too many shop proprietors have concluded from this that about all that is necessary to do such work is to buy the ma- chine, hire a boy to run it, have him shown how for an hour or so by one of the lathe hands, and then let the boy and the machine work out their own salvation. No greater mistake could possibly be made, and it is in such a shop that a milling machine man findc: the machine working often at less than half of its capacity, with an apology for a cut- ter, ground by hand in every shape but the right one, two or three only of its superabundant ieeth touching the work, and they, with a distinct thump and knock, indicating anything but a real cutting action, while the boy stands by, and occa- sionally — when it occurs to him to do so — squirting a few drops of black lubricating oil onto the chips with which the spaces between the teeth are tightly jammed. The proprietors of such shops are not usually very enthusiastic regarding the use of milling machines, and it would be a wonder if they were. Where a Universal Milling Machine is used upon tool work or for other purposes requiring a constant change from one job to another, it is a mistake to suppose that there is economy in the employment of a boy or cheap man to operate it. Many of those who think they are saving money in that way would be greatly surprised to see the work turned out from such machines by good mechanics who thoroughly understand them, and are capable of earning good wages upon them. Experience has proved that it pays as well to put first- 162 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. class mechanics upon such machines as upon any other ma- chine tools. Where milling machines are used for regular manufactur- ing operations, and the same cycle of movements is to be repeated for a large number of pieces, boys or men who are not skilled mechanics, answer every purpose ; but the skilled supervision must be there, and it must be seen to that the machine is as well taken care of, the cutters as well made and ground, and in fact, everything as well done as though a good mechanic actually operated the machines. In fact, in the shops in which the best results are obtained from the use of milling machines in regular manufacturing operations, all changes of the machines from one job to another, all adjust- ments, and the grinding and replacing of cutters, are done by, or at least under the direct supervision of, a skilled mechanic, responsible for the work of the machines, and who thoroughly understands and appreciates them. In this way only can the full benefits of the machine be realized. It is far too common to go into the tool room and find a splendid Universal Milling Machine standing idle, while per- haps two or three men are doing at the shaper,planer or vise, jobs which could be done by an expert milling machine hand in one- fourth down to one-tenth of the time and a great deal better. One fault, which is far more common than would readily be believed in some quarters, is a failure to recognize the fact that a milling machine necessarily calls for some sort of a machine for grinding cutters, and that a machine upon which cutters are used, that are ground by holding the edges one after the other against an emery wheel by hand, is at a decided disad- vantage, and will do no work which either in quality or cost will make a favorable showing when compared with that which is done by properly ground cutters. It should be much more generally recognized that in mill- ing machine practice, as in other things, there is a right way and a wrong way, and that skilled, intelligent labor pays best. When these facts are more generally recognized, it will be better for both the builders and for the users of the machines. THE MACHINIST AND TOOIi MAKER'S INSTRUCTOR. 1G3 For most purposes, it is well to have mills or cutters as small in diameter as the work or their strength will admit. The reason is shown by figure 41. Suppose the piece IDCJE is to be cut from I J to D B. If the large mill A is used, it will 164 THE MACHINIST AND TOOI, MAKER'S INSTRUCTOR. strike the piece first at I, when its center is at K, and will finish its cnt when its center is at M. The line G shows how far the mill must travel to cut off the stock I J D K. If the small mill B is used, however, it travels only the length of the line H. It can also be seen that a tooth of B travels through a shorter distance between the lines D E) and I J than a tooth of A. This is true of all ordinary work, or where the depth of cut I D is not more than half the diameter of the small mill. The advantage of small mills has been illustrated in our own works where a difference of y 2 an inch in the mills has made a difference of 10 % in the cost of the work. In short, small mills do more and better work, cut more easily, keep sharp longer and cost less than large mills. When it is possible the mill should be wider than the work, and the hole in the mill should be as small as the strength of the arbor will admit. The stock around the hole, however, should not be less than % of an inch thick. A mill is not necessarily too soft because it can be scratched with a file, for sometimes when cutters are too hard, or brittle, and trouble is caused by pieces breaking out of the teeth, they can be made to stand well and do good work by starting the temper. Of late years mills have been made with coarser teeth than formerly; the advantages being more room for the chips and less friction between the teeth and the work. When the teeth are so fine that the mill drags, or the stock is powdered, the mill heats quickly and does not cut freely. The friction may also be reduced, especially in large mills taking heavy cuts, by nicking or cutting away parts of the teeth, which breaks the chips and allows heavier cuts and feeds to be taken. Knowing the conditions under which a mill is to be used, in our own practice, we modify the number of teeth as seems expedient, usually making the special mills coarser in pitch than the stock mills, for our observation indicates that there are more mills with too many teeth than with too few. But sometimes we relatively increase the number of teeth, as for THE MACHINIST AND TOOL, MAKER'S INSTRUCTOR. 165 instance, large mills, in some cases, can advantageously be designed to have more than one tooth cutting all the time on broad surfaces and in deep cuts. The adoption of the most suitable cutting angle should receive the same close attention that is now universally be- stowed upon the ordinary tools for turning and plan- ing. But in practice, while in many instances adopting the angle according to the material to be used, yet taking into con- sideration all the conditions of using and caring for the cutters, we have generally found it satisfactory to make the cutting edges of the teeth radial. The relief or clearance of mills we think should usually be about three degrees, and the land at the top of the teeth from .02 inches to .04 inches wide before the clearance is cut or ground. Mills to cut grooves should be hollowed about five one- thousandths in one inch for clearance; that is, a grooving mill should be about one one-hundreth of an inch thinner at one inch from its edge or circumference than it is at the edge. Our grooving mills are given a limit of two one-thous- andths in thickness. Mills made to exact thickness are very ' expensive. In cutting grooves that are to have some parts of their sides nearly or quite parallel, it is well to leave con- siderable stock for the finishing cut, as mills, like taps, do better work when they get well into the stock. A dull mill wears away rapidly and does poor work. Ac- cordingly care must be taken to keep mills sharp. In sharp- ening them it is necessary to be very careful that the temper should not be drawn. The emery wheel should be of the proper grade as to hardness and as to the size of the emery. The wheel should be soft enough so that it can be easily scratched with a pocket-knife blade, and the emery should not be finer than 90, nor coarser than 60. As a rule, the coarser and softer the wheel, the faster it should run, although the periphery speed should not exceed 5,000 feet per minute. A wheel of the proper grade should be used with the face, not to exceed %" wide. If the wheel glazes, the temper of the 166 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. cutter will be drawn. In such a case, if the wheel is not altogether too hard, it can sometimes be remedied b}' reducing the face of the wheel to about Jg", or by reducing the speed, or by both. Before using, a wheel should be turned off so that it will run true. A wheel that glazes immediately after it has been turned off, can sometimes be corrected by loosening the nut and allowing the wheel to assume a slightly different posi- tion, when it is again tightened. Another method of preventing a wheel's glazing is to use a piece of emery wheel, a few grades harder than the wheel in use, on the face of the wheel, whereby the cutting surface of the wheel is made more open and less apt to glaze. Mills that have their teeth ground for clearance are par- ticularly apt to have their temper drawn in sharpening, espe- cially at the edge of the teeth, and often when the temper has been drawn and the teeth are polished, they will look as usual after being ground. Oil is used in milling to obtain smoother work, to make the cutters last longer, and where the nature of the work re- quires, to wash the chips from the work or from the teeth of the cutters. It is generally used in milling a large number of pieces of steel, wrought iron, malleable iron ? or tough bronze. When only a few pieces are to be milled it frequently is not used, and some steel castings are milled without oil; also in cut- ting cast iron it is not used. For light, flat cuts it is put on the cut- ter with a brush, giving the work a thin covering like varnish; for heavy cuts it should be led to the cutter from the drip can sent with each machine, or it should be pumped upon or across the cutter in cutting deep grooves ; in milling several grooves atone time, or indeed, in milling any work, where, if the chips should stick, they might catch between the teeth and sides of the groove and scratch or bend the work. Generally we use lard oil in milling, but any animal or fish oils may be used. The oil may be separated from the chips by a centrifugal separator, or by the wet process, so that a large amount may be used with but little waste. THE MACHINIST AND TOOL MAKER S INSTRUCTOR. 167 Some manufacturers prefer to mix mineral oil with lard or fish oil, and state the mixture is less expensive and works well. Prof. J. B. Denton has made experiments with mixtures and thinks that mineral or coal oil can be advantageously used. An excellent lubricant to use with a pump is by mixing together and boiling for one-half hour, 1^ pound sal soda, % pint lard oil, J^ pint soft soap, and water enough to make ten quarts. There is a difference of opinion as to whether the work should be moved against the cutter, or with it. But in most cases our experience and experiments show it is best for the work to move against the cutter, as shown at A, Figure 413^. When it moves in this way the teeth of the cutter, in com- mencing their work, as soon as the hard surface or scale is once broken, are immediately brought into contact with the softer metal, and when the scale is reached it is pried or broken off. Also when a piece moves in this way the cutter cannot dig into the work, as it is liable to do when the bed is moved in the direction indicated at B. When a piece is on the side of a cutter that is moving downwards, the piece should, as a rule, have a rigid support and be fed by raising the knee of the machine. Some work, however, is better milled by moving with the cutter. For example : To dress both sides of a thick piece (D) with a pair of large straddle mills, it might be well to move the piece toward the left, as the cutters then tend to keep it down in place instead of lifting it. 168 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. Again, in milling deep slots with a thin cutter or saw, it may be better to move the work with the cutter, as the cutter is then less likely to crown sidewise and make a crooked slot. When the work is moving with the cutter the table gib screws must be set up rather hard, for if the work moves too easily the cutter may catch, and the cutter orwork be injured. A counter weight to hold back the table is excellent in such milling. It is impossible to give definite rules for the speed and feed of cutters, and what is here said is only in the way of suggestions. The judgment of the foreman or man in charge of the machine should determine what is best in each instance. The average speed on wrought iron and annealed steel is perhaps forty feet a minute, which gives about sixty turns a minute for mills 2J£ inches in diameter. The feed of the work for this surface speed of the mill can be about 1% inches a minute, and the depth of cut say 1/16 of an inch. In cast iron a mill can have a surface speed of about fifty feet a minute, while the feed is 1% inches a minute, and the cut 3/16 of an inch deep ; in tough brass the speed may be eighty feet, the feed as before, and the chip 3/32 of an inch. As a small mill cuts faster than a large one, an end mill, for example, % inch in diameter, can be run about 400 revolu- tions with a feed of 4 inches a minute. For example of what may regularly be done under suita- ble conditions, we may mention that cutters 2^ inches in diameter used in cutting annealed cast iron in our works are run at more than 200 turns, or at a surface speed of more than 125 feet, while the work is fed more than 8 inches a minute. The cuts are light, not more than 1/32 of an inch deep, and the work is short, from %. inch to 1 inch long. Two side mills 5 inches in diameter running 50 turns a minute, dress both edges of cast iron bars % of an inch thick, with a feed of more than 4 inches a minute. An English authority, Mr. Geo. Addy, gives as safe speeds for cutters of 6 inches diameter and upwards : Steel, 36 feet per minute, with feed oi y?." per minute. THE MACHINIST AND TOOT, MAKER' S INSTRUCTOR. 169 Wrought Iron, 48 feet per minute with feed of 1" per minute. Cast Iron, 60 feet per minute with feed of 1%" per minute. Brass, 120 feet per minute with feed of 2% 7/ per minute. And he gives as a simple rule for obtaining the speed: Number of revolutions which the cutter spindle should make when working on cast iron = 240 divided by the diameter of the cutter in inches. Mr. John H. Briggs, another English authority, states for cutting wrought iron with a milling cutter, taking a cut of one inch depth, which was a different thing from mere surface cutting, a circumferential speed of from 36 to 40 feet per min- ute was the highest that could be obtained with due considera- tion to economy, and to the time occupied in grinding and changing cutters; the feed would be at the rate of % inches per minute. Upon soft, mild steel, about 30 feet per minute was the highest speed, with 34 inch depth of cut and % inch feed per minute. Upon tough gun metal, 80 feet per minute, with J£ inch depth of cut and %inch feed. For cutting cast iron gear wheels from blanks pre- viously turned, and using in this case comparatively small milling cutters of only 3J£ inches diameter, the speed was 26J^ feet per minute, with J£ inch depth of cut and % inch feed per minute. Slotting cutters may often be run at a higher speed than other cutters of the same diameter, but with a wider face. Angular cutters must in some instances be used with a fine feed to prevent breaking the points of the teeth. Castings that are to be milled should be free from sand. They should be well pickled, and in some cases it is an advan- tage to have them rattled after being pickled. Where they are small and are to be finished rapidly it is also well to have them annealed. Forgings should be free from scale. They can be pickled in ten minutes in one part sulphuric acid and twenty-five parts boiling water, and if then rinsed in boiling water, they will dry before becoming rusty. 170 INDEX TABLE. NUM- NUMBER OF NUM- NUMBER OF BER OF HOLES IN NUMBER OF BER OF HOLES IN NUMBER OF DIVI- THE INDEX TURNS OF THE DIVI- THE INDEX TURNS OF THE SIONS. CIRCLE. CRANK. SIONS. CIRCLE. CRAINK. 2 ANY 20 65 39 24-39 3 39 13 13-39 66 33 20-33 4 ANY 10 68 17 10-17 5 ANY 8 70 49 28-4£ 6 39 6 26-39 72 27 15-27 7 49 5 35-49 74 37 20-37 8 ANY 5 75 15 8-15 9 27 4 12-27 76 19 10-lfr 10 ANY 4 78 39 20-39 H 33 3 21-33 80 20 10-20 12 39 3 13-39 82 41 20-41 13 39 3 3-39 84 21 10-21 14 49 2 42-49 85 17 8-17 15 39 2 26-39 86 43 20-43 16 20 2 10-20 88 33 15-33 17 17 2 6-17 90 27 12-27 18 27 2 6-27 92 23 10-23 19 19 2 2-19 94 47 20-47 20 ANY 2 95 19 8-19 21 21 1 19-21 98 49 20-49 22 33 1 27-33 100 20 8-20 23 23 1 17-23 104 39 15-39 24 39 1 26-39 105 21 8-21 25 20 1 12-20 108 27 10-27 26 39 1 21-39 110 33 12-33 27 27 1 13-27 115 23 8-23 28 49 1 21-49 116 29 10-29 29 29 1 11-29 120 39 13-39 30 39 1 13-39 124 31 10-31 31 31 1 9-31 128 16 5-16 32 20 1 5-20 130 39 12-39 33 83 1 7-33 132 33 10-33 34 17 1 3-17 135 27 8-27 35 49 1 7-49 136 17 5-17 36 27 1 3-27 140 49 14-49 37 37 1 3-37 144 18 5-18 38 19 1 1-19 145 29 8-29 39 39 1 1-39 148 37 10-37 40 ANY 1 150 15 4-15 41 41 40-41 152 19 5-19 42 21 20-21 155 31 8-31 43 43 40-43 156 39 10-39 44 33 30-33 160 20 5-20 45 27 24-27 164 41 10-41 46 23 20-23 165 33 8-33 47 47 40-47 168 21 5-21 48 38 15-18 ]70 17 4-17 49 49 40-49 172 43 10-43 50 20 16-20 180 27 6-27 52 39 30-39 184 23 5-23 54 27 20-27 185 37 8-37 55 33 24-33 188 47 10-47 56 49 35-49 190 19 4-19 58 29 20-29 195 39 8-39 60 39 26-39 196 49 10-49 62 31 20-31 200 20 4-20 64 16 10-16 205 41 8-41 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 171 INDEXING. The first office of the indexing head stock or spiral head is to divide the periphery of a piece of work into a number of equal parts, and in connection with the foot stock; it also enables the milling machine to be used for work sometimes done on planer centers and on gear cutting machines. As the index spindle may be revolved by the crank, and as forty turns of the crank make one revolution of the spin- dle, to find how many turns of the crank are necessary for a certain division of the work, or, what is the same thing, for a certain division of a revolution of the spindle, forty is di- vided by the number of the divisions which are desired. The rule then may be said to be, divide forty by the number of the divisions to be made and the quotient will be the number of turns, or the part of a turn, of the crank, which will give each desired division. Applying this rule — to make forty divisions, the crank would be turned completely around once to obtain each division; or, to obtain twenty divisions, it would be turned twice. When, to obtain the necessary divisions, the crank has to be turned only part of the way around, an index plate is used. For example: — If the work is to be divided into eighty divisions, the crank must be turned one- half way around, and an index plate with an even number of holes in one of the circles would be selected, it being necessary only to have two holes opposite to each other in the plate. If the work is to be divided into three divisions an index plate should be selected which has a circle with a number of holes that can be divided by three, as fifteen, eighteen, twenty- seven, etc., the numbers on the index plates indicating the number of holes in the various circles. The sector is of service in obviating the necessity of counting the holes at each partial rotation or turn of the crank, and to illustrate its use it may be supposed that it is desired to divide the work into 144 divisions. Dividing 40 by 144 the result, 5-18, shows that the index crank must be moved 5-18 of a turn to obtain each of the 144 divisions. An index plate with a circle containing eighteen holes or a. 172 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. multiple of eighteen, is selected and the sector is set to meas- ure off five spaces, or the corresponding multiple, ten spaces for example, in a circle with thirty-six holes. In setting the sector it should be remembered that there must always be one more hole between the arms than there are spaces to be counted or measured off. The required number of turns of the crank for a large number of divisions may be readily ascertained from the accompanying index tables, which correspond to the Brown & Sharpe Manufacturing Company's Milling Machines. If the angle of elevation of the spiral head spindle is changed during the progress of the work, the work must be rotated slightly to bring it back to the proper position, as when the spindle is elevated or depressed the worm wheel is rotated about the worm, and the effect is the same as if the worm were turned in the opposite direction, CUTTING SPIRALS WITH UNIVERSAL MILLING MACHINES. The indexing head stock or spiral head, as indicated in connection with the descriptions of the Universal Milling Machines, is used for cutting spirals, the flutes of twist drills, for example, as well as for indexing or dividing. A positive rotary movement is given to the work -while the spiral bed is being moved lengthways by the feed screw, and the velocity ratios of these movements are regulated by four change gears, shown in position in Figure 42 and known as the gear on worm or worm gear, first gear on stud, second gear on stud and gear on screw or screw gear. The screw gear and first gear on stud are the drivers and the others the driven gears. Usually those gears are of such ratio that the work is advanced more than an inch while making one turn, and thus the spirals, cut on milling ma- chines, are designated in terms of inches to one turn, rather than turns, or threads per inch; for instance, a spiral is said to be of 8 inches lead, not that its pitch is 1-8 turn per inch. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 173 The feed screw of the spiral bed has four threads to the inch, and forty turns of the worm make one turn of the spiral head spindle; accordingly, if change gears of equal diameters are used, the work will make a complete turn while it is moved lengthways 10 inches; that is, the spiral will have a lead of 10 inches. But this lead is practically the lead of the machine, as it is the resultant of the action of the parts of the machine that are always employed in this work, and it is so regarded in making the calculations used in cut- ting spirals. In principle, these calculations are the same as for the change gears of a Screw Cutting Lathe. The compound ratio of the driven to the driving gears equals in all cases the ratio of the lead of the required spiral to the lead of the 174 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. machine. This can be readily understood by changing the diameters of the gears. Gears of the same diameter produce, as explained above, a spiral with a lead of 10 inches, which is the same lead as the lead of the machine. Three gears of equal diameter and a driven gear double this diameter produce a spiral with a lead of 20 inches, or twice the lead of the machine; and with both driven gears twice the diameters of the drivers, the ratio being compound, a spiral is produced with a lead of 40 inches, or four times the machine's lead. Conversely, driving gears twice the diameter of the driven produce a spiral with a lead equal to J4 the lead of the machine or 2y 2 inches. EXPRESSING THE RATIOS AS FRACTIONS. Driven Gears Lead of Required Spiral. = ; or, as the product of Driving Gears Lead of Machine each class of gears determines the ratio, the head being double-geared, and as the lead of the machine is ten inches Product of Driven Gears Lead of Required Spiral Product of Driving Gears 10 that is, the compound ratio of the driven to the driving gears may always be represented by a fraction whose nu- merator is the lead to be cut and whose denominator is 10; or, in other words, the ratio is as the required lead is to 10; that is, if the required lead is 20 the ratio is 20:10: or, to ex- press this in units instead of tens, the ratio is always the same as one-tenth of the required lead is to one. Fre- quently this is a very convenient way to think of the ratio; for example, if the ratio of the lead is 40, the gears are 4:1. If the lead is 25, the gears are 2.5:1, etc. To illustrate the usual calculations, assume that a spiral of 12-inch lead is to be cut. The compound ratio of the driven to the driving gears equals the desired lead divided THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 175 by 10, or it may be represented by the fraction 12-10. Re- solving this into two factors to represent the two pairs of 12 3 4 •change gears, — = — X — • Both terms of the first factor are 10 2 5 multiplied by such a number (24 in this instance) that the resulting numerator and denominator will correspond with the number of teeth of two of the change gears furnished with the machine (such multiplications not affecting the value 3 24 72 of a fraction), — X — = — • The second factor is similarly 2 24 48 4 8 32 treated, — X — = — > and the gears with 72 and 32 and 48 and 5 8 40 12 r 72X32 ) 40 teeth are selected, — = < > The first two are the driv- 10 (48X40) en and the last two the drivers, the numerators of the frac- tions having represented the driven gears, and the 72 is placed as the worm gear, the 40 as the first on stud, 32 the second on stud and 48 as the screw gear. The two driving gears might be transposed and the two driven gears might also be transposed without changing the spiral. That is, the 72 could be used as second on stud and the 32 as the worm gear, if such an arrangement was more convenient. From what has been said, the rules are plain. Note the ratio of the required lead to 10. This ratio is the compound ratio of the driven to the driving gears. Example: — If the lead of a required spiral is 12 inches, 12 to 10 will be the ratio of the gears, or divide the required lead by 10 and note the ratio between the quotient and 1. This ratio is usually the most simple form of the compound ratio of the driven to the driving gears. Example: — If the required lead is 40 inches the quotient, 40-^-10 is 4, and the ratio 4 to 1. Having obtained the ratio between the required lead and 10 by one of the preceding rules, express the ratio in the form of a 176 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. fraction, resolve this fraction into two factors; raise these factors to higher terms that correspond with the teeth of gears that can be conveniently used. The numerators will represent the driven and the denominators the driving gears that produce the required spiral. For example: — What gears shall be used to cut a lead of 27 inches? f 3 16) (9X8) 48X72 27-10=3-2X9-5==^ -X— [ X \ \= ( 2 16 J Ux8j 32X40 From the fact that the product of the driven gears divided by the product of the drivers equals the lead divided by 10, or one-tenth of the lead, it is evident that ten times the product of the driven gears divided by the product of the drivers will equal the lead of the spiral. Hence, the rule: — Divide ten times the product of the driven gears by the prod- uct of the drivers, and the quotient is the lead of the resulting spiral in inches to one turn. For example: — What spiral will be cut by gears with 48, 72, 32 and 40 teeth, the first two being used as driven gears? Spiral to be cut equals 10X48X72 =27 inches to one turn. 32X40 Cuts that have one face radial are best made with an angular cutter, for cutters of this form readily clear the radial face of the cut, and so keep sharp longer and produce a smoother surface than when the radial face is cut in a verti- cal plane with a cutter where the teeth can have no side clear- ance from the work. The setting for these cuts must also be made before swinging the spiral bed to the angle of the spiral. THE UNIVERSAL MILLING MACHINE— Continued. (by the author ) From the preceding method of obtaining spirals as prac- ticed by the Brown & Sharpe Manufacturing Company, we find that the same rule will also apply to the milling ma- chine as constructed by others. For example: — Figure 43 is THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 177 178 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. another style of the Universal Milling Machine, a portion of the index head only being shown. The feed screw A has two threads per inch, but there is a pair of bevel gears as shown at F G that drive the change gears, and F is twice as large as G; these bevel wheels are not to be changed, and B is called the screw gear. Now it can be seen that this gear near B will make four revolutions while the screw moves the table one inch, and as the index crank C makes forty turns to one turn of the spindle D, if all four of the change gears were of the same size, the table of the machine would move through a space of ten inches, while the index crank made 40 turns or the spindle of the Universal head made but one turn, and consequently, the lead of the machine (figure 43) is also ten inches. Now, suppose we want to cut a spiral mill that is to have a lead of one turn in 60 inches; dividing 60 by 10 we have the ratio of the change gears, 6 to 1. Now, if we select for one pair of these gears 12 and 36 or 24 and 72, or any number that is 3 to 1, dividing the ratio 6 by 3=2; then any pair of gears that is 2 to 1 will answer for the second pair. In compound gearing always remember when you find the ratio of the gears wanted, and select one pair, that in order to find the second pair you must divide the ratio of the change gears by the ratio of the first pair selected; that is, we found the ratio of the gears were 6 to 1, and, taking 24 and 72 for the first pair, which is 3 to 1, we divide 6 by 3 and we have 2, which means that any number 2 to 1 will be right for the next pair, and therefore 64 and 32 will answer. Now, we can place the 32 on the screw and 64 on the stud to mesh with this gear, and the 72 on the worm with the 24 in gear with it; or, we can put the 24 on the screw and the 72 in mesh with it and the 64 on the worm with 32 to engage with it; either way will be right. Let us take another example:— Suppose we want to mill a spiral of one turn in 28 inches, what gears should we use? We found the lead of the machine to be 10 inches, therefore dividing 28 by 10 we have 2 8-10, or 2 4-5, for the ratio of the gears. We will now try a pair that is 2 to 1, and see how we come out. Say 20 and 40, or 32 and 64 for the first pair of gears; now, 2 4-5=14-5, and any pair 2 to 1, or one gear THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 179 being twice as large as its mate =2-1, which is equal to 10-5, and dividing 14-5 by 10-5 we have 1 4-10 for the ratio of the next pair, and taking 40 for one gear and multiplying this by 1 4-10, we have 56 for the next gear, so that 64 and 35 will answer for one pair and 56 and 40 for the other; these gears are shown in position in the figure (43). Now let us prove this to be one turn in 28 inches: — As the screw is the driver, so is 40 the driver of 56, and as 32 is on the same stud with 56, then 32 is the driver of 64; then multiplying the product of the two driven gears by 10 and dividing by the product of the drivers, will be the lead of the spiral, thus: 64=Driven 32=Driver 56=Driven 40=Driver 384 1280 320 3584 10 1280)35840(28", the answer, or the lead of spiral 2560 10240 10240 Example :— What spiral can we obtain from 72 and 48, as driven gears, and 28 and 40 as drivers? 72=Driven 28=Driver 48=Driven 40=Driver 576 1120 288 3456 10 1120)34560(30.86", or nearly, the answer. 3360 9600 8960 6400 6720 180 THE MACHINIST AND TOOL MAKERS INSTRUCTOR. ^^K54*w£*Z!*z4£, The next thing is to find the angle to set the machine when cut- ting spirals, which is found in the following simple manner: Find the circumference of the work that is to be cut spiral, no matter what it is; take the outside diameter (except in '^p^case of a spiral gear, and then take the pitch diameter) and divide it by the length of the piece, which will give the tangent of the angle, and, in the table of tangents you will find the angle, or as follows: Figure 44 represents a piece of work that is 28" long and 2%" diameter, that is to have a spiral of one turn in its length. The circumference of 2^"=9.032", and divided by 28=.3225; in the ta- ble of tangents, we find the angle 17° 53' to correspond to this num- ber, which is the correct angle to' swivel the table. If the piece was- V/z" in diameter and the same length, we should then have a circumference of 7.854", nearly, and, di- vided by 28, we should have a tangent of .2805, which would then be 15° 40', as per table. Remember that this is a right angled triangle, or, in other words, the two lines marked circumference and length, as in figure 44, should always be at right angles to each other. A spiral mill 2>4" diameter is large enough for ordinary work and should have from 22 to 24 teeth, and should have a spiral of about one turn in 36". A cutter of this size and number of teeth will have about Yz" solid stock with a one- inch hole. For a spiral mill 3^" diameter I should give about 26 teeth, and one turn in not less than 48 inches; a cutter of this size can be cut coarser than one of V/2" diameter, be- THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 181 cause there will be more stock around the hole. If you want a good spiral mill for roughing purposes (that is, one that will cut fast), after the size has been finished in the lathe, take an ordinary cutting off tool, about one-eighth of an inch wide, with the corners rounded, and cut a thread of about two or three threads per inch in the opposite direction that you intend to mill the spiral. That is, if you wish to have a right hand mill, then you should cut this coarse thread in the lathe left hand. This is the best style of cutters we have for heavy or fast milling. If a spiral mill of about 2" diameter with a solid shank was required I should give it a spiral of about one turn in 36", and should also make it right hand for the reason that, instead of pulling out of the spindle while finishing a cut, the tendency would be to force itself tighter in the spindle, and therefore less danger of marking the work. An angle of 10° to 12° will answer for most any ordinary size of spiral mills. In small end mills the teeth will have to be finer than large ones, to give them greater strength. I should give in end mills of y 2 " and */%' diameter eight teeth each; %", %" and 1" ten teeth each; \%* and l l / 2 " each twelve teeth, etc., and for a 1-34" diameter thin cutter, say 3-16" to y thick, not less than sixteen teeth. If a pair of twin or straddle mills were wanted for gen- eral work, such as milling, bolt heads, etc., I would make them 4 inches diameter, y 2 " thick and not more than 32 teeth, and for a pair of 5" diameter for general work, I should give them 36 teeth and make them Y% face; for a 60° angular cutter, V$* diameter at the large end, I should give it 12 teeth, or, one of the same kind (60°) 2J4" largest diameter, 16 teeth; and one of 3" largest diameter (60°), 20 teeth, etc. Always remember to have the teeth of cutters coarse enough so that in grinding the clearance the emery wheel will not touch the cutting edge of the next tooth. On most cutters the teeth on the periphery can be sharpened with 182 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. wheels as large as 8" to 12" diameter, but on the sides of small cutters sometimes a wheel 2" diameter is too large. Milling cutters larger than 4 inches diameter and one inch face, and larger than 2]/ 2 inches diameter and 4-inch face, should have key ways: y%" wide and 3-32" deep is large enough for most purposes; the key ways should be slightly rounded in the corners to lessen the danger of crack- ing while hardening. I will now show you the shapes of cutters for a variety of purposes, and also the sizes, angles, etc., for ordinary work. I say ordinary, because we sometimes have to make a cutter twice the size or may be only one-half the size for some special class of work that we may have on hand. Fig- ure 45 represents a cutter for cutting the teeth of spiral mills, It is 12° on one side and 40° on the other side, as shown; the side marked 12° should always point to the center of the work, and, if two pieces having different diameters are to be milled, the table with the work will have to be moved in or out to suit that particular size, because setting it for any certain size it will not be right for any other diameter of work. The best way to set the work for this cutter is to raise the table THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 183 until you think it high enough for the cutter when at the proper depth in the work, then, with a short scale or straight edge placed directly under the center of the cutter arbor and resting on the side marked 12°, with the center of tail stock also moved directly under the center of cutter arbor, the scale should point to the center of tail stock. The teeth of spiral mills on top should not be much over 1-32" thick when milled. Figure 46 represents an angular cutter of 60° for cutting the teeth in other cutters. This is generally used for cutting the teeth on the periphery of all ordinary cutters, except when the angles are very great. For instance, if the cutter shown at figure 48 was cut on both sides with a cutter of 60°, the teeth would be easily broken, and for this reason we will have to be governed accordingly. For the various shapes of cutters we use 60°, 65°, 70° and 75° cutters to cut them with, depending, as stated, on the acuteness of the angles. All cutters should have the teeth cut rather coarse when they are strong enough to stand it, for the reason that it will give more room for chips, and this is a very desirable matter with milling cutters. I would give the cutters (figures 45-46) each 26 teeth for the diameter as shown. Figure 47 represents the neatest style of cutters for fluting taps; five sizes of cutters will answer for nearly all sizes. I would make them }i", y 2 " , %", J4" and 1" thick, and all of them 2^" diameter, but the corner of the small one should be rounded very little and the large one about as shown in the figure; the others should have the corners pro- portioned between these two. I would also give them each 24 teeth. The width of land on all regular taps should be 1-32 inch wide to every %" in diameter, as shown at B. This refers to standard sizes only. Figure 48 represents the shape of cutters for cutting the teeth in formed cutters, worm wheel-hobs, etc. Figure 49, represents the best style of milling cutters for milling the teeth in screw machine dies. I would give the former (figure 48) 32 teeth and the latter (figure 49) 24 teeth, for the diameters given. 184 THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. 1 s$ I v V ate THE MACHINIST AND TOOL MAKER S INSTRUCTOR. 185 Figure 50, which is also the same as figure 49, is shown in position for milling a hollow mill for screw machines. After set- ting the cutter in position, as shown, the table should be swiv- eled from about 20° for a y 2 " mill to about 30° for a iy 8 " mill. • For fluting reamers we some- i times use the cutter shown at fig- _ If ure 47, but only when the flutes gj^ are to be very coarse. For hand 50 reamers they should be shaped as shown by the dotted line at A. All milling cutters should have the holes ground after hardening, and for this purpose I always allow five one-thousandths of an inch in the diameter; the sides always should be ground, as otherwise they will not run true. In these illustrations of cutters I have marked the thick- ness and also the diameters and angles as suitable for gen- eral purposes. For small shank milling I use a chuck as shown at figure 51; it is composed of three pieces, A, B, C. The long taper of A is fitted to the machine spindle and bored large enough to suit a 34" shank mill nearly through. The back end is tapped %" and the spindle is bored large enough to admit a rod of that size extending from the back end to hold this chuck from any danger of slipping out and spoiling the work while in operation. These small mills are all made of Cres- cent tool steel (drill rods) and for a y 2 " mill you have only to cut off a piece of l / 2 " stock, say 5 inches long, as this is quite long enough for most purposes; cut the teeth on one end, say l 1 /*" , and, as the mills are parallel, you have the advantage of having them project just long enough for any particular class of work. In order to suit the different sizes, however, you will have 186 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. to have one of these pieces, shown at B, for each size mill. This piece is reamed parallel, as shown by the dotted lines, and turned taper to suit the body of chuck nearly three-fourths of its length, and from there to the front end it is tapered in the opposite direct front, in order to draw the shell B tight on the shank of the mill; this shell B on and the nut C is also bored taper in s sawed in six equal spaces to within say THE MACHINIST AND TOOL, MAKEE's INSTRUCTOR. 187 *4" from the end, each alternately starting from opposite ends, and in this manner the shell will close parallel on the work. The body of the chuck A and also the shell B should be made of tool steel. Figure 52 represents a special chuck that I made several years ago for milling bolt heads. The body of the chuck, which is about 2y 2 ,r through contains six 1*4" holes, a por- tion of which is cut off in the drawing for want of space. The lower portion of the chuck shown in dotted lines is screwed on the spindle of the Universal Head, and, being split, is held firmly in position by a screw. In order to get good re- sults with this chuck it is necessary that the holes should be accurately spaced; this is quite an easy matter, as they can be drilled, bored and reamed by indexing on the milling machine. These holes are 1J4" and we have split bushes of different sizes, so that all sizes of bolts can be milled up to and including V/\" . The bolts are held in position by the screw A and collar B, of which only one is shown. These collars and screws are all fitted in place before the holes C are bored, after which the bottom of B should have about one sixty-fourth of an inch taken off, so that they may clamp hard on the bush or bolt, as the case may be, so that they will not mark the work. In using this chuck the spindle of the Universal Head is placed in a vertical position with the face of the chuck up- right, as shown in the drawing, and six straddle mills about Zy 2 " or 4" diameter are used. The manner of using is as follows: The bolts, six in number, are placed in position with the heads projecting a little above the face of the chuck and the cutters set to just pass over without marking the chuck. The bolts 1, 2 and 3 are now milled on one side as shown by the dotted lines, and one-sixth of a turn is then made to the next bolt, and numbers 1 and 2 will then be milled on two sides, after which another one-sixth of a turn is made, and by this time No. 1 will be finished. At every one-sixth of a turn we can now take out a -finished bolt, and, of course,. 188 THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. / H / ^/f^p / /<^ <-- i i i i 7 r-r-j ' v. ' ' ' > ~~- — ■ — £3 1 ' _ V °* \ \ \ \ \ / ^» ^_ \ \ \ *\ 7 *1_ -j4 _ 7K_Z_ Y a nv CO i* _-^f - ,5o *(. \y THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 189 we also put in a blank every time we take out a finished bolt. Suppose that we want to mill some bolt heads 134" hex- agon (which can be represented by the holes) and the cutters are ground just V^' hub and face alike in width. The next thing to do is to find a spacing collar that will be' the right thickness to separate the two cutters shown at F. As there are six divisions represented by these holes, an angle, as shown, will be 60°, and G E being the radius of this angle, then D E is the sine of the angle of 60°, and the sine of 60°=. 866 to a radius of 1"; then three times this would be the distance D E, or 2.598", and, taking from this the diameter of one of these bolts and the two one-half inch cutters, will leave .348" for the width of collar at F, as shown. STRAIGHT LINE INDEX DRILLING. It sometimes happens that we want a certain number of holes per inch drilled in a straight line; now, if it is possible to clamp this work on the milling machine bed or to hold it in the vise, we can very easily do a piece of work of this kind and know that it is right when done. Suppose that we want some holes spaced off to be exactly four per inch. As the screw has four threads per inch we know that every time the screw makes just one turn or y±" that it would be right for four holes, but in order to get this distance exact, we must gear up the machine similar to what we would if cutting a spiral, except that instead of turning the worm shaft crank for spacing we pull out the pin, being careful not to move it, but moving the index plate until the proper hole is reached and then securing it by the pin until the next move is to be made, etc. In other words, we move the table by compound gearing by the index plate, instead of the crank. If we have five holes per inch to drill, dividing 5 by 4 threads on the screw, we have the ratio of the gears, or 1^4> which means that if we select a gear, say 40 teeth, and an- other one with V/\ times 40=50 teeth, then the next pair will be alike, and just one turn of the index plate will be correct. 190 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. If we want to drill ten holes per inch, dividing 10 by 4 threads per inch, we have a ratio of 2y 2 for the gears. Now we will take 64 and 32 for one pair, which is 2 to 1, and divid- ing V/ 2 by 2 we have 1J4 for the ratio of the next pair, or say 48 and 60 teeth, etc. If seven holes per inch were required, dividing 7 by 4, we have 1J4 for the compound ratio; now we will take 32 and 40 for one pair, which is a ratio of 1*4, or, in other words, one gear is one and one-quarter times as large as the other, or has a difference of one and one-quarter times as many teeth as the other; then for the next pair of gears we divide the compound ratio 1^4 by the ratio of the pair already se- lected and we have as follows: 1.75-^1.25=1.4 for the ratio of the next pair, then taking 40 for one of these gears and mul- tiplying it by 1.4 we have 56 for the last gear, and one turn of the index plate will move the table 1-7 of an inch. Suppose we prove this example. If we were to drill 100 holes we would turn the index plate 100 times around in order to get that number of holes, and we have two pair of gears 32 and 40 and 40 and 56. Now it is plain to see that the small gears are the drivers in both cases, and it makes no difference so long as we keep them in pairs as found, where we place them; then as we turn the index plate, we will put say the 40 tooth gear back of it for the first driver, and its mate, 56, on the stud in gear with it; then the next small gear, 32, on the same stud in mesh with the 40 tooth gear on the screw; then 100 holes=100 turns. 40=first driver First driven=56)4000(71.4285 392 80 56 240 224 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 191 160 112 480 448 320 280 71.4285 32=second driver. 1428570 2142855 Second driven=40)2285.7120 Threads on screw=4)57.1428==:revolutions of screw. 14.2857=the number of inches the table has moved in drilling 100 holes. Holes per inch=r7)100=holes to be drilled. 14.2857=the number of inches the table has moved. In calculating the gears for drilling 14 holes per inch, we divide 14 by 4, which is 3 J />, or the compound ratio of gears; then if we take any pair that is 2 to 1, say 64 and 32, and divide the compound ratio Syi by the ratio of the pair se- lected, which is 2 in this case, we have 1^4 as the ratio for the second pair, or say 48 and 84. It can readily be seen that if we should gear up for drilling seven holes to the inch with one turn, that we could also drill fourteen holes to the inch if we moved the index plate but one-half of a turn, or fifty-six holes per inch if moved only % of a turn, etc. Graduations can also be done in the same manner, but great care would be necessary if accuracy was required to take up all back lash in the running I parts, while the screw itself might not be in good condition to give the desired result. 192 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. EXTRACTS FROM THE BROWN & SHARPE MFG. CO.'S TREATISE ON THE UNIVERSAL GRINDING MACHINE. (By permission). As the durability of a machine depends very largely upon the care of the operator, however well a machine may be constructed, if it is not properly cared for it will soon be- come unreliable. In all cases we recommend a good quality of oil in pref- erence to one of low grade. Sperm oil should be used on the internal grinding fixtures. The machine should be kept as clean as possible, and in no case should the bearings be allowed to "gum up." When bearings are opened and exposed for any purpose whatever, they should be carefully wiped off before they are closed again, to free them from any grit that may have found its way upon the surfaces. A loose fit between the wheel spindle and its boxes will produce imperfect work, and when very fine work is required the bearings should run nearly, if not quite, metal to metal. This necessarily will cause the boxes to heat, but in this case the heat is not injurious, for as the bearings are hard and the boxes bronze, the belt will slip, unless it is exceedingly tight, before abrasion can occur. All end motion should be taken out of the wheel spindle before the wheel is used on the work. A satisfactory emery wheel is an important factor in the production of good work. Too much, however, must not be expected of one wheel. A variety of shapes, sizes and grades of wheels are necessary to bring out all the possibilities of the grinding machine, the same as a variety of shapes and sizes of tools are necessary to obtain the best results from the lathe or milling machine. Our aim in grinding is usually to obtain an accurate or true surface, but as a true surface is almost always a ^ood surface it should be remembered that generally the same THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 193 methods are employed, whether an exact size or a fine finish is the object desired. In selecting and using a wheel, we are governed by the character of the metal to be operated upon, the shape and size of the work and the degree of accuracy desired. We have to consider the size of the particles of emery in the wheel, the hardness of the wheel and its width. We also have to determine the speed at which the work is to travel or be revolved, and whether or not water is to be used. For the sake of clearness we refer separately to the various characteristics of wheels, but it should be borne in mind that a wheel should not be selected for a single charac- teristic, but that each of the essential elements is importantly affected by the others, and that all should be considered in choosing or using a wheel for any desired work. Wheels are numbered from coarse to fine; that is, a wheel made of No. 60 emery is coarser than one made of No. 100. Within certain limits, and other things being equal, a coarse wheel is less liable to change the temperature of the work and less liable to glaze than a fine wheel. As a rule^ the harder the stock the coarser the wheel required to produce a given finish. For example, coarser wheels are required to produce a given surface upon hardened steel than upon soft steel, while finer wheels are required to produce this surface upon brass or copper than upon either hardened or soft steel- Wheels are graded from soft to hard and the grade is denoted by the letters of the alphabet, A denoting the softest grade. A wheel is soft or hard chiefly on account of the amount and character of the material combined in its manu- facture with emery or corundum, but other characteristics being equal, a wheel that is composed of fine emery is more compact and harder than one made of coarser emery. For instance, a wheel of No. 100 emery, grade B, will be harder than one of No. 60 emery, same grade. The softness of a wheel is generally its most important characteristic. 194 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. A soft wheel is less apt to cause a change of temperature in the work or to become glazed than a harder one. It is best for grinding hardened steel, cast iron, brass, copper and rubber, while a harder or more compact wheel is better for grinding soft steel and wrought iron. As a rule, other things being equal, the harder the stock the softer the wheel required to produce a given finish. Generally speaking, a wheel should be softer as the sur- face in contact with the work is increased. For example, a wheel 1-16 inch face should be harder than one y 2 inch face. If a wheel is hard and heats or chatters it can often be made somewhat more effective by turning off a part of its cutting surface; but it should be clearly understood that while this will sometimes prevent a hard wheel from heating or chatter- ing the work, such a wheel will not prove as economical as one of the full width and proper grade, for it should be borne in mind that the grade should always bear the proper relation to the width. The width should be in proportion to the amount of ma- terial to be removed with each revolution, and as a wheel cuts in proportion to the number of particles in contact with the work, less stock will ordinarily be removed by a narrow wheel than by one that is of full width. The feed will also have to be finer if a narrow wheel is used. The quality of the work as a rule is improved by using a wheel of full width if the wheel is soft in proportion. Judgment should be exercised in deciding upon the width of wheel to be used, as sometimes the work is of such size and shape as to make it necessary to use a wheel with a narrow face. Where this is the case the wheel should, where strength will admit, be only that width throughout, and care should be taken that the grade is kept in the proper relation to the width. A wheel is most efficient in grinding just at the point before it ceases to crumble. The faster it is run up to this point the more stock will be removed and the more econom- ically the work will be produced. Occasionally, however, THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 1 95 it is necessary to run a wheel rather slowly, as the more slowly it runs the coarser it cuts and the less likelv it is to change the temperature of the work. As a general rule, on any stock, the softer the wheel the faster it should be run.. Should a wheel heat or glaze it can often be made some- what more effective by being run more slowly. On the other hand, if it be too soft, it can often be made to somewhat better hold its size and grind straight by being run more rapidly. The surface speed of the work should be proportional to the speed of the wheel, that is, other things being equal, if the speed of the wheel is reduced the speed of the work should be reduced also. The desire is to nave the work revolve at such a speed as to allow time for the wheel to cut away the high points on the work. If the work is run so fast that there is no time given for the wheel to cut, but the work is simply crowded against the wheel, the tendency is for the wheel to follow the inequalities in the form of the work, and straight or round surfaces are not obtained. When the wheel is not free cutting and the pressure of the wheel against the work is sufficient to cause the work itself to spring cr to cause a slight movement of the oil upon the centers, the accuracy of the result is impaired. The iesire in accurate grinding is to have a free cutting wheel and to obtain the proper speeds so that the stock may be removed with the least possible amount of pressure, thus preventing a change of temperature in the work and allowing the high parts to be most speedily reduced. Thus far we have had in mind the selection and use of wheels for the comparatively small or medium sized work ordinarily ground on our machines. The requirements in grinding extremely large or long pieces are somewhat differ- ent. For example, in grinding a piece of steel thiee inches long, one inch diameter, on a Universal Grinding Machine we have indicated that the most absolutely accurate work would be accomplished by selecting a wheel only just hard enough to retain its size while passing six or eight times 196 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. over the surface of the piece, and we have suggested that such a wheel should be run at a high rate of speed. We have con- sidered rapidity of production as more important than econ- omy of emery. If, however, we should attempt to use such a wheel to grind a piece of steel one inch diameter and three feet long, it is clear that before the wheel had passed over two of the three feet it would have ceased to cut. The problem now is to maintain the diameter of the wheel so as to take a uniform cut over a large area. Each particle of emery must be used as long as possible before be- ing thrown away. The particles may be used a longer time and are not so rapidly thrown away in a hard as in a soft wheel. Accordingly, one expedient in grinding large areas is to increase the grade of the wheel as the area increases, the speed of the wheel being reduced as the grade is increased. The loss of fine particles will not decrease the diameter of the wheel as rapidly as the loss of coarser or larger par- ticles. Thus another expedient is to use a finer wheel. A fine wheel can be relatively softer than a coarser wheel, and so with a fine one there need be less pressure between the wheel and the work, and there is more certainty of ob- taining an accurate surface. If a wheel is run rapidly the particles of emery soon be- come dull and have to be thrown away. To retard this loss it is w T ell to run the wheel more slowly as the length or area of the work increases. As the length or area of the work increases, the feed should be coarser, so that the wheel may travel the entire length or area of the piece while its diameter is practically unchanged. Water should be used on such classes of work as are in- juriously affected by a change in temperature caused by grinding. It should be used upon work revolved upon centers, as in this work a slight change in temperature will cause the wheel to cut on one side of the piece, after it has been ground ap- parently round. THE MACHINIST AND TOOL MAKEK's INSTRUCTOR. 197 In very accurate grinding water is especially useful, for it should be remembered that the exactness of the work will be affected by a change in temperature which is not percep- tible to the touch. In very accurate grinding it is also well to use the water over and over again, as by so doing there is less difference between the temperature of the water and that of the work than if fresh water is used. For many purposes soda water is the most satisfactory, as it has less tendency to rust the work or the machine. For internal grinding it is especially important that a wheel should be free cutting and the work revolved so slowly as to enable the wheel to readily do its work. The wheels should generally be softer than for external grinding, as a much larger portion of the periphery is in contact with the work. Their small diameters make it impossible for the proper periphery speed to be obtained, and this must be considered in regulating the speed of the work. The wheels listed at the end of this chapter are those which our experience has shown to be suitable for the various purposes specified. Special cases may demand changes in the grade letter, but under ordinary circumstances the list should be accepted as a guide. The speed, diameter and width of wheels, and the num- ber of the emery cannot be changed without changing the grade and cutting qualities of the wheels. In mounting emery wheels there should always be elastic washers placed between the wheel and the flanges. Sheet rubber is best for this purpose, but soft leather will answer very well. In some cases manufacturers of emery wheels attach a thick, soft paper washer to each side of the wheel for this purpose, in which case no further attention is required in this direction. In all kinds of grinding the work should move in a direc- tion opposite to that of the wheel at the cutting point. To obtain good results when the grinding is done on the 198 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. centers it must be remembered that true center holes in the work, as well as true centers, are absolutely necessary. The face plates should be kept true. They can be readily ground in place on the machines. When slight tapers are desired for either external or in- ternal grindings, the swivel table is set to the proper angle. When more abrupt tapers are wanted for work ground on centers or for internal surfaces, the wheel slide is set to the proper angle. By placing the wheel slide and the swivel table at the proper angles, two tapers, for either external or internal work, may be obtained without changing the set- tings of the machine, the one automatically by the longitu- dinal movement of the table, the other by operating the cross feed by hand. When an abrupt taper, similar to that shown in Figure 1, is to be ground, the swivel table remains parallel to the ways of the bed as in plain grinding, but the wheel bed is set to the angle, which brings the line of motion of the wheel slide, when operated by the cross feed, parallel with the taper to be obtained The wheel platen is set at right angles with the line of »i ovement of the wheel slide, indicated by the arrow, and the face of the wheel is thus brought parallel with the line of the desired taper. The work is revolved by the dead center pully, as shown in cut, and the wheel is moved over the sur- face of the work by the cross-feed. The method of grinding two tapers with one setting of the machine when one of the tapers is not more than ten degrees is shown in Figure 2. For grinding the slight taper the swivel table is set over, and for grinding the more abrupt taper the wheel bed is set as in Figure 1, but the wheel platen is here set to bring the face of the wheel parallel with the longest surface to be ground. Were the abrupt taper longer than the slight taper it would be well to set the wheel platen as in Fgure 1, so that the face of the wheel would be parallel with the line of taper. In obtaining the angle at which the wheel bed is to be set * THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 199 200 THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. Wg2 when the swivel table has been set over, it should be remem- bered that the angle must equal the sum of the two tapers. The abrupt taper is ground by feeding the wheel across the work by hand. The slight taper is ground while the table is fed automatically. When, as suggested by the cut, a spindle and box are both to be ground, the box is ground first and the spindle is fitted to it. For convenience in fitting the work, the box may be placed as shown by the dotted lines and supported so that it will not touch the spindle as the latter revolves. The THE MACHINIST AND T00E MAKER'S INSTRUCTOR. 201 ^^ 3 spindle thus need not be removed from the centers when it is necessary to try it in the box. In grinding the box the machine is set as shown in Figure 3. Provision is made for grinding the slight taper by setting the swivel table, and for grinding the abrupt taper by swivel- ing the wheel bed. Thin washers and cutters are conveniently mounted on a 202 THE MACHINIST AND TOOD MAKEE'S INSTRUCTOR. special chuck furnished with the machine. This chuck holds the cutters by the hole in the center and should be used in all disk grinding where both sides of the pieces must be parallel. For most disk grinding it is also more convenient than a common chuck and more accurate results are generally ob- tained by its use. Thin saws are held this way and are ground concave or thinner at the center than at the teeth, to give the proper clearance. Angular or taper cutters may be ground by holding them on a mandrel, but the swivel table must be set over for the proper taper. When the taper of the cutter is greater than two inches per foot, the feeding may be done with the cross feed, and the wheel set as shown and explained in connec- tion with Figure 1. The side teeth of straddle mills may be ground on a Universal Grinding Machine, but the machine is not recom- mended for such a purpose, as, in order to obtain the proper clearance, the wheel stand must be elevated so that the center of the wheel will be above the tooth rest, the tooth rest in this case having to be set as high as the center of the head stock spindle. Where many cutters of this class have to be ground, or, in fact, for manufacturing establishments having a great deal of tool grinding, we would recommend the Brown & Sharpe's special tool and cutter grinding machines as more convenient and economical. The following list of emery wheels, as made by the Nor- ton Emery Wheel Company of Worcester, Massachusetts, to be used in connection with the Number 2 Universal Grind- ing Machine Improved (Brown & Sharpe), is recommended. To grind hardened steel spinales, etc., shape 3 (plain 12" diameter, J^" face), emery 60, grade K, mixture 14- A, speed 1,300 to 2,000; also emery 60, grade M, mixture 13- A, speed 1,300 to 2,000. To grind soft steel, shape 3, emery 100, grade L, mixture 14-A, speed 2,000. To grind long or large soft steel pieces, shape as before, THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 203 emery 100, grade N, mixture 5- A; also emery 100, grade O, mixture 6-A. To grind collars, disks, saws, etc., shape 1, 13 or 14 (plain 6" to 7" diameter, and from 34" to y 2 " face), Georgia corun- dum 36, grade J, mixture X-15-C, speed 3,000. To grind cast iron spindles (solid), shape 3, (12" diameter y 2 " face), emery 60, grade M, mixture 13- A, speed 1,300 to 2,000. To grind hollow cast iron rolls, shape 3, emery 60, grade F, mixture 15-A, speed 1,300 to 2,000. WHEELS FOR INTERNAL GRINDING, TO ROUGH OUT FOR ACCURATE WORK BUT CANNOT BE CROWDED. FOR HARD AND SOFT STEEI* AND CAST IRON. Shapes 82, 83 and 84 (from y 2 " to J4" diameter, Y A " face and yi" hole), emery 46, grade G, mixture X-12-A, speed 13,400. Shapes 84, 85 and 86, (from J4" to 1* diameter, %" hole and 34" face), emery 46, grade G, mixture X-12-A, .speed 12,200. Shapes 87, 88 and 89, (from 1" to iy 2 " diameter, y 8 " face and from y to y%" holes), emery 46, grade G, mixture X-12- A, speed 11,200. Shapes 90 and 91, (from 2" to 2^" in diameter, y s " face and J4" hole), emery 46, grade G, mixture X-12-A, speed 8,050. To grind for finishing after roughing wheels have been used: Shapes 82, 83 and 84, emery 120, grade F, mixture X-10- A, speed 13,400. Shapes 84, 85 and 86, emery 120, grade F, mixture X-10- A, speed 12,200. Shapes 87, 88 and 89, emery 120, grade F, mixture X-10- A, speed 11,200. Shapes 90 and 91, emery 120, grade F, mixture X-10-A, speed 8,050. For internal grinding the Special Fixtures are used. .204 THE MACHINIST AND T00X, MAKER'S INSTRUCTOR. CHAPTER V. MECHANICS. On this subject I shall treat more particularly on such matters as concern the mechanic in his surroundings. QUESTIONS UPON THE PRINCIPLES OF THE LEVER. At one extremity of a straight lever, whose length is 6 feet, a weight of 10 pounds is suspended; at the distance of 5 feet from the point of suspension a fulcrum is placed. What weight must be suspended from the other extremity of the lever to keep it in equilibrium (balanced)? If 10 pounds is suspended 5 feet from the fulcrum, then at one foot from the fulcrum on the opposite end it will take 5 times as much to balance it, or 50 pounds, the answer. A lever is 30 feet long; at what distance from the fulcrum must a weight of 200 pounds be placed so that it may be supported by a power able to sustain 50 pounds? 200-^50=4, then 30-^4=7^2 feet, the answer. In a lever, as shown in figure 1, a weight of 100 pounds is suspended at A. What will be the required weight at B to hold this weight in equilibrium? Both arms being of the same length from the fulcrum C, but while A is horizontal B is inclined at an angle of 45°, as shown. The cosine of 45 o =.707, then 1-f-. 707X100= 141. 3 pounds, the answer. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 205 U. 30" ~^ — 50" € In the lever, as shown at figure 2, A is the fulcrum, B the weight, and the power is applied at C in the direction of the arrow. Required, the power at C to support a weight of 60 pounds in equilibrium at B, as shown. In this case the weight at B multiplied by the distance A B must equal the power at C multiplied by the distance A C, or thus: Weight at B=60 pounds, distance A B= 30", then 60X30=1800; the distance A C is 30"-f50", or 80", then 1800-f-80"=22.5 pounds, the power at C to bal- ance a weight of 60 pounds, as shown at B. The pressure on the fulcrum at A is equal to the difference between the weight suspended at B and the power at C; then, as the weight at B=60 pounds and the power at C— 22.5 pounds, 60—22.5=37.5 pounds as the pressure on the fulcrum. !«e- 30". 50" In figure 3 the power is represented between the fulcrum and the weight. Required, the power at B to support a weight of 60 pounds suspended at C, as shown. In this, or similar examples, we multiply the weight by 206 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. the distance A C and divide by the power multiplied by the distance A B; or thus: 60x80"=4800; now, as the power is not known we divide 4800 by the distance A B, or 30", thus: 4800-^30"=160 pounds, or the power at B to just balance a weight of 60 pounds at C. In this example, as the power is greater than the weight, the pressure at the fulcrum is up- wards, and is equal to the difference between the power and the weight, thus: 160—60=100 pounds. The safety valve also comes under the head of levers, and the calculations are similar to figure 3. In figure 4 let A be the fulcrum of the lever, B the valve, 3" from the fulcrum, and D a weight of 40 pounds suspended 25" from the valve or 28" from the fulcrum; the valve is 3" diameter. Required, the steam pressure on the valve to just balance the weight. (The weight of valve and lever is not taken into considera- tion). The distance A C is 28", and the weight 40 pounds, then 28X40=1120; the valve is 3" in diameter, 3X3=9X-7854 =7.07 square inches, or the area of the valve; this being 3" from the fulcrum, we multiply it thus: 7.07X3=21.21, and 1120-^21.21=53.2 pounds pressure, the answer. Now, suppose that our safety valve is 4" diameter and Sy 2 ff from the fulcrum, and we want to blow off at 100 pounds pressure, the weight is 90 pounds; how far from the fulcrum will it have to be suspended? The valve is 4" diameter, then 4X4=16, and .7854X16= 12.56, or the number of square inches of area in the valve, and this, multiplied by the distance from the fulcrum, 3^", THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 207 thus: 12.56X3.5=43.9, and again by the pressure required, 43.9X100=4390; then 4390-^90=48.8 inches, nearly. This weight, of course, is supposed to balance the pressure on the valve, and any excess in the pressures will lift it. The weight of valve and lever has not been taken into consideration. We have another safety valve 3^" diameter and placed 254" from the fulcrum, and we wish to maintain a boiler pres- sure of 65 pounds; the extreme length of lever from the fulcrum is 35 inches. Required, the weight, if suspended at the end of the lever. The valve is 3.5" diameter, then 3.5"X3.5=12.25X.7854= 9.62, the area in square inches of the valve; then, 9.62X2.75, the distance from the fulcrum, =26.45, and multiplied by the pressure, 65 pounds, =1719.25; this divided by the length of lever 35' =49.1 pounds, as the required weight on the lever. It must be remembered that the diameter of the valve in all cases must be taken from where the steam acts directly against it, and not from the largest part of the valve. h 3 We have a safety valve (Fig. 5); the fulcrum A is 3 inches from the valve B, the weight C is 65 pounds, and the weight of the lever D is 10 pounds and its center of gravity (of the lever) is 9 inches from the fulcrum A; the valve is 3^ inches diameter at the point of pressure, and we wish it to lift when the pressure is 70 pounds per square inch; how far from the fulcrum will we have to place the weight? The valve being 3^ inches diameter, then 3.5X3. 5X-T854 =9.62 inches, area of the valve, or the number of square 208 THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. inches; then, 9.62x70, the steam pressure, and again by 3", the distance A to B, =2020.2. The iever weighs 10 pounds, and its center of gravity is 9 inches from A; then, 10x9=90, and 2020.2—90=1930.2, and this divided by the weight, thus: 1930.2-^65=29.7 inches, nearly, the answer. Or, if the lever had a notch filed into it 29.7" from the fulcrum and our weight had been lost and we wished to know how large a weight would be required under the same condi- tions, we would divide 1930.2 by 29.7 inches, and this would give the answer, or the weight required. V \\ #V /6 16 n* L c \ // s i \ i i i \ \ \ \ \ / +5° i \ \ > s *-— / \ i \ ^W— ? i V "" i jb: -— ^^5> V-.-- - - - aa" - i K 1 1 i t 1 We have a reel, attached to a lever at B, (Fig. 6), that swings on a stud shown at A, the long arm of which is 38" from the fulcrum. The short arm is in the form of a segment of a gear and is 16" from the pitch line to the fulcrum. To turn this lever, we have a crank attached to the pinion D, 16" long; the pinion is 2 1-6" diameter at the pitch line. What force will be required at E to balance a weight of 153 pounds suspended at B in a horizontal position, disregarding friction and weight of the moving parts? In this example the force required at C will be as much greater than at B as the length from A B is greater thar THE MACHINIST AND TOOL MAKERS INSTRUCTOR. 209 A C; therefore, dividing 38 by 16, which is 2j^, and, as the weight at B is 153 pounds, multiplying this by 2j£ we have 363.4 pounds to be overcome at C. But as the crank E is 16" long, and the pinion that it turns is but 1 1-12" radius (one-half the diameter), we gain a leverage equal to the dif- ference between the radius of the pinion and the radius of the crank; then, dividing 16" by 1 1-12", or 1.083", and the result, 14.7, is the ratio of the power gained by the difference in the radius of the crank to that of the pinion; then, dividing 363.4 by 14.7 and the result, 24.7 pounds, will be the force required on the crank E to balance 153 pounds suspended at B. (Dis- regarding friction, etc.) A force of 36 pounds would be a safe estimate in this case, including friction, to lift the weight. It must be remembered that if the weight is in any other than a horizontal position from the fulcrum on which it moves that the distance A B, will be equal to the cosine of the angle, as shown in the figure. Thus, suppose that B was in- clined at an angle of 45° (shown at F), then as the cosine of 45° =.707, and multiplied by 38" will be 26.8", for the distance to be considered for A B instead of 38". THE INCLINED PLANE. We have a barrel weighing 300 pounds, (Fig. 7), and we want to roll it into a wagon that is 45 inches high by using a plank (A) which is ten feet long; what force will be necessary to roll it up the inclined plane as shown when the pull is par- allel with the line A? 300X45=13500 and 13500-^120" (10 feet) =112.5 pounds. 210 THE MACHINIST AND TOOD MAKEE's INSTRUCTOR. This is the force necessary to hold the barrel on the plank, and if there was no friction, any excess of pressure ever 112.5 pounds would roll the barrel. Or, multiply the weight by the perpendicular of the angle formed and dividing by the hypothenuse will give the answer. From the preceding example, knowing how much force you could exert without fatigue, and having a barrel of oil, or anything similar, to roll into a wagon, if the weight was known, you would know how long a plank to select for the purpose. Suppose you were alone and wished to roll a barrel weighing 250 pounds into a wagon that was 45 inches high, and 100 pounds was about as much as you cared to exert; what would be the length of the board required? 250-^-100=2^, then 45x2^=112.5 inches, or nearly 10 feet. This would not be quite enough to overcome the fric- tion, and we would get one a little longer for the purpose. On the same principle, if you were to draw a wagon weighing 200 pounds up a hill that was 60 feet high in 300 feet in length, you would not only have the friction to over- come, the same as on a level road, but you would also add to this 60-300 of 200 pounds, or 40 pounds. THE SCREW. We have a screw, the pitch is 1 inch (or the distance be- tween two threads) and a lever 30 inches long, measuring from the center of the screw (or its axis); what pressure can be produced by this screw, if we apply a force of 45 pounds at the end of the lever? In this or similar examples we find the circumference of the circle described by the end of the lever, or where the power is applied, which in this case would be, thus: 30 inches radius =60 inches diameter, then 60X3.1416=188.5, and 188.5 X45, the force at the end of the lever, =8482.5, and this divided by the pitch will give the answer. As the pitch in this case is one inch, then 8482.5 pounds is the answer. Now, if this screw was two inches pitch then we would have a pressure of one-half as much, or 4241.25 pounds. But if the screw THE MACHINIST AND TOCTL MAKER'S INSTRUCTOR. 211 was % inch pitch, or four threads per inch, then we would have a pressure of four times as much, or 33930 pounds. WHEEL AND AXLE. In this example, Fig. 8, we have two pulleys or drums on the same shaft. A is 24" diameter and B is 7" diameter; a weight of 40 pounds is suspended from the largest pulley, shown at C; what weight will be required at D to balance the weight at C? A being 24" and B 7", then 24-^7=3 3-7, and 40 times 3 3-7=about 137 pounds, the answer. WHEEL GEARING. In this example, Fig. 9, (compound gearing), A is the driving gear with 20 teeth, B 62 teeth and C 12 teeth 212 THE MACHINIST AND TOOT, MAKER* S INSTRUCTOR. (B and C are both on the same shaft or stud), and D has 100 teeth; now if we turn the gear A with a force equal to 15 pounds on the pitch line, what will be the pressure on the pitch line of D? If A exerts a pull of 15 pounds on the circumference of B, then the force acting on the circumference of D will be as 62 is to 12 times 15 pounds, thus: 62-4-12=5 1-6, which mul- tiplied by 15=77.5 pounds, nearly, the answer. It can easily be seen that if the gear C was of the same diameter as B, that the force exerted on the periphery of D would not change. In other words, it would be 15 pounds, and if it was one-half as large we would double the leverage, and consequently would have a pressure of 30 pounds on D. But if C was twice as large as B, then we would have but one-half of 15 pounds, or iy 2 pounds. Suppose we have a machine on which there is a com- bination of gears, and also a drum operated by a crank, as follows: On the first shaft we have a pinion 4" diameter and a crank 16" long (or radius). On the second shaft there is a gear 15" diameter and also a pinion 4" diameter, and on the third shaft there is a gear 15" diameter, and a drum 5 inches diameter, with a rope wound around it. These 4" pin- ions in both cases mesh with the 15" gears. If we exert a force of 50 pounds on the crank, what will be the weight that we could hold in suspension attached to the rope? In this example we will first find the ratio between each pair of gears, thus: 15-^4=3 J4, and as there are two pairs alike, then we multiply these two ratios together, thus: 3.75 X3.75=about 14 for the compound ratio of the gears. We next find the difference between the radius of the crank and the radius of the drum, thus: 16-^-2^=6.4, and multiplied by the compound ratio 14=89.6, and this again multiplied by 50, the force on the crank =4480 pounds, the answer. Or, as follows, which is the usual way. 15"Xl5"=225", and 4"X4"=16"; then 225"-^-16"=14 ratio, which multiplied by 16", the leneth of crank=224, and agrain by 50, the force on the crank=11200, divided by one half the diameter (radius) of the drum=4480 pounds. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 213 THE SCREW AND GEAR. We have a machine combining the following parts, as shown in Fig. 10, in which A is a crank, 16" radius, o- length; B is a worm of 4 threads per inch; C a worm wheel 12 inches diameter on the pitch line. On the same shaft with the worm wheel is a pinion D, 3 inches diameter (pitch line) in mesh with the gear E, 16 inches in diameter. On the same shaft with this gear (E) is a small pulley F, 5 inches diameter, with a rope wound around it, as shown at G, making no allow- ance for either the friction or diameter of the rope; what weight can we support at G by a force of 30 pounds applied on the crank at A? The crank A being 16" long will describe a circle of 32" in diameter, the circumference of which is 100.5 inches. Now, as the worm is Y\" pitch (4 threads) then the crank A will pass through four times as great a distance, or 402 inches, while the worm, or what is the same thing, the worm wheel, moves but one inch (on the pitch line). As the worm wheel C is 12" diameter and the pinion D but 3" diameter, we gain a leverage of 3 to 1; then 462X3= 214 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 1206. It is plain that whatever may be the power gained at D will be imparted to E, since D drives E. But as E is 16 inches in diameter and F is but 5 inches, we gain accordingly; thus: 16—5 =3 1-5; then 1206X3 1-5=3859.2, which multiplied by the force on the crank of 30 pounds=U5776 pounds, the answer. THE HYDRAULIC PRESS. Suppose we have a small hydraulic press, the lever of which is 24 inches in length to the fulcrum; this lever is worked by hand and is attached to the plunger, which is but three inches from the fulcrum and also between it and the power. Suppose this plunger to be 24 of an inch in diameter and the ram to be 12 inches diameter; by exerting a force of 40 pounds on the lever, what pressure can be obtained by the ram? As the power applied is 24 inches from the fulcrum and the plunger is but 3 inches, we gain a ratio of 8 to 1 by moving the lever 8 inches to obtain 1 inch of plunger. The diameter of the plunger is $4", then .75X-75X3. 1416=442 thousandths of an inch (area of the plunger), and the ram 12"X12"X3.141«3=113" (area of the ram); then 113-=-.442== 255.6; or, in other words, the area of a cross section of the ram is more than 255 times as great as that of the plunger; then multiply 255.6 by the ratio of 8 and again by the force of 40 pounds on the lever, and we have as follows: 40X8X 255.6=81792 pounds, the answer. TO FIND THE BRAKE HORSE-POWER OF AN ENGINE. To find the bra*ve horse-power of an engine proceed as follows: Clamp the friction brake on a pulley placed on the crank shaft of the engine with the arms A A in such a posi- tion that the ends B of the arms which meet will be in a horizontal position with the center of pulleys shown in figure 11; the end of arms at B are to rest on a scale blocked up in order to be in this position. Now we find that the pressure THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 215 on the scale is iust 70v pounds, and the distance from B to the center of pulley or brake is 6 feet; then multiplying to- gether, thus: 700 y 6=4200, and dividing this by the radius of the pulley on which the brake is secured, we have as follows: 4200-1-2=2100; this is the resistance upon the cir- cumference of the pulley. Now, if this pulley revolves 125 times per minute, we must find how many feet per minute the circumference of the pulley is making. Then 3.1416X4= 12.56, the number of feet around the pulley, and this multi- plied by the speed, thus: 12.56X1-5=1570; this equals the number of feet per minute the pulley travels at the circum- ference, and this again multiplied by the resistance on the pulley as follows: 1570X2100=3297000, or the foot pounds per minute; then dividing by 33000 we have as follows: 3297000-^33000=99.9 horse-power of the engine (brake). It must be remembered that the brake should be tight- ened sufficiently on the pulley until the speed of the engine is about to slack; this can easily be seen by the aid of a speed indicator. HORSE-POWER. The horse-power of an engine is found in the following manner. Example: We have an engine 24 inches diameter of cylinder, 40 inches stroke of piston, making 120 revolutions per minute, with steam at 75 pounds pressure per square inch; required, the horse-power. We first find the area of the piston 216 THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. by squaring the diameter and multiplying it by .7854, thus: 24X24X.7854=452, and multiplying this by the pressure of steam=:33900; multiplying this again by the stroke in feet per minute, which is as follows: as the stroke is 40 inches, at every revolution of the engine the piston will travel 80 inches, then multiplying this by 120 (revolutions per minute) and we have 9600 inches, or 800 feet per minute; then 33900 X800=27120000 foot pounds per minute. Then as a horse- power is equal to the work of 33000 pounds done in one min- ute, we divide 27120000 by 33000 and we have 822 horse-power for the answer. This is called the nominal horse-power; the indicated horse-power is found in the same manner, excepting that the steam pressure (the mean effective) is found by means of the indicator, which is always less than the boiler pressure. PRESSURE ON THE GUIDES OF AN ENGINE. Suppose we have an engine the cylinder oi which is 30 inches bore, and 40 inches stroke, and we want to know the greatest pressure on the guides during a revolution, using 100 pounds pressure of steam per square inch and calling the connecting rod 10 feet long between centers. In figure 12, let A B, 10 feet or 120 inches be the con- necting rod; B C=the length of crank, or one-half of the stroke. In the figure B C is drawn perpendicular to the line A C 5 because the greatest pressure is on the guides when the crank is at half stroke, as shown. Now, the cylin- der is 30 inches bore, and consequently the area is 30X30X THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 217 7854=706 inches; multiplying this by the pressure of 100 pounds=70600 pounds pressure on the piston, and conse- quently on the line A C; 70600 multiplied by the distance (20") B C, and divided by the length A C, will give the pressure on the guides. In the figure (12) we have a right angled triangle, as shown, the radius of which is 120 inches; B C is the sine of the angle B A C; then 20 inches (the sine) divided by the radius of 120 inches=.166, or the sine to a radius of 1 inch, and, in the table of sines, we find this to correspond with the angle of 9° 33'. The cosine of this angle=.986; then multiplying this by the radius of 120", thus: .986X120'' =118.3 inches, or the length oi A C. Then 70600X20= 1412000 and divided by 118.3=11935+ pounds, the answer. TO FIND THE DIAMETER OF A SHAFT FOR TRANSMITTING POWER, WHEN THE HORSE-POWER OF THE ENGINE IS KNOWN. Example: Suppose in a factory we have an engine 150 horse-power which is to run at a speed of 175 revolutions per minute. Required, the diameter of the main line of shafting. 175)150.00000(.857142 140 10 00 1250 1225 250 175 750 700 500 218 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. .857142(.94+ 729 128142 9 2 X300=24300 9X30X4 = 1080 4 2 = 16 101584 .95+ 2.9 25396 26558 855 190 2.755 inches, the answer. We first divide the horse-power of the engine by the revolutions it is to run per minute and extract the cube root, which in this case is .95, then multiply this by the constant 2.9, which gives about 2J4 inches for the diameter of the shaft. This is for good ordinary steel, but if it is to be of wrought iron, it should be one-eighth (J^) larger, thus: y 8 of 2.755"— 344", and, added together =3.099", the answer, if made of wrought iron. IMPACT OR COLLISION OF BODIES. Let us now for a moment examine into the force of a shot fired from one of our modern 12-mch guns. Suppose a solid shot, weighing 1,100 pounds, is fired at an object and strikes it with a velocity of 1,500 feet per second; then the mass of this body, which is the weight divided by 32.2, mul- tiplied by the velocity in feet per second will be the answer, provided it takes just one second from the time the shot strikes the object until the momentum of the ball is stopped, or, as we say, destroyed; but we know that it does not take a second to stop it, and we will suppose that it takes but one- fiftieth (1-50) part of a second to stop it, then the force will THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. 2lS k be as follows: 1100-^-32.2=34.16, which multiplied by the velocity equals 41240, which would be the answer, as explained if it required one second of time from the moment of striking^ until the momentum was destroyed; but if it only takes one- fiftieth of a second, then the force (average) will be 50 times as much, or 2062000 pounds, the answer. Or, if it takes the 1-100 part of a second, then the answer will be 41240X100= 4124000 pounds. The force of a railway train or any other object can also be found in a similar manner if we know the weight (or mass) of the body, the velocity in feet per second, and the duration of the impact in time. Remember that the mass of any body is its weight divided by 32.2; that is, anything that weighs just 96.6 pounds, we would say that its mass was 3. CENTRIFUGAL FORCE. The centrifugal force of a fly wheel of an engine, or any- thing of similar shape, is the force exerted outward from the center, like the sparks from an emery wheel; the force of which can be found (approximately) as follows: We have an engine with a fly wheel, the rim of which weighs 60 tons, or 120000 pounds, is 18 feet in diameter at the center of the rim and has a velocity of 80 revolutions per minute. Required, the centrifugal force. 18 feet the diameter=56.5 feet in circumference, which multiplied by the number of revolutions 80=4520 feet per minute, and divided by 60=75.3 feet per second; 75.3 squared=5670, and multiplied by the weight in pounds= 680400000, and dividing this by the radius in feet, multiplied by 32.2, is the answer. Thus the radius of the wheel is 9 feet, which multiplied by 32.2=about 290; then 68040000O di- vided by 290=2346206 pounds, the answer. 220 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. CHAPTER VI. SCREW CUTTING. In nearly all modern lathes threads can be cut of most any pitch by simple gearing, by which we mean that there is one driver and one driven gear, and usually but one inter- mediate gear placed between them; but it makes no differ- ence how many there are so long as they connect one with another in a direct train. The number of teeth in the inter- mediate gears also do not affect it; only the driver, as it is called, on the spindle and the driven on the leading screw are to be considered. If the leading screw has eight threads per inch, and we wish to cut eight threads on a piece of work, then, as usually made, any two gears with the same numbers of teeth, one on the spindle and the other on the leading screw will cut it. If the leading screw has 4 threads per inch and we want to cut 8 threads, it is plain that the leading screw should turn only one-half as fast as the work, for the reason that while the leading screw makes four revolutions, and consequently the carriage travels one inch, the piece of work on the centers must make twice as many revolutions, or eight turns, in order to have eight threads per inch. In simple gearing, then, if the leading screw has four threads per inch and we wish to cut four threads per inch on a piece of work, any two gears with the same numbers of teeth will answer. And, if the leading screw has four threads and we want to cut five threads per inch, multiplying these numbers by any other number will answer, thus: 4X6=24 and 5X6=30; then 24 and 30 will answer; or, 4X7=28 and 5X7=35; then 28 and ■35 will cut it, and so on. Remember to multiply the number of threads on the lead- ing screw and the number of threads you wish to cut by the THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 221 same number, no matter what the number may be; if you have the gears, it will be correct. Sometimes we find the pair of gears required thus : lead- ing screw has two (2) threads per inch and we wish to cut six (6) threads per inch; then 2x10^—21 and 6XlO>4=63. It is plain to see that one is just three times as coarse as the other, and consequently, one gear must have just three times as many teeth as the other. It is also plain that if we are to cut threads finer than the leading screw that the largest gear should always be placed on the leading screw, and when coarser than the leading screw, the largest gear should go on the spindle. SCREW CUTTING BY COMPOUND GEARING. When cutting screws that are of a. very coarse pitch or very fine it is usually done by compound gearing, for the reason that it is difficult to find a pair of gears in which the difference in the numbers of teeth is great enough to answer the purpose by simple gearing. In compound gearing we generally have four (4) gears, arranged in two pairs as follows: Gear on spindle connects with one on stud, and this is one pair; by the side of this gear on the stud is another gear that meshes with the gear on the leading screw, this is the second pair. Always, in com- pound gearing, these four gears have to be considered both in regard to the numbers of teeth, as well as being placed in their proper positions. Suppose, for instance, that the leading screw of a lathe has four (4) threads per inch and we want to cut a coarse thread on a worm, say one thread or one turn in three (3) inches; then, while the worm is making just one turn on the centers the screw will have to make twelve turns, and con- sequently, one is exactly twelve (12) times as coarse as the other, or, as we usually say, the compound ratio of the gears to cut this worm is 12 to 1. Now we can take a pair, say 20 and 60 for one pair, which is a ratio of 3 to 1, and dividing the compound ratio 12 by 222 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. the simple ratio of 3 we have four for the answer, which means that the next pair must be 4 to 1, and it makes no dif- ference what the numbers of their teeth may be in either case so that they are kept in pairs and proportioned 3 to 1 and 4 to 1. It is plain to see, however, that in both cases the largest wheels are the drivers; thus 60 can be placed on the spindle and 20 on the stud for one pair, and 72 on the stud by the side of the 20 and 18 on the leading screw; or, 72 can be placed on the spindle connecting with 18 on the stud and 60 along- side of the 18 in mesh, with 20 on the leading screw. Or, we can take a pair of gears 2 to 1, say 40 and 80, and dividing 12 by 2, we have 6 for the ratio of the next pair; then any pair 20 and 120 or 30 and 180 will answer. Suppose that the leading screw on a lathe has four (4) threads per inch and we want to cut a thread of Y% inch pitch. In a case of this kind we generally find how many threads there will be in a certain number of inches, both on the leading screw and the work that is to be threaded. Let us take three (3) inches in this example, and we find that there are 24 eighths, and the pitch is to be three-eighths; then there will be just eight full threads in three inches, while in the leading screw of four threads per inch there will be just twelve full threads in three inches, then the gears to cut this screw will be as 8 is to 12; that is, in simple gearing where only two gears are changed, multiply 8 and 12 by 2, 3, 4, 5, 6, or any other number, sometimes by 2 l / 2i 3 J /2, 4j4, and so on, and if you have such gears they will be correct. Or, if you compound the gears, take one pair alike, say 60 and 60 or 50 and 50, and the second pair will be as just described in simple gearing (8 to 12); for instance, 24 and 36. Or, suppose you want to cut a screw of % inch pitch, the leading screw being four threads per inch; then we find that in 7 inches there are 56 eighths, and as each thread takes up a space of % inches, dividing 56 by 7 we have just eight full threads in the seven inches, while in the leading screw there will be 7 times 4, or 28 full threads; then our ratio will be as 8 is to 28, or 3J4 to 1. In simple gearing we THE MACHINIST AND TOOI, MAKER' S INSTRUCTOR. 223 can take any gear, say 20 for one and S J / 2 times 20 for the next and so on, while in compound gearing we can take a pair of 2 to 1 and dividing 3 l / 2 by 2, thus: 3.5-^2=1.75, or 1J4 to 1, for the next pair; thus any pair 2 to 1, call it 30 and 60 for one pair, and 40 and 70 for the next. Suppose the leading screw has three (3) threads per inch, and we want to cut a screw of Y% inch pitch. In three inches there are 24 eighths, and dividing 24 eighths by three-eighths we have eight full threads in three inches; while in the lead- ing screw there would be 9 full threads in three inches, and the gear should be as 8 is to 9; then in simple gearing mul- tiply these numbers by any other number, say 3, and we have 24 and 27; or, in compound gearing, these same gears will answer for one pair if the next pair are alike. If we should want to cut 27 threads to the inch and the leading screw has 3 threads per inch, then 27-^3=9 for the ratio; then any pair 3 to 1, say 20 and 60 for one pair, and dividing the ratio of 9 by 3, we then have two pairs of gears 3 to 1; while if we should take for one pair 4 to 1, say 22 and 88, or 20 and 80, then dividing the ratio of 9 by 4, we have for the second pair 2J4 to 1; that is, any number in which one gear has 2% times as many teeth as the other, for instance, 24 and 54. Again, suppose our leading screw has four threads per inch and we want to cut a coarse thread on a worm of 2^ inch pitch; then while the worm makes four revolutions the carriage will have to move four times 2% inches or 10y 2 inches, and, as the leading screw has four threads per inch, it will turn four times 10y 2 , or 42 revolutions; the compound ratio of the gears will then be 42 revolutions divided by 4 revolutions, or 10^; we will now take 30 and 90 for the one pair, which is 3 to 1, and dividing 10.5 by 3 and we will have thus: 10.5-^-3=3.5, or S J / 2 to 1 for the second pair; thus 20 and 70 or 24 and 84 will answer. 224 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. CHAPTER VII. A Method of Calculating the Speed and the Diame- * ter of Pulleys. The countershaft of a lathe, or any other machine, has a pulley 16" diameter and should run 80 revolutions per min- ute; the main shaft runs at 100 revolutions per minute. What should be the size of pulley on the main shaft? Multiply the diameter of the driven pulley by the speed it should run per minute and divide the product by the speed of the main shaft, thus: 16"=di. of pulley on counter shaft. 80=revolutions per minute. Revolutions of main shaft=100)1280(12.8:=diameter of pulley on 100 main shaft. 280 200 800 800 We have an emery wheel that should run 3,600 revolu- tions per minute; the pulley on the emery wheel spindle is 3 inches diameter and the pulley on the countershaft is 14 inches diameter that is to drive the 3-inch pulley; alongside this 14" pulley on the counter shaft is a pulley 6 inches in diameter; the main shaft runs at 120 revolutions per minute. Required, the diameter of pulley on main shaft. Dividing the 14" pulley on the countershaft by the 3" pulley on the emery wheel spindle we have as follows: 14-f- 3=4 2-3, or 4.66, which means that the speed of the counter- shaft will be as 4.66 is contained in 3600, which is 772.5; then dividing 772.5 by 120, the speed of the main shaft, and we have 6.43, which means that the pulley on the main shaft THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 225 should be 6.43 times as large as the pulley on the counter- shaft which it is to drive; then multiplying tne 6-inch pulley by 6.43 we have 38.5+, or 38y 2 inches for the size of the pulley required. Another way: Multiply the speed required by the diam- eter of the driven pulley on the emery wheel spindle and di- vide by the driver on the countershaft; multiply this by the driven pulley on the countershaft and divide by the speed of the main shaft, thus: 3600X3=10800, and 10800-5-14=771.4; 771.4X6=4628.4; then 4628.4-7-120=38.5, the answer. The countershaft of a machine has no pulley on it, and we want it to run at a speed of 240 revolutions per minute; the main shaft has a pulley 18 inches diameter and has a speed of 130 revolutions per minute. Required, the diameter of pulley on the countershaft. Multiply the diameter of the driver by its number of revolutions per minute and divide this by the number of revolutions required. Thus: 130 revolutions of main shaft. 18 diameter of pulley. 1040 130 240)2340(9.75 inches, the answer. 2160 1800 1680 1200 1200 We have a machine that should run at a speed of 800 revolutions per minute; the countershaft has no pulleys, but we have a pulley 20 inches diameter on the main shaft which runs at a speed of 130 revolutions per minute; our machine has a pulley 5 inches diameter on the spindle. What size pulleys may we select for the countershaft in order to get the speed required? Dividing the speed required for the machine by the 'speed 226 THE MACHINIST AND TOOL MAKER' S INSTBUCTOR. of the main shaft, thus: 800-^-130=6.15, we find that the ratio of speed is as 6.15 to 1. Dividing the diameter of pulley on the main shaft by the one on the spindle, thus: 20 -r-5=4, and we find the ratio of the two pulleys is as 4 to 1. Therefore, dividing the ratio of the speed by the ratio of the pulleys, thus: 6.15-^4=1.53. Then we may select any pulley, say 16 inches for the large one, and dividing 16 bv 1.53= 10.4 inches, which will be the diameter of the other pulley. That is, a belt connecting a 20-inch pulley on the main shaft (130 per minute) with the 10.4 inch pulley on the countershaft, and another belt connecting the 16-inch pulley alongside of the 10.4" pulley on the countershaft with the 5" pulley on the ma- chine will drive it with a speed of 800 revolutions per minute. We have a machine that should run at a speed of 240 revo- lutions per minute; it has no pulley on the spindle, and we wish to use a countershaft that has two pulleys, one of 12 and the other 8" diameter; there is also a pulley 14" diameter on the main shaft, the speed of which is 140 revolutions per min- ute; required, the diameter of the pulley on the machine. Dividing the speed of machine by the speed of the main shaft, thus: 240-^140=about 1.7 for the ratio of speed required. As the speed of the machine should be greater than the main line, we will drive from the 14" pulley on the main shaft to the 8" pulley on the countershaft. Then dividing 14 by 8 we have a ratio of 1.75, and, as this is greater than what we want, (1.7), then the pulley on the machine should be a little larger than the one that is to drive it (the 12") and dividing the ratio of speed that the 14" pulley would drive the 8" (1.75) by the ratio required (1.7), and the answer 1.03Xl2"=12.3" about, for the size required. Or thus, 140X14=1960-^8= 245X12=2940-^-12.3=239-)-, the speed of the machine per minute. The simplest manner, then, to find the diameter of one or more pulleys when the speed of the main shaft and also the machine that you wish to drive is given, is as follows : The main shaft revolves at a speed of 80 revolutions per minute, and we wish to drive the spindle of the machine with THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. Tli a velocity of 2300 revolutions per minute; required, the size of pulleys. 2300-f-80=about 28.7. Now if we want only two (2) pulleys, one of them would be 28.7 times as large as the other; it would not make any difference what it was, so that the large one was 28.7 times the size of the small pulley. But it would be much better to use four pulleys, one on the main shaft, two on the countershaft and one on the spindle. In such a case we should try not to have them vary in size any more than possible, and, therefore, we would take for the first pair a ratio of 5 to 1; that is, say one of 6 inches and the other 30 inches diameter, for the one pair, and dividing 28.7 by the ratio of 5, we have for the next pair a ratio of 5.7+; that is, say the small pulley is 5 inches diameter, then the large pulley will be 5.7 times 5, or about 28.5 inches in diameter. CHAPTER VIII. Condensed Suggestions for Steei, Workers. Crescent Steei, Co., Pittsburg, Pa. (by permission.) ANNEALING. Owing to the fact that the operations of rolling or hammer- ing steel make it very hard, it is frequently necessary that the steel should be annealed before it can be conveniently cut into the required shapes for tools. Annealing or softening is accomplished by heating steel to a red heat and then cooling it very slowly, to prevent it from getting hard again. The higher the degree of heat the more will steel be sof- tened, until the limit of softness is reached, when the steel is melted. It does not follow that the higher a piece of steel is heated the softer it w T ill be when cooled, no matter how slowly it may be cooled; this is proved by the fact that an ingot is always harder than a rolled or hammered bar made from it. 228 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. Therefore, there is nothing gained by heating a piece of steel hotter than a good, bright, cherry red; on the contrary, a higher heat has several disadvantages; first, if carried too far, it may leave the steel actually harder than a good red heat would leave it; second, if a scale is raised on the steel, this scale will be harsh, granular oxide of iron, and will spoil the tools used to cut it. It often occurs that steel is scalded in this way, and then because it does not cut well, it is custom- ary to heat it again, and hotter still, to overcome the trouble, while the fact is, that the more this operation is repeated, the harder the steel will work, because of the hard scale and the harsh grain underneath; third, a high scaling heat, continued for a little time, changes the structure of the steel, destroys its crystalline property, makes it brittle, liable to crack in hardening and impossible to refine. Again, it is common practice to put steel into a hot fur- nace at the close of a day's work and leave it there all night. This method always gets the steel too hot, always raises a scale on it, and worse than either, it leaves it soaking in the fire too long, and this is more injurious to steel than any other operation to which it can be subjected. A good illustration of the destruction of crystalline struc- ture by long continued heating may be had by operating on chilled cast iron. If a chill be heated red hot and removed from the fire as soon as it is hot, it will, when cold, retain its peculiar crystal- line structure; if now it be heated red hot, and left at a moder- ate heat for several hours — in short, if it be treated as steel often is, and be left in the furnace over night, it will be found when cold, to have a perfect amorphous structure, every trace of chill crystals will be gone and the whole piece will be non- crystalline gray cast iron. If this is the effect upon coarse cast iron, what better is to be expected from fine cast steel? A piece of fine tap steel, after having been in a furnace over night, will act as follows: It will be harsh in the lathe and spoil the cutting tools. When hardened it will almost certainly crack; if it does THE MACHINIST AND TOOT, MAKER'S INSTRUCTOR. 229 net crack it will be a remarkably good steel to begin with. When the temper is drawn to the proper color and the tap is put into use, the teeth will either crumble off or crush down like so much lead. Upon breaking the tap, the grain will be coarse and the steel brittle. To anneal any piece of steel, heat it red hot; heat it uni- formly and heat it through, taking care not to let the ends and corners get too hot. As soon as it is hot, take it out of the fire, the sooner the better, and cool it as slowly as possible. A good rule for heating is to heat it at so low a red that when the piece is cold it will still show the blue gloss of the oxide that was put there by the hammer or the rolls. Steel annealed in this way will cut very soft; it will harden very hard, without cracking, and when tempered it will be very strong, nicely refined, and will hold a keen, strong edge. HEATING TO FORGE. Fully as much trouble and loss are caused by improper heating in the forge fire as in the tempering fire, although steel may be heated safely very hot for forging if it be done properly; but any high degree of heat, no matter how uniform it may be, is unsafe for hardening. The trouble in the forge fire is usually uneven heat, and not too high heat. Suppose the piece to be forged has been put into a very hot fire, and forced as quickly as possible to a high yellow heat, so that it is almost up to the scintillating point. If this be done, in a few minutes the outside will be quite soft and in nice condition for forging, while the middle parts will be not more than red hot. The highly heated soft outside will have very little tenacity; that is to say, this part will be so far advanced toward fusion that the particles will slide easily over one another, while the less highly heated inside parts will be hard, possessed of high tenacity, and the particles will not slide so easily over each other. The soft outside will yield so much more readily than 230 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. the hard inside that the outer particles will be torn asunder, while the inside will remain sound, and the piece will be pitched out and branded "burned." Suppose the case to be reversed, and the inside to be much hotter than the outside; that is, that the inside shall be in a state of semi-fusion, while the outside is hard and firm. If now the piece be forged, the outside will be all sound and the whole piece will appear perfectly good until it is cropped, and then it is found to be hollow inside, and it is pitched out and branded "burned." In either case, if the piece had been heated soft all through, or if it had been only red hot all through, it would have forged perfectly sound and good. If it be asked, why then is there ever any necessity for smiths to use a low heat in forging, when a uniform high heat will do as well: We answer: In some cases a high heat is more desirable to save heavy labor, but in every case where a fine steel is to be used for cut- ting purposes, it must be borne in mind that very heavy forging refines the bars as they slowly cool, and if the smith heats such refined bars until they are soft, he raises the grain, makes them coarse, and he cannot get them fine again unless he has a very heavy steam hammer at command, and knows how to use it well. In following the above hints there is a still greater danger to be avoided; that is by letting the steel lie in the fire after it is properly heated. When the steel is hot through it should be taken from the fire immediately and forged as quickly as possible. "Soaking" in the fire causes steel to become dry and brittle, and does it more injury than any bad practice known to the most experienced. HEATING. Owing to the varying instructions on a great many differ- ent labels, we find at times a good deal of misapprehension as to the best way to heat steel; in some cases this causes too THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 231 much work for the smith, and in other instances disasters fol- low the act of hardening. There are three distinct stages or times of heating: First, for forging, Second, for hardening, Third, for tempering. The first requisite for a good heat for forging is a clean fire and plenty of fuel, so that jets of hot air will not strike the corners of the piece; next, the fire should be regular, and give a good uniform heat to the whole part to be forged. It should be keen enough to heat the piece as rapidly as may be, and allow it to be thoroughly heated through, without being so fierce as to overheat the corners. Steel should not be left in the fire any longer than is nec- essary to heat it clear through, as "soaking " in fire is very injurious; and, on the other hand, it is necessary that it should be hot through to prevent surface cracks, which are caused by the reduced cohesion of the overheated parts, which overlie the colder center of an irregularly heated piece. By observing these precautions a piece of steel may al- ways be heated safely up to even a bright yellow heat when there is much forging to be done on it, and at this heat it will weld well. The best and most economical of welding fluxes is clean, crude borax, which should be first thoroughly melted and then ground to a fine powder. Borax prepared in this way will not froth on the steel, and one-half of the usual quantity will do the work as well as the whole quantity unmelted. After the steel is properly heated, it should be forgred to shape as quickly as possible, and just as the red heat is leaving the parts intended for cutting edges these parts should be refined by rapid light blows, continued until the red disap- pears. For the second stage of heating, for hardening, great care should be used; first, to protect the cutting edges and working parts from heating more rapidly than the body of the piece; next, that the whole part to be hardened be heated 232 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. uniformly through, without any part becoming visibly hotter than the other. A uniform heat, as low as will give the re- quired hardness, is the best for hardening. BEAR IN MIND, That for every variation of heat which is great enough to be seen there will result a variation in grain, which may be seen by breaking the piece; and for every such variation in temperature there is a very good chance for a crack to be seen; many a costly tool is ruined by inattention to this point, The effect of too high heat is to open the grain; to make the steel coarse. The effect of an irregular heat is to cause irregular grain, irregular strains and cracks. As soon as the piece is properly heated for hardening, it should be promptly and thoroughly quenched in plenty of the cooling medium; water, brine, or oil, as the case may be. An abundance of the cooling bath, to do the work quickly and uniformly all over, is very necessary to good and safe work. To harden a large piece safely, a running stream should be used. Much uneven hardening is caused by the use of too small baths. For the third stage of heating, to temper, the first im- portant requisite is again uniformity. The next is time. The more slowly a piece is brought down to its temper, the better and safer is the operation. When expensive tools, such as taps, reamers, etc., are to be made, it is a wise precaution, and one easily taken, to try small pieces of steel at different temperatures, so as to find out how low a heat will give the necessary hardness. The low- est heat is the best for any steel; the test costs nothing, takes very little time, and very often saves considerable loss. TEMPER. The word temper, as used by the steel maker, indicates the amount of carbon in steel; thus, steel of high temper is steel containing much carbon; steel of low temper is steel THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. 233 containing little carbon; steel of medium temper is steel con- taining carbon between these limits, etc., etc. Between the highest and the lowest we have some twenty divisions, each representing a definite content of carbon. As the temper of steel can only be observed in the ingot, it is not necessary to the needs of the trade to attempt any description of the mode of observation, especially as this is purely a matter of education of the eye, only to be obtained by years of experience. Likewise, the quality of steel cannot be determined from the appearance of the fracture of a bar as it comes from the hands of the manufacturer. This appearance is determined in the main, by the heat at which the bar is finished, and therefore, one end of a long bar (and especially of a hammered bar) may show a coarse, and the other end, a fine grain, where the whole bar will be well suited for the purpose intended. Two tools properly heated, forged and hardened (one from each end of such a bar) will, if broken, show fractures sirnlar in color and grain. The act of tempering steel is the act of giving to a piece of steel, after it has been shaped, the hardness necessary for the work it has to do. This is done by first hardening the piece, generally a good deal harder than is necessary, and then toughening it by slow heating and gradual softening, until it is just right for work. A piece of steel properly tempered should always be fin- ished finer in grain than the bar from which it is made. If it is necessary, in order to make the piece as hard as is required, to heat it so hot that after being hardened it will be as coarse, or coarser, in grain than the bar, then the steel itself is of too low temper for the desired work. In a case of this kind, the steel maker should at once be notified of the fact, and could immediately correct the trouble by furnishing higher steel. Sometimes an effort is made to harden fine steel without removing (by grinding or other method) the scale formed in rolling, hammering or annealing. The result will generally be disappointing, as steel which would harden through such 234 THE MACHINIST AND TOOL MAKERS INSTRUCTOR. a coating would be of too high a temper where the scale was removed. This surface scale is necessarily of irregular thickness and density, is oxide of iron — not steel — and, therefore, it will not harden, and is to a certain extent a bad conductor of heat. It should, therefore, be removed in every case to insure the best results. If a great degree of hardness is not desired, as in the case of taps, and most tools of complicated form, and it is found that at a moderate heat the tools are too hard and are liable to crack, the smith should first use a lower heat in order to save the tools already made, and then notify the steel maker that his steel is too high, so as to prevent a recurrence of the trouble. In all cases where steel is used in large quantities for the same purpose, as in the making of taps, reamers, drills, etc., there is very little difficulty about temper, because, after one or two trials, the steel maker learns what his customer requires and can always furnish it to him. In large general works, however, such as a rolling mill and nail factory, or large machine works, or large railroad shops, both the maker and worker of the steel labor under great disadvantages from want of a mutual understanding. The steel maker receives his order and fills the sizes, of temper best adapted to general work, and the smith usually tries to harden all tools at about the same heat. The steel maker is right, because he is afraid to make the steel too high or too low, for fear it will not suit, and so he gives an average adapted to the size of the bar. The smith is right, because he is generally the most hurried and crowded man about the es- tablishment. He must forge a tap for this man, a cold nail •knife for that one, and a lathe tool for another, and so on; and each man is in a hurry. Under these circumstances he cannot be expected to stop and test every piece of steel he uses, and find out exactly at what heat it will harden best and refine properly. He needs steel that will all harden properly at the same heat, and this he usually gets from the general practice among steel makers of making each bar of a certain temper, according to its size. THE MACHINIST AND TOCXL MAKERS INSTRUCTOR. 235- But if it should happen that he were caught with only one bar of say, inch and a quarter octagon, and three men should come in a hur-y, one for a tap, another for a punch and an- other for a chilled roll plug, he would find it very difficult to make one bar of steel answer for all of these purposes even if it were of the very be#t quality. The chances are that he would make one good tool and two bad tools; and when the steel maker came around to inquire, he would find one friend and two enemies, and the smith puzzled and in doubt. There is a perfectly easy and simple way to avoid all of this trouble; and that is, to write after each size the purpose for which it is wanted, as for instance, lathe tools, taps, dies, hot or cold punches, shear knives, etc. This gives very little trouble in making the order, and it is the greatest relief to the steel maker. It is his delight to get hold of such an order, for he knows that when it is filled he will hardly ever hear a. complaint. Every steel maker worthy of the name knows exactly what temper to provide for any tool, or if it is a new case, one or two trials are enough to inform him, and as he always should have twenty odd tempers on hand, it is just as easy, and far more satisfactory to both parties, to have it made right as to have it made wrong. For these reasons we urge all persons to specify the work the steel is to do, then the smith can harden all tools at about the same heat, and he will not be annoyed by complaints, or hints that he does not do his work well. MISCELLANEOUS. (From The Pratt & Whitney Co.) For comprehensive information regarding the subject of standard pipe and pipe threads, as applied to American prac- tice, we would refer all who may be interested to the Excerpt Minutes of Proceedings of the Institution of Civil Engineers 236 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. -of Great Britain, Vol. LXXI, Session 1882-3, Part I, contain- ing the papers of the late Robert Briggs, C. E., presented and read after his death, on "American Practice in Warming Buildings by Steam.'' The following extracts from the paper of Mr. Briggs (in- cluded more fully in the report of the committee on standard pipe and pipe threads, American Society of Mechanical Engi- neers, Vol. VIII, transactions,) are here presented, giving data upon which the Briggs standard pipe-thread sizes are oased. The taper employed for the conical tube ends is uniform with all makers of tubes or fittings, namely, an inclination of 1 in 32 to the axis. Custom has also established a particular length of screwed end for each different diameter of tube. Tubes of the several diameters are kept in stock by manu- facturers and merchants, and form the basis of a regular trade in the apparatus for warming by steam. A knowledge of all these particulars is therefore essential for designing apparatus for the purpose. The ruling dimensions in wrought iron tube work is the external diameter of certain nominal sizes, which are designated roughly according to their internal diameter. These nominal sizes were mainly established in the English tube trade between 1820 and 1840, and certain pitches of screw-thread w r ere adopted for them, the coarseness of the pitch carrying roughly with the diameter, but in an arbitrary way utterly devoid of regularity. The length of the screwed portion on the tube end varies with the external diameter of the tube according to an arbitrary rule of thumb, whence results, for each size of tube, a certain mini- mum of thickness of metal at the outer extremity of the taper- ing screwed tube-end. It is the determination of this mini- mum thickness of metal, for the tapering screwed end of a wrought-iron tube, which constitutes the question of mechani- cal interest. The thread employed has an angle of 60°; it is slightly rounded off, both at the top and at the bottom, so that the height or depth of the thread, instead of being exactly equal THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 237 to the pitch, is only four-fifths (4-5) of the pitch, or equal to 0.08 1/n, if n be the number of threads per inch. For the length of tube-end throughout which the screw-thread con- tinues perfect, the empirical formula used is (0.8D+4.8) Xl/n, where D is the actual external diameter of the tube throughout its parallel length, and is expressed in inches. Farther back, beyond the perfect threads, come two having the same taper at the bottom, but imperfect at the top. The remaining im- perfect portion of the screw-thread farthest back from the extremity of the tube is not essential in any way to this sys- tem of joint; and its imperfection is simply incidental to the process of cutting the thread at a single operation. STANDARD DIMENSIONS OF WROUGHT IRON WELDED TUBES. (Briggs Standard. n DIAMETER OF TUBE. ") SCREWED ENDS. NO. OF THREADS j LENGTH OF NOMINAL ACTUAL ACTUAL THICKNESS PERFECT INSIDE. INSIDE. OUTSIDE. OF METAL. THREAD AT PER INCH. BOTTOM. Vs in. 0.270 in. 0.405 in. 0.068 in. 27 0.19 in. H " 0.364 " 0.540 44 0.088 M 18 0.29 M fk " 0.494 M 0.675 44 0.091 44 18 0.30 44 { A " 623 M 0.840 4 * 0.109 44 14 039 " % " 0.824 " 1.050 ' 4 0113 44 14 0.40 M 1 " 1.048 " 1.315 44 0.134 44 1V4 1VA 0.51 44 Ui" 1.380 " 1.660 44 0.140 44 0.54 4< VA ' 1.610 " 1.900 4< 0.145 M IVA 0.55 44 2 " 2.067 " 2.375 4I 0.154 44 11H 0.58 44 2*4 M 2.468 M 2.875 " 0.204 44 8 ^.89 44 3 M 3.067 " 3.500 '■ .217 " 8 0.95 4 * PA" 3.548 M 4.000 44 .226 ,4 8 1.0 ,4 4 4t 4.026 l- 4.500 44 .237 44 8 1.05 44 4 1 /* " 4.508 " 5.000 44 .246 44 8 1.10 " 5 " 5.045 4t 5.563 44 .259 44 8 1.16 * 4 6 " 6.065 44 6.625 44 .280 44 8 1.26 " 7 M 7.023 44 7.625 44 .301 44 8 1.36 M 8 M 7.982 4i 8.625 44 .322 44 8 1.46 4t *9 M 9.000 " 9.688 M .344 44 8 1.57 * 4 10 " 10.019 4< 10.750 44 .366 ' 4 8 1.68 ,s *By the action of the Manufacturers of Wrought Iron Pipe and Boiler Tubes, at a meeting held in New York, May 9, 1889, a change in size of actual outside diameter of 9 238 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. inch pipe was adopted, making the latter 9.625 instead of 9.688 inches, as given in the table of Briggs standard pipe ■diameters. The sizes of Twist Drills to be used in boring holes to be reamed with Pipe Reamer, and threaded with Pipe Taps are as follows : SIZE OF TAP. DIAM. OF DRILL SIZE OF TAP. DIAM. of DRILL. y% inch. i/ <« 21/64 inch. 29/64 " 19/32 " 23/32 " 15/16 " 1 inch. IX " 2 " 2/ 2 " 3 1 T \ inch. 115 « if! " 2A " Ol 1 " The angle for turning or grinding pipe taps and reamers is 1° 47'=(angle from center of axis.) THE SELLERS SYSTEM. Recommended by the Franklin Institute of Philadelphia, has been adopted by the United States Government, the Mas- ter Mechanics and Master Car Builders Associations, Locomo- tive Works, Machine Bolt Makers and by many manufacturing establishments throughout the country. The thread has an angle of 60 degrees, with flat top and bottom equal to one- eighth of the pitch. The advantages of this form of thread over the sharp V are that, in the tap, the edges of the thread are less liable to accidental injury, and will wear and retain their size and form longer, and, in the bolt, the flat top and bottom give increased strength and an improved appearance, while the greater facility with which practical uniformity and consequent interchangeability is now attained by it's use, as compared with tne Witworth form, will commend it to the at- tention of every user of taps and dies, wherever its application may be possible. The accompanying table gives the standard diameter and number of threads per inch for all usual sizes, from one-quar- ter inch to six inches, inclusive. THE MACHINIST AND TOOL. MAKER'S INSTRUCTOR. 239 SEINERS OR U. S. STANDARD. Diameter X A H A % A # ^ ^ Ko. Threads per inch 20 18 16 14 13 12 11 10 9 Diameter 1 1% 1% 1# 1% IjK 1% 1# 2 No. Threads per inch 877665^55 4^ Diameter 2% 2% 2)i 2% 2/ s $U 2% 3 No. Threads per inch , 4^ 4^ 4 4 4 i Z% Z% Diameter 3^ 3X 3^ 3^ 3>£ 3^ 3^ 4 No. Threads per inch 3>£ %y 2 3X 3^ 3^ 3 3 3 Diameter 4/ s ±% 4}i 4 l A 4# 4# 4^ 5 No. Threads per inch 2^ 2j£ 2^ 2# 2>£ 2^ 2# 2% Diameter h% b% 5^ b l / z Q'/& o% oji 6 No. Threads per inch 2/ 2 2% 2j/ s 2^ 2y % 2yi 2% 2% A 6 4, A 50 > # 40 > A 36 > A 32 and A 28 are al so according to the Sellers system „ Screws and bolts 11-16, 13-16, and 15-16-inch diameter are usually made, having 11, 10 and 9 threads per inch, respect- ively, but under the Sellers formula, strictly followed, they should be 10, 9 and 8, respectively. DIMENSIONS OF PRATT & WHITNEY CO. REAMERS FOR MORSE STANDARD TAPER TWIST DRILIv SOCKET. No Diam. small end. Diam. large end. Gauge Diam. large end. Gauge length. Length of Flutes. Total length Taper per foot. Center Angle or Angle to grind. in. in. in. in. in. in. in. 1 0.365 0.525 0.475 t% 3 5tf 0.605=1°27 / — 2 0.573 0.749 0.699 i% s% S* 0.600=1°26'— 3 0.779 0.982 0.936 3^ 4 ny 2 0.605=1°27 / — 4 1.026 1.283 1.231 4 5 $u 0.615=1°28 / 5 1.486 1.796 1.746 5 6 10 0.625=1°29 / + 6 2.117 2.566 2.500 7X sy 2 ny 2 0.634=1°31 / — 240 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. USEFUL INFORMATION. STEAM. (By permission of the Geo. F. Blake Mfg. Co., New York.) A cubic inch of water evaporated under ordinary atmos- pheric pressure is converted into 1 cubic foot of steam (ap- proximately). The specific gravity of steam (at atmospheric pressure) is .411 that of air at 34° Fahrenheit, and .0006 that of water at the same temperature. 27.222 cubic feet of steam weigh one (1) pound; 13,817 cubic feet of air weight one (1) pound. Locomotives average a consumption of 3000 gallons of water per 100 miles run. The best designed boilers, well set, with good draft, and skillful firing, will evaporate from 7 to 10 pounds of water per pound of first class coal. In calculating horse-power of tubular or flue boilers, con- sider 15 square feet of heating surface equivalent to one nomi- nal horse-power. On one square foot of grate can be burned on an average from 10 to 12 pounds of hard coal, or 18 to 20 pounds soft coal, per hour, with natural draft. With forced draft nearly double these amounts can be burned. Steam engines, in economy, vary from 14 to 60 pounds of feed water and from \y 2 to 7 pounds of coal per hour per in- dicated H.-P. See table following for duty of high grade engines. Condensing engines require from 20 to 30 gallons of water at an average low temperature, to condense the steam represented by every gallon of water evaporated in the boilers supplying engines — approximately for most engines, we say, from 1 to iy 2 gallons condensing water per minute per in- dicated horse-power. Surface condensers should have about 2 square feet of tube (cooling) surface per horse-power for a compound steam THE MACHINIST AND TOOL MAKEK's INSTRUCTOR. 241 engine. Ordinary engines will require more surface accord- ing to their economy in the use of steam. It is absolutely necessary to place air pumps below condensers to get satis- factory results. RATIO OF VACUUM TO TEMPERATURE (Fahrenheit) OF FEED WATER. INCHES VACUUM. 00 212° 11 190° 18 170° 22^ 150° *25 135° 27 tf 112° 28# 92° 29 72° 29J£ 52° ♦Usually considered the standard point of efficiency— Condenser and Air Pump being well proportioned. WEIGHT AND COMPARATIVE FUEL VALUE OF WOOD. 1 cord air-dried hickory or hard maple weighs ^ about 4,500 pounds, and is equal to about 2,000 pounds of coal. 1 cord air-dried white oak weighs about 3,850 pounds, and is equal to about 1,715 pounds of coal. 1 cord air-dried beech, red oak and black oak, weighs about 3,250 pounds, and is equal to about 1,450 pounds of coal. 1 cord air-dried poplar (white wood), chestnut and elm, weighs about 2,350 pounds, and is equal to about 1,050 pounds of coal. 1 cord air-dried average pine, weighs about 2,000 pounds and is equal to about 925 pounds of coal. From the above it is safe to assume that 2% pounds of dry wood is equal to 1 pound average quality of soft coal, and that the full value of the same weight of different woods is very nearly the same; that is, a pound of hickory is worth no 242 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. more for fuel than a pound of pine, assuming both to be dry. It is important that the wood be dry, as each 10 per cent, of water or moisture in wood will detract about 12 per cent, from its value as fuel. DUTY OF STEAM ENGINES. A well known engineer of high authority gives the fol- lowing comparative figures, showing the economy of high grade steam engines in actual practice: Cost per I. H. P. per hour sup- posing coal at $6 per ton. TYP3 OF KNGINK. Tempera- ture of feed water Pouuds of water evaporated per lb. of Cumber- land Coal. Pounds of steam per I. H. P. used per hour. Pounds of Cumber- land Coal used per I. H. P. per hour. Non- Condensing, Condensing, Compound Jacketed, Triple Expansion Jacketed, 210° 100° 100° 100° 10.5 9.4 9.4 9.4 29. 20. 17. 13.6 2.75 2.12 1.81 1.44 $0.0073 0.0056 0.0045 0.0036 The effect of a good condenser and air pump should be to make available about 10 pounds more mean effective pres- sure, with the same terminal pressure; or to give the same mean effective pressure with a correspondingly less terminal pressure. When the load on the engine requires 20 pounds M. E. P., the condenser does half the work; at 30 pounds, one-third of the work; at 40 pounds, one-fourth, and so on. It is safe to assume that practically the condenser will save from one-fourth to one-third of the fuel, and it can be applied to any engine, cut-off, or throttling, where a sufficient supply of water is available. THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 243 USEFUL INFORMATION.— WATER. (Geo. F. Blake Mfg. Co. Doubling the diameter of a pipe increases its capacity- four times. Friction of liquids in pipes increases as the square of the velocity. The mean pressure of the atmosphere is usually estimated at 14.7 pounds per square inch, so that with a perfect vacuum it will sustain a column of mercury 29.9 inches, or a column of water 33.9 feet high at sea level. To find the pressure in pounds per square inch of a col- umn of water, multiply the height of the column in feet by .434. Approximately, we say that every foot elevation is equal to y 2 pound pressure per square inch; this allows for ordinary friction. To find the diameter of a pump cylinder to move a given quantity of water per minute (100 feet of piston being the standard of speed) divide the number of gallons by 4, then ex- tract the square root, and the product will be the diameter in inches of the pump cylinder. To find quantity of water elevated in one minute running at 100 feet of piston speed per minute, square the diameter of the water cylinder in inches and multiply by 4. Example : Capacity of a 5-inch cylinder is desired. The square of the diameter (5 inches) is 25, which multiplied by 4 gives 100, the number of gallons per minute (approximately.) To find the horse-power necessary to elevate water to a given height, multiply the weight of the water elevated per minute in pounds by the height in feet, and divide the pro- duct by 33,000 (an allowance should be added for water fric- tion and a further allowance for loss in steam cylinder, say from 20 to 30 per cent.) The area of the steam piston, multiplied by the steam pres- sure, gives the total amount of pressure that can be exerted. The area of the water piston multiplied by the pressure of water per square inch, gives the resistance. A margin must 244 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. be made between the power and the resistance to move the pistons at the required speed — say from 20 to 40 per cent., ac- cording to speed and other conditions. To find the capacity of a cylinder in gallons, multiply- ing the area in inches by the length of stroke in inches will give the total number of cubic inches; divide this amount by 231 (which is the cubical contents of a U. S. gallon in inches) and the product is the capacity in gallons. WEIGHT AND CAPACITY OF DIFFERENT STANDARD GALLONS OF WATER. Cubic inches Weight of a Gal- Gallons in a in a Gallon. Ion in pounds. Cubic Foot. Imperial or English 277.274 10.00 6.232102 United States 231 8.33111 7.480519 Weight of a cubic foot of water, English Standard, 62.321 lbs. Avoirdupois. Weight of Crude Petroleum, 6)4 lbs. per U. S. gallon. Weight of Refined " 6*4 lbs. per U. S. gallon. 42 gallons to the barrel. A "miner's inch" of water is approximately equal to a supply of 12 U. S. gallons per minute. THE MACHINIST AND TOOL MAKERS' INSTRUCTOR. 245 TABLE OF SQUARES AND HEXAGONS, EXACT SIZE, ACROSS FLATS AND CORNERS. STAND- ACROSS CORNERS. STAND- LARGEST LARGEST STAND- ACROSS ARD SQUARE. ARD ROUND. INSCRIBED SQUARE. INSCRIBED HEXAGUN. ARD HEXAGON. CORNERS. 1/8 .1767 1/8 .0S83 .108 1/8 .1443 5/32 .2209 5/32 .1104 .1352 5/32 .1803 3/16 .2651 3/16 .1325 .1623 3/16 .2165 7/32 .3093 7/32 .1546 .1893 7/32 .2526 1/4 .3535 1/4 .1767 .2165 1/4 .2887 5/16 .4419 5/16 .2209 .2706 5/16 .3608 3/8 .5303 3/8 .2651 .3247 3/8 .4330 7/16 .6187 7/16 .3093 .3788 7/16 .5052 1/2 .7071 1/2 .3535 .4330 1/2 .5773 9/16 .7955 9/16 .3977 .4871 9/16 .6495 5/8 .8839 5/8 .4419 .5412 5/8 .7217 11/16 .9723 11/16 .4861 .5953 11/16 .7938 3/4 1.0606 3/4 .5303 .6495 3/4 .8660 13/16 1.1490 13/16 .5745 .7036 13/16 .9382 7/8 1.2374 7/8 .6187 .7577 7/8 1.0104 15/16 1.3258 15/16 .6629 .8118 15/16 1.0825 1 1.4142 1 .7071 .8660 1 1.1547 1A 1.5026 ItV .7513 .9201 iA 1.2269 l* 1.5910 H .7954 .9742 H 1.2990 1A 1.6794 1A .8396 1.0283 iA 1.3712 246 THE MACHINIST AND TOOL MAKERS INSTRUCTOR. TABLE OF SQUARES AND HEXAGONS, EXACT SIZE, ACROSS FLATS AND CORNERS. STAND- ACROSS CORNERS. STAND- LARGEST LARGEST STAND- A fIDACQ ARD SQUARE. ARD ROUND. INSCRIBED SQUARE. INSCRIBED HEXAGON. ARD HEXAGON AUKUoo CORNERS. H 1.7677 It .8838 1.0825 n 1.443 1A 1.8561 lr 5 s .9280 1.1366 1A 1.5155 if 1.9445 If .9722 1.1907 if 1.5877 1A 2.0329 i* 1.016 1.2448 1A 1.6599 n 2.1213 if 1.060 1.2990 ii 1.7320 1 9 X J<5 2.2097 i& 1.104 1.3531 1 9 1 T¥ 1.8042 1 5 2.2981 if 1.149 1.4072 1 5 1.8764 113 2.3865 m 1.193 1.4613 111 1.9486 if 2.4748 it 1.237 1.5155 If 2.0207 HI 2.5632 m 1.281 1.5696 1H 2.0929 1 7 2.6516 n 1.325 1.6237 if 2.1651 115 J-Te 2.7400 m 1.370 1.6778 itf 2.2372 2 2.8284 2 1.414 1.7320 2 2.3094 2i 3.0052 2i 1.502 1.8402 2f 2.4537 2i 3.1819 2i 1.590 1.9485 n 2.5980 2f 3.3587 2f 1.679 2.0567 2f 2.7424 » 3.5355 2i 1.767 2.1650 2| 2.8867 2f 3.7123 2f 1.856 2.2732 2 5 - 3.0311 2| 3.8890 2| 1.944 2.3815 H 3.1754 21 4.0658 2J 2.033 2.4897 2f 3.3197 3 4.2426 3 2.121 2.5980 3 3.4641 3J 4.5961 H 2.298 2.8145 H 3.7528 3^ 4.9497 3i 2.474 3.0310 H 4.0414 3| 5.3032 81 2.651 3.2475 3-i 4.3301 4 5.6568 4 2.828 3.4640 4 4.6188 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 247 TAPERS PER INCH AND CORRESPONDING ANGLES, ALSO THE ANGLES TO TURN OR GRIND. TAPER PER INCH. INCLUDED ANGLE. ANGLE TO TURN OR GRIND. TAPER PER INCH. INCLUDED ANGLE. ANGLE TO TURN OR GRIND. 1-64" 0° 54' 0° 27'- 33-64" 28° 54' 14° 27'+ 1-32" 1° 48' 0° 54'- 17-32 29° 46' 14° 53'— 3-64 2° 42' 1° 21'— 35-64 30° 36' 15° 18'— 1-16 3° 34' 1° 47'+ 9-16 31° 24' 15° 42'+- 5-64 4° 28' 2° 14' 37-64 32° 14' 16° 7'+ 3-32" 5° 22' 2° 41' 19-82" 33° 4' 16° 32'+ 7-64" 6° 16' 3° 8'- 39-64 33° 54' 16° 57'- 1-8 7° 10' 3° 35'- 5-8 34° 42' 17° 21'+ 9-64 8° 2' 4° 1'+ 41-64 35° 32' 17° 46'— 5-32 8° 56' 4° 28' 21-32 36° 20' 18° 10'- 11-64" 9° 50' 4° 55'— 43-64" 37° 8' 18° 34'+ 3-16" 10° 42' 5° 21'+ 11-16 37° 56' 18° 58'+ 13-64 11° 36' 5° 48'- 45-64 38° 44' 19° 22'+ 7-32 12° 28' 6° 14'+ 23-32 39° 32' 19° 46' 15-64 13° 22' 6° 41' 47-64 40° 20' 20° 1C- 1-4 " 14° 16' 7° 8'— 3-4 " 41° 6' 20° 33'+ ft-04" 15° 8' 7° 34'- 49-64 41° 54' 20° 57'- 9-32 16° 0' 8° 0'+ 25-32 42° 40' 21° 20'+ 19-64 16° 54' 8° 27'- 51-64 43° 26' 21° 43'4 5-16 17° 46' 8° 53'— 13-16 44° 14' 22° 7'— 21-64" 18° 38' QO 19' f 53-64" 44° 58' 22° 29'+ 11-32" 19° 30' 9° 45'+ 27-32 45° 44' 22° 52'+ 23-64 20° 22' 10° 11'+ 55-64 46 a 30' 23° 15'+ 3-8 21° 14' 10° 37'+ 7-8 47° 16' 23° 38' 25-64 22° 6' 11° 3' 57-64 48° 0' 24° 0'+ 13-32" 22° 58' 11° 29'— 29-32" 48° 46' 24° 23'- 27-64" 23° 48' 11° 54'+ 59-64 49° 30' 24° 45'- 7-16 24° 40' 12° 20'+ 15-16 50° 14' 25° 7'- 29-64 25° 32' 12° 46'— 61-64 50° 58' 25° 29'- 15-32 26° 22' 13° 11'+ 31-32 51° 42' 25° 51'- 31-64" 27° 14' 13° 37'— 63-64" 52° 24' 26° 12'+ 1-2" 28° 4' 14° 2'+ 1 53° 8' 26° 34' 248 THE MACHINIST AND TOOL MAKER S INSTRUCTOR. TAPERS PER FOOT AND CORRESPONDING ANGLES, ALSO THE ANGLES TO TURN OR GRIND. TAPER PER FOOT. INCLUDED ANGLE. ANGLE TO TURN OR GRIND. TAPER PER FOOT. INCLUDED ANGLE. ANGLE TO TURN OR GRIND. 1-8 " 0° 34' 0° 17'+ 1 " 4° 46' 2° 23'+ 9-64" 0° 40' 0° 20'+ 1 1-16 5° 4' 2° 32'+ 5-32 0° 44' 0° 22'+ 1 1-8 5° 22' 2° 41' 11-64 0° 50' 0° 25'- 1 3-16 5° 40' 2° 50'— 3-16 0° 54' 0° 27'- 1 1-4 5° 58' 2° 59'— 13-64" 0° 58' 0° 29'+ 1 5-16" 6° 16' 3° 8'- 7-32" 1° 2' 0° 31'+ 1 3-8 6° 34' 3° 17'- 15-64 1° 8' 0° 34'— 1 7-16 6° 52' 3° 26'- 1-4 1° 12' 0° 36'— 1 1-2 7° 10' 3° 35'— 17-64 1° 16' 0° 38'+ 1 9-16 7° 26' 3° 43'+ 9-32" 1° 20' 0° 40' + 1 5-8 " 7° 44' 3° 4° 4° 4° 4° 52'+ 1'+ 10'+ 19'+ 28' 19-64" 1° 26' 0° 43'- 1 11-16 8° 2' 5-16 1° 30' 0° 45'— 1 3-4 8° 20' 21-64 1° 34' 0° 47' 1 13-16 8° 38' 11-32 1° 38' 0° 49'+ 1 7-8 8° 56' 23-64" 1° 42' 0° 51'+ 1 15-16" 9° 14' 4° 37'— 3-8 " 1° 48' 0° 54'— 2 9° 32' 4° 46'- 25-64 1° 52' 0° 56'- 2 1-8 10° 8' 5° 4'- 13-32 1° 56' 0° 58'+ 2 1-4 10° 42' 5° 21'+ 27-64 2° 0' 1° 0'+ 2 3-8 11° 18' 5° 39'+ 7-16" 2° 4' 1° 2'f 2 1-2 " 11° 54' 5° 57'- 29-64" 2° 10' 1° 5'- 2 5-8 12° 30' 6° 15'-- 15-32 2° 14' 1° 7'+ 2 3-4 13° 4' 6° 32'+ 31-64 2° 18' 1° 9'+ 2 7-8 13° 40' 6° 50'— 1-2 2° 24' 1° 12'— 3 14° 16' 7° 8'— 17-32" 2° 32' 1° 16'+ 3 1-8 " 14° 50' 7° 25'+ 9-16" 2° 42' 1° 21'- 3 1-4 15° 26' 7° 43'- 19-32 2° 50' 1° 25' 3 3-8 16° 0' 8° 0'+ 5-8 2° 58' 1° 29'+ 3 1-2 16° 36' 8° 18'- 21-32 3° 8' 1° 34'- 3 5-8 17° 10' 8° 35'+ 11-16" 3° 16' 1° 38'+ 3 3-4 " 17° 46' 8° 53'- 23-32" 3° 26' 1° 43'- 3 7-8 18° 20' 9° 10'+ 3-4 3° 34' 1° 47' + 4 18° 56' 9° 28'- 25-32 3° 41' 1° 52'— 4 1-8 19° 30' 9° 45'+ 13-16 3° 52' 1° 56'+ 4 1-4 20° 4' 10° 2'+ 27-32" 4° 2' 2° 1'— 4 1-2" 21° 14' 10° 37'+ 7-8" 4° 10' 2° 5'+ 4 3-4 22° 24' 11° 12'- 29-32 4° 20' 2° 10'- 5 23° 32' 11° 46' 15-16 31-32 4° 4° 28' 38' 2° 2° 14'+ 19'— THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 249 TAP DRILLS FOR SHARP V AND U. S. STANDARD TIJREADS. The sizes of Twist Drills given in these Tables are correct for general purposes. NO. OF THREADS PES INCH DIAMETER OF SIZE OF DIAMETER OF TAP AT BOTTOM OF U. S. STAND. THREAD. SIZE OF DIAMETER OF TAP. TAP AT BOTTOM OF V THREAD. DRILL FOR V THREAD. DRILL FOR U.S. STAND. THREAD. f 16 .142" 11-64" ) 1-4" { f 18 20 16 .154 .163 .173 3-16 [ 3-16 ) 13-64 .185" No. 11. 9-32 { 18 20 .185 .194 7-32 7-32 5-16 I 16 18 .204 .216 15-64 1 1-4 J .240 1-4" 11-32 \ 16 18 14 .235 .247 .251 17-64 9-32 19-64 ) 3-8 ( 16 18 14 .267 .279 .282 19-64 y 5-16 J 21-64 .294 19-64 13-32 X 16 18 .298 .310 21-64 11-32 7-16 \ 14 16 .313 .329 23-64 X 23-64 J .344 23-64 15-32 { ( 14 16 12 .344 .360 .356 25-64 25-64 13-32 ) 1-2 f 13 14 12 .366 .376 .387 27-64 Y 27-64 J 7-16 .399 13-32 17-32 1 13 14 .397 .407 7-16 29-64 9-16 I 12 14 .418 .438 15-32 \ 31-64 j .454 15-32 19-32 { 12 14 .449 .469 1-2 33-64 5-8 1 10 11 .452 .468 1-2 ) 33-64 Y .507 33-64 12 .481 17-32 ) ! 10 .483 17-32 21-32 11 .499 35-64 12 .512 9-16 250 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. TAP DRILLS FOR SHARP V AND U. S. STANDARD THREADS. [CONTINUED.] DIAMETER OF TAP. NO. OF THREADS PER INCH. DIAMETER OF TAP AT BOTTOM OF V THREAD. SIZE OF DRILL FOR V THREAD. DIAMETER OF TAP AT BOTTOM OF U. S. STAND. THREAD. SIZE OP DRILL FOR U.S. STAND THREAD. 11-16" { 11 12 .53( " .543 37-64" 19-32 H { 10 11 ' 12 .577 .593 .606 5-8 ) 41-64 y 21-82 j .62 5-8" 25-32 -j 10 11 12 .608 .624 .637 21-32 43-64 11-16 13-16 { 9 10 .620 .639 43-64 11-16 7-8 { 9 10 .683 .702 47-64 1 3-4 ; .731 47-64 29-32 { 9 10 .714 .733 49-64 25-32 15-16 j 8 9 .720 .745 25-32 51-64 1 8 .783 27-32 .837 27-32 If { 7 8 .878 .908 61-64 "I 31-32 J .940 61-64 1 5 J 7 8 .909 .939 63-64 1 u 7 1.003 1 5 1.065 1& 1 9 7 1.034 1*¥ 11 6 1.086 m 1.158 m 113 1^2 6 1.117 igf H 6 1.211 m 1.283 hi 117 -I 35 6 1.242 m « { 5 5| 1.279 1.311 Hi I iii J 1.389 m 11 5 1.404 ill 1.490 H « i 5 1.491 1.529 HI 1 1.615 if 2 ih 1.616 Iff 1.712 Iff THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 251 TAP DRILLS FOR MACHINE SCREWS. The sizes of Drills given in this Table correspond to the Brown & Sharpe Twist Drill and Steel Wire Gauge. DIAMETER OF NUMBER THREADS PER DIAMETER TAP AT BOTTOM SIZE OF DRILL OF TAP. INCH. OF TAP. OF THREAD. FOR TAPPING. 4 32 .110 .056 No. 45 4 36 .110 .062 44 44 4 40 .110 .067 44 44 6 30 .137 .080 No. 37 6 32 .137 .083 * 35 6 36 .137 .089 " 35 6 40 .137 .094 44 as 8 24 .163 .091 No. 30 8 30 .163 .106 44 29 8 32 .163 .109 " 29 8 36 .163 .115 44 29 10 20 .189 .103 No. 29 10 22 .189 .111 44 28 10 24 .189 .117 " 24 10 30 .189 .132 44 22 10 32 .189 .135 " 20 12 20 .216 .130 No. 20 12 22 .216 .138 44 19 12 24 .216 .144 44 15 14 16 .242 .134 No. 19 14 18 .242 .146 44 16 14 20 .242 .156 44 13 14 22 .222 .164 44 10 14 24 .242 .170 44 5 16 16 .268 .160 No. 11 16 18 .268 .172 44 6 16 20 .268 .182 •• 3 16 22 .268 .190 44 2 18 16 .295 .187 No. 2 18 18 .295 .199 44 1 18 20 .295 .209 11 15-64 20 16 .321 .213 No. 1-4 20 18 .321 .225 44 1-4 20 20 .321 .235 44 17-64 22 16 .347 .239 No. 17-64 22 18 .347 .251 44 9-32 24 14 .374 .250 No. 19-64 24 16 .374 .266 44 19-64 24 18 .374 .278 44 5-16 26 16 .400 .292 No. 21-64 l 252 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. TABLE OF DECIMAL EQUIVALENTS OF STUBS STEEI WIRE GAUGE. The Crescent Drill Rods made by Miller, Metcalf and Parkin, Pitts T)urgh, Pa., also correspond to these sizes. LETTER. SIZE OF LETTER IN DECI- MALS. NO. OP WIRE GAUGE. SIZE OF NUMBER IN DECI- MALS. NO. OP WIRE GAUGE. SIZE OF NUMBER IN DECI- MALS. NO. OF wirh: GAUGE. SIZE OF NUMBER IN DECI- MALS. z .413 1 .227 28 .139 55 .050 Y .404 2 .219 29 .134 56 .045 X .397 3 .212 30 .127 57 .042 W .386 4 .207 31 .120 58 .041 V .377 5 .204 32 .115 59 .040 TJ .368 6 .201 33 .112 60 .039 T .358 7 .199 34 .110 61 .038 S .348 8 .197 35 .108 62 .037 B .339 9 .194 36 .106 63 .036 Q, .332 10 .191 37 .103 64 .035 P .323 11 .188 38 .101 65 .033 O .316 12 .185 39 .(199 66 .032 N .302 13 .182 40 .097 67 .031 M .295 14 .180 41 .095 68 .030 L .290 15 .178 42 .092 69 .029 K .281 16 .175 43 .088 70 .027 J .277 17 .172 44 .085 71 .026 I .272 18 .168 45 .081 72 .024 H .266 19 .164 46 .079 73 .023 G .261 20 .161 47 .077 74 .022 F .257 21 .157 48 .075 75 .020 E .250 22 .155 49 .072 76 .018 D .246 23 .153 50 .069 77 .016 C .242 24 .151 51 .066 78 .015 B .238 25 .148 52 .063 79 .014 A .234 26 .146 53 .058 80 .013 27 .143 54 .055 THE BROWN AND SHARPE TWIST DRILX AND STEEL, WIRE GAUGE. Size of Numbers. SIZE OF SIZE OF SIZE OF SIZE OF NO. NUMBER IN NO. NUMBER IN NO. NUMBER IN NO. NUMBER IN DECIMALS. DECIMALS. DECIMALS. DECIMALS. 1 .2280 16 .1770 31 .1200 46 .0810 2 .2210 17 .1730 32 .1160 47 .0785 3 .2130 18 .1695 33 .1130 48 .0760 4 .2090 19 •1660 31 .1110 49 .0730 5 .2055 20 .1610 35 .1100 50 .0700 6 .2040 21 .1590 36 .1065 51 .0670 7 .2010 22 .1570 37 .1040 52 .0635 8 .1990 23 .1540 38 .1015 53 .0595 9 .1960 24 .1520 39 .0995 54 .0550 10 .1935 25 .1495 40 .0980 55 .0520 11 .1910 26 .1470 41 .0960 56 .0465 12 .1890 27 .1440 42 .0935 57 .0430 13 .1850 28 .1405 43 .0890 58 .0420 14 .1820 29 .1360 44 .0860 59 .0410 15 .1800 30 .1285 45 .0820 60 .0400 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 253 DIFFERENT STANDARDS FOR WIRE GAUGE IN USE IN THE UNITED STATES. Dimensions of Sizes in Decimal Parts of an Inch. (Brown & Sharpe Mfg. Co.) 03 K . fH 05 W w fe 2 53 &■ 2 tf O fc Ph «og3 fcOgJH 1 *** y, 6 oq 55 2 . •« £ C (25 gss» ce ^ ^ m. * 000000 .4687 «~ *, & 00000 .45 .4375 •g<»5 0000 .46 .454 .3938 .4 .4062 je a '-& 000 .40964 .425 .3625 .36 .375 00 .3648 .38 .3310 .33 .3437 .32486 .34 .3065 .305 .3125 MCC-tf 1 .2893 .3 .2830 .285 .227 .2812 IJjy 2 2 .25763 .284 .2625 .265 .219 .2656 3*3 3 .22942 .259 .2437 .2JL5 .212 .25 4 .20431 .238 .2253 .225 .207 .2343 % tn^ 5 .18194 .22 .2070 .205 .204 .2187 ~*o 6 .16202 .203 .1920 .19 .201 .2031 s« £ 7 .14428 .18 .1770 .175 .199 .1875 a Er 8 .12849 .165 .1620 .16 .197 .1718 $■-$ 9 .11443 .148 .1483 .145 .194 .1562 £- 10 .10189 ,134 .1350 .13 .191 .1406 M £ ii . 11 .090742 .12 .1205 .1175 .188 .125 12 .080808 .109 .1055 .105 .185 .1093 g pj 13 .071961 .095 .0915 .0925 .182 .0937 8 fc i 14 .064084 .083 .0800 .08 .18) .0781 29 g.-3 lo .057068 .072 .0720 .07 .178 .0703 1 S ~ -5 36 .05082 .065 .0625 .061 .175 .0625 8«SPa 17 .045257 .058 .0540 .0525 .172 .0562 18 .040303 .049 .0475 .045 .168 .05 s*n 19 .03589 .042 .0410 .04 .164 .0437 * « u 20 21 .031961 .f 28462 .035 .032 .0348 .03175 .035 .031 .161 .157 .0375 .0343 22 .025347 .028 .0286 .028 .155 .0312 23 .022571 .025 .0258 .025 .153 .0281 bjO ^ Ik 24 .0201 .022 .0230 .0225 .151 .025 p'd'd u 25 .0179 .02 .0204 .02 .148 .0218 /11 ctf .„ « 26 .01594 .018 .0181 .018 .146 .0187 27 .014195 .016 .0173 .017 .143 .0171 2 ..2>» 28 .012641 .014 .0162 .016 .139 .0156 29 .011257 .013 .0150 .015 .134 .0140 30 .010025 .012 .0140 .014 .127 •Q25 0^"^ 31 .008928 .01 .0132 .013 .120 .0109 £°s£ 32 .00795 .009 .0128 .012 .115 .0101 - Sc-g 33 .00708 .008 .0118 .011 .112 .0i.93 3.3 « 3 34 .006304 .007 .0104 .01 .110 .0086 35 .005614 .005 .0095 .0095 .108 .0078 36 .005 .004 .0090 .009 .106 .0070 37 38 39 40 .004453 .003965 .003531 .003144 .0085 .008 .0075 .007 .103 .101 .099 .097 .0066 .0062 ^J V* ? '■" ^Sfc *8? So 3 1 254 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. CIRCUMFERENCES AND AREAS OF CIRCLES. CIRCUMFER- CIRCUMFER- DIAMETER. ENCE. AREA. DIAMETER. ENCE. AREA. 1/16 .1963 .00307 4 12.5664 12.5664 % .3927 .01227 4^ 12.9591 13.3641 H .7854 .04908 m 13.3518 14.1863 % 1.1781 .11045 Ws 13.7445 15.0330 Mi 1.5708 .19635 V/2 14.1372 15.9043 Yb 1.9635 .30679 4% 14.5299 16.8002 % 2.3562 .44179 4% 14.9226 17.7206 % 2.7489 .60132 m 15.3153 18.6655 1 3.1416 .7854 5 15.7080 19.6350 m 3.5343 .9940 Ws 16.1007 20.6290 1M 3.9270 1 .2272 fyi 16.4934 21.6476 Wa 4.3197 1.4849 r o% 16.8861 22.6907 m 4.7124 1.7671 % 17.2788 23.7583 m 5.1051 2.0739 &/s 17.6715 24.8505 i% 5.4978 2.4053 *% 18.0642 25.9673 Ws 5.8905 2.7612 5% 18.4569 27.1086 . 2 6.2832 3.1416 6 18.8496 28.2744 2% 6.6759 3.5466 V/s 19.2423 29.4648 2M 7.0686 3.9761 614 19.6350 30.6797 Ws 7.4613 4.4301 6% 20.0277 31.9191 m 7.8540 4.9087 6^ 20.4204 33.1831 2% 8.2467 5.4119 &/s 20.8131 34.4717 2% 8.6394 5.9396 6% 21.2058 35.7848 2% 9.0321 6.4918 6% 21.5985 37.1224 3 9.4248 7.0686 7 21.9912 38.4846 Ws 9.8175 7.6699 m 22.3839 39.8713 3M 10.2102 8.2958 7M 22.7766 41.2826 m 10.6023 8.9462 Ws 23.1693 42.7184 zy* 10.9956 9.6211 m 23.5620 44.1787 3% 11.3883 10.3206 Ws 23.9547 45.6636 m 11.7810 11.0447 m 24.3174 47.1731 Wa 12.1737 11.7933 7% 24.7401 48.7071 THE MACHINIST AND TOOL. MAKER'S INSTRUCTOR. 255 CIRCUMFERENCES AND AREAS OF CIRCLES [continued.] CIRCUMFER- CIRCUMFER- DIAMETER. ENCE. AREA. DIAMETER. ENCE. AREA. 8 25.1328 50.2656 12 37.6992 113.097 m 25.5255 51.8487 im 38.0919 115.466 *u 25.9182 53.4563 im 38.4846 117.859 8% 26.3109 55.0884 12% 38.8773 120.276 m 26.7036 56.7451 12^ 39.270 122.719 8% 27.0963 58.4264 12% 39.6627 125.185 m 27.4890 60.1322 12% 40.0554 127.676 s% 27 8817 61.8625 12J/8 40.4481 130.192 9 28.2744 63.6174 13 40.8408 132.733 9% 28.6671 65.3968 ! 13% 41.2335 135.297 9% 29.0598 67.2008 ! 13M 41.6262 137.887 9% 29.4525 69.0293 ] 13% 42.0189 140.501 9% 29.8452 70.8823 1 13% 42.4116 143.139 9% 30.2379 72.7599 ! 13% 42.8043 145.802 9% 30.6306 74.6621 1 13% 43.1970 148.490 9% 31.0233 76.5888 13% 43.5897 151.202 10 31.4160 78.5400 14 43.9824 153.938 mi 31.8987 80.5188 14% 44.3751 156.698 iom 32.2014 82.5161 14M 44.7678 159.485 10% 32.5941 84.5409 14% 45.1605 162.296 io% 32.9868 86.5903 im 45.5532 165.130 10% 33.3795 88.6643 14% 45.9459 167.990 10% 33.7722 90.7628 14% 46.3386 170.874 10% 34.1649 92.8858 14% 46.7313 173.782 11 34.5576 95.0334 15 47.1240 176.715 11% 34 9503 97.2055 15% 47.5167 179.673 11M 35 3430 99.4022 1534 47.9094 182.655 n% 35.7357 101.6234 15% 48.3021 185.661 n% 36.1284 103.8691 15% 48.6948 188.692 n% 36.5211 106.1394 15% 49.0875 191.748 n% 36.9138 108.4343 15% 49.4802 194.828 n% 37.3065 110.7537 15% 49.8729 197.933 256 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. CIRCUMFERENCES AND AREAS OF CIRCLES [continued.] CIRCUMFER- CIRCUMFER- DIAMETER. ENCE. AREA. DIAMETER. ENCE. AREA. 16 50.2656 201.062 20 62.8320 314.160 16% 50.6583 204.216 20^ 63.2247 318.099 16% 51.0510 207.395 20M 63.6174 322.063 16% 51.4437 210.598 20% 64.0101 326.051 16% 51.8364 213.825 2oy 2 64.4028 330.064 16% 52.2291 217.077 20% 64.7955 334.102 16% 52.6218 220.354 20% 65.1882 338.164 16% 53.0145 223.655 20% 65.5809 342.249 17 53.4072 226.981 21 65.9736 346.361 17% 53.7999 230.331 21% 66.3663 350.497 17% 54.1926 233.706 2114 66.7590 354.657 17% 54.5853 237.105 21% 67.1517 358.842 17% 54.9780 240.529 213^ 67.5444 363.051 17% 55.3707 243.977 21% 67.9371 367.285 17% 55.7634 247.450 21% 68.3298 371.543 17% 56.1561 250.948 21% 68.7225 375.826 18 56.5487 254.469 22 69.1152 380.134 18% 56.9415 258.0 J 6 22% 69.5079 384.466 18% 57.3342 261.587 22M 69.9006 388.822 18% 57.7269 265.183 22% 70.2933 393.203 18% 58.1196 268.803 22%£ 70.6860 397.609 18% 58.5123 272.448 22% 71.0787 402.038 18% 58.9050 276.117 22% 71.4714 406.494 18% 59.2977 279.811 22% 71.8641 410.973 19 59.6904 283.529 23 72.2568 415.477 19% 60.0831 287.272 23% 72.6495 420.004 19% 60.4758 291.040 2&A 73.0422 424.558 19% 60.8685 294.832 23% 73.4349 429.135 19% 61.2612 298.648 23^ 73.8276 433.737 19% 61.6539 302.489 23% 74.2203 438.364 19% 62.0466 306.355 23% 74.6130 443.015 19% 62.4393 310.245 23% 75.0056 447.690 THE MACHINIST AND TOOI, MAKER'S INSTRUCTOR. 257 CIRCUMFERENCES AND AREAS OF CIRCLES [continued.] CIRCUMFER- CIRCUMFER- DIAMETER. AREA. DIAMETER. AREA. ENCE. ENCE. 24 75.3984 452.39 28 87.9648 615.754 24% 75.7911 457.115 28% 88.3575 621.264 24% 76.1838 461.864 28U 88.7502 626.798 24% 76.5765 466.638 28% 89.1429 632.357 24% 76.9692 471.436 28% 89.5356 637.941 24% 77.3619 476.259 28% 89.9283 643.549 24% 77.7546 481.107 28% 90.3210 649.182 24% 78.1473 485.979 28% 90.7137 654.840 25 78.540 490.875 29 91.1064 660.521 25% 78.9327 495.796 29% 91.4991 666.228 25% 79.3254 500.742 29M 91.8918 671.959 25% 79.7181 505.712 29% 92.2845 677.714 25% 80.1108 510.706 29% 92.6772 683.494 25% 80.5035 515.726 29% 93.0699 689.299 25% 80.8962 520.769 29% 93.4626 695.128 25% 81.2889 525.838 29% 93.8553 700.982 26 81.6816 530.930 30 94.2480 706.860 26% 82.0743 536.048 30% 94.6407 712.763 26% 82.4670 541.190 30M 95.0334 718.6^0 26% 82.8597 546.356 3<% 95.4261 724.642 26% 83.2524 551.547 30% 95.8188 730.618 26% 83.6451 556.763 30% 96.2115 736.619 26% 84.0378 562.003 30% 96.6042 742.645 26% 84.4305 567.267 30% 96.9969 74S.695 27 84.8232 572.557 31 97.3896 754.769 27% 85.2159 577.870 31% 97.7823 760.869 27% 85.6086 583.209 mi 98.1750 766.992 27% 86.0013 588.571 31% 98.5677 773.140 27% 86.3940 593.959 31% 98.9604 779.313 27% 86.7867 599.371 31% 99.3531 785.510 27% 87.1794 604.807 31% 99.7458 791.732 27% 87.5721 610 268 31% 100.1385 797.979 258 THE MACHINIST AND TOOL KAKEE's INSTRUCTOR. CIRCUMFERENCES AND AREAS OF CIRCLES. [continued.] CIRCUMFER- CIRCUMFER- DIAMETER. ENCE. AREA. DIAMETER. ENCE. AREA. 32 100.5312 804.250 36 113.098 1017.878 32% 100.9239 810.545 36% 113.490 1024.960 32% 101.3166 816.865 36^ 113.883 1032.065 32% 101.7093 823.210 36% 114.276 1039.195 32% 102.1020 829.579 36^ 114.668 1046.349 32% 102.4947 835.972 36% 115.061 1053.528 32% 102.8874 842.391 36% 115.454 1060.732 32% 103.2801 848.833 36% 115.846 1067.960 33 103.6728 855.301 37 116.239 1075.213 33% 104.0655 861.792 37% 116.632 1082.490 33% 104.4592 868.309 37M 117.025 1089.792 33% 104.8509 874.850 37% 117.417 1097.118 33% 105.2436 881.415 373^ 117.810 1104.469 33% 105.6363 888.005 37% 118.203 1111.844 33% 106.0290 894.620 37% 118.595 1119.244 33% 106.4217 901.259 37% 118.988 1126.669 34 106.814 907.922 38 119.381 1134.118 34% 107.207 914.611 38% 119.773 1141.591 34% 107.600 921.323 38M 120.166 1149.089 34% 107.992 928.061 38% 120.559 1156.612 34% 108.385 934.822 38% 120.952 1164.159 34% 108.778 941.609 38% 121.344 1171.731 34% 109.171 948.420 38% 121.737 1179.327 34% 109.563 955.255 38% 122.130 1186.948 35 109.956 962.115 39 122.522 1194.593 35% 110.349 969. 39% 122.915 1202.263 35% 110.741 975.909 39M 123.308 1209.958 35% 111.134 982.842 39% 123.700 1217.677 35% 111.527 989.800 39% 124.093 1225.420 35% 111.919 996.783 39% 124.486 1233.188 35% 1133.312 1003.790 39% 124.879 1240.981 35% 1110.705 1110.822 39% 125.271 1248.798 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. 259 CIRCUMFERENCES AND AREAS OF CIRCLES [continued.] CIRCUMFER- CIRCUMFER- DIAMETER. ENCE. AREA. DIAMETER- ENCE. AREA, 40 125.664 1256.64 44 138.230 1520.5 40% 126.057 1264.51 44% 138.623 1529.2 40% 126.449 1272.40 44^ 139.016 1537.9 40% 126.842 1280.31 44% 139.408 1546.6 40% 127.235 1288.25 uy 2 139.801 1555.3 40% 127.627 1296.22 44% 140.194 1564.1 40% 128.020 1304.21 44% 140.587 1572.8 40% 128.413 1312.22 44% 140.979 1681.6 41 128.806 1320.26 45 141.372 1590.4 41% 129.198 1328.32 45% 141.765 1599.3 41% 129.591 1336.41 45^ 142.157 1608.2 41% 129.984 1344.52 45% 142.550 1617.1 41% 130.376 1352.66 45^ 142.943 1625.9 41% 130.769 1360.82 45% 143.335 1634.9 41% 131.162 1369.00 45% 143.728 1644.0 41% 131.554 1377.21 45% 144.121 1652.9 42 131.947 1385.5 46 144.514 1661.9 42% 132.340 1393.7 46% 144.906 1670.9 42% 132.733 1401.9 46M 145.299 1680.0 42% 133.125 1410.3 46% 145.692 1689.1 42% 133.518 1418.6 46y 2 146.084 1698.2 42% 133.911 1426.9 46% 146.477 1707.4 42% 134.303 3435.4 46% 146.870 1716.5 42% 134.696 1443.8 46% 147.262 1725.7 43 135.089 1452.2 47 147.655 1734.9 43% 135.481 1460.7 47% 148.048 1744.2 43% 135.874 1469.1 47^ 148.441 1753.4 43% 136.267 1477.6 47% 148.833 1762.7 43% 136.660 1486.2 47}£ 149.226 1772.1 43% 137.052 1494.7 47% 149.619 1781.4 43% 137.445 1503.3 41'% 150.011 1790.7 43% 137.838 1511.9 47% 150.404 1800.1 260 THE MACHINIST AND TOOL MAKER'S INSTRUCTOR. CIRCUMFERENCES AND AREAS OF CIRCLES. [continued.] CIRCUMFER- CIRCUMFER- DIAMETER. ENCE. AREA. DIAMETER. ENCE. AREA. 48 150.796 1809.6 49 153.938 1885.7 48% 151.189 1819.0 49^ 154.331 1895.4 48% 151.582 1828.5 491/ 4 154.723 1905.0 48% 151.975 1837.9 49^ 155.116 1914.7 48% 152.367 1847.5 49^ 155.509 1924.4 48% 152.760 1856.9 49^ 155.902 1934.2 48% 153.153 1866.5 49% 156.294 1944.0 48% 153.545 1876.1 49% 156.687 1953.7 50 157.08 1963.5 THE MACHINIST AND TOOL MAKEE'S INSTRUCTOR. 261 INDEX. CHAPTER I. ARITHMETIC 5 to 28 Addition of whole numbers and signs of 5 Multiplication of whole numbers and signs of 5 Division of whole numbers and signs of 6 DECIMALS— Addition of 6 Subtraction of ? Multiplication of 7 Division of 8 FRACTIONS— Addition of 9 Subtraction of 10 Multiplication of 12 Division of 13 INVOLUTION, or— Squares, cubes, etc 14 EVOLUTION, or— Square and cube root 14 to 18 MENSURATION— Angles 18 to 22 Solids ! 22 to 28 TRIGONOMETRY 28 Tangents, Sines, Cosines, and secants, denned. 29 to 33 Angles of taper work, How to find 34-36 GEOMETRICAL FIGURES— Hexagon, to find the long and short diameters of 37-38 Squares, to find the long and short diameters of 39 Octagons, to find the long and short diameters of 39 Decagon and Dodecagon, to find the long and short diame- ters of 40 SCREW THREADS— V threads, to find the depths of 41-42 U. S. standard, to find the depths of 42-43 Whitworth, or English standard 44 CRANKS— When two are to be keyed on the same shaft to locate the correct angle 44 to 48 BALLS— To find the groove measurements for 48-50 BELTS— To find the length of a cross belt accurately 50 To find the length of an open belt accurately 52-54 262 THE MACHINIST AND TOOL MAKER* S INSTRUCTOR. JIGS AND TEMPLETS— To lay out and finish accurately 54 to 60, 62 to 65 Pulley, to find the size of from a broken section of the rim. 61 Machines, to set at right angles to the main shaft 62 Natural sines, tangents, cosines and secants, tables of.65 to 110 CHAPTER II. GEARING— Classification of 110 Diametral pitch Hi Sizing- of blanks 112-113 Table of Tooth Dimensions 114 Distances between centers 115 Circular Pitch, with Dimensions 115 Generating Circles 116 Pin Gearing 117-118 Single Curve Gears 119 Double Curve Gears 121-122 Double Curve Gears, Tables 123 Rack and Pinion 124-125 Involute, the perfect 127-128 Epicycloid, the perfect 129 to 132 BEVEL WHEELS— Mitre gears, how to draw 132-123 Ang-ular gears, how to draw 134-135 Bevel gears, shapes of the teeth 135, 137 Bevel gears, how to find the angles of for turning and cut- ting 137-143 Bevel gears, how to cut the teeth 143, 144 Bevel gears, how to find the correct angle for swiveling the Table to cut the teeth 144 to 147 spiral gears- how to find the diameter of Blanks when cut with stand- ard cutters 147 How to find the thickness of cutter when the Blank is of standard size 150 Right hand and left hand spirals 151 To find the angle for setting the Table when cutting 151 WORM GEARING— To find the pitch and thickness of thread of a worm 152 To find the Dimensions of Worm W^heels 152 to 156 Tooth Dimensions, Table of 154 How to find the width of thread at top and bottom 156 How to find the angle for roughing out the teeth of Worm Wheels i57 Attachment for cutting Worm Wheels complete 157-159 How to cut the teeth of Worm Wheels. 156-157 THE MACHINIST AND TOOL MAKER' S INSTRUCTOR. 263 CHAPTER III. THE MILLING MACHINE 160 Advantages of small mills over large ones 163 Relief or clearance of mills 165 Width of land at top of teeth 165 Grooving mills should be hollowing 165 Emery Wheels for grinding 165 Oil, its use in milling 166 Lubricant for milling 167 Work moving with or against the cutters 167 Speed and feed of cutters 168-169 Castings and forgings to be pickled 169 Index Table 170 Indexing, explanation of 171 Cutting spirals with Universal Milling Machines 172 to 176 Selection of gears for cutting spirals 178-179 To find the correct angle for setting the machine 180 The best form of spiral mills for fast cutting 181 Angles, shapes and dimensions of cutters for general pur- poses 181 to 185 Shank mills and special chuck for 185-186 Special chuck for milling bolt heads 187-188 Straight line index drilling 189-191 CHAPTER IV. THE UNIVERSAL GRINDING MACHINE— Position of the Table for grinding Tapers 198-200 Position of machine while grinding two Tapers 201 The manner of holding thin saws and cutters while grind- ing 202 Position in grinding Angular or Taper Cutters 202 Grinding Twin or Straddle Mills 202 A list of Emery Wheels for outside grinding 202 Wheels for internal grinding 203 CHAPTER V. MECHANICS— Questions upon the principles of the Lever 204, 209 Safety Valves 206-208 The Inclined Plane 209 The Screw 210 Wheel and Axle 211 Wheel Gearing 211-212 The Screw and Gear 213 The Hydraulic Press 214 Brake Horse-Power of an Engine 214 264 THE MACHINIST AND TOOL MAKEE's INSTRUCTOR. The Horse-Power of an Engine 215 Pressure on the Guides of an Engine 216 To find the diameter of a shaft for transmitting power 217 Impact or Collision of Bodies 218 Centrifugal Force 219 CHAPTER VI. Screw Cutting by Simple and Compound Gearing 220 to 224 CHAPTER VII. A method of calculating the speed and diameter of Pul- leys 224-227 CHAPTER VIII. THE TREATMENT OF STEEL- Annealing ; 227 Heating to Forge 229 Heating 230 Temper 232 MISCELLANEOUS— The Manufacture of Pipe and Pipe Threads 235 Table of standard dimensions of Wrought Iron Tubes 237 The size of Twist Drills to be used in boring holes to be reamed and threaded with Pipe Taps 238 The Sellers System or the U. S. Standard shape of threads. 238 Diameters of Taps and number of threads from ^4" to 6", inclusive 239 Dimensions of the Pratt & Whitney Co.'s Reamers for Morse Standard Taper Twist Drill Socket 239 Useful Information— Steam 240 to 243 Useful Information — Water 243 to 245 Table of dimensions across flats and corners of Squares and Hexagons 245-24G Tapers per Inch and corresponding Angles, Table of Tapers per Foot and corresponding Angles, Table of c*--248 Tap Drills, &" to 2" 249-250 Tap Drills for Machine Screws 251 Table of Decimal Equivalents of Stubs and the Crescent Steel Co.'s Drill Rods .252 Size of numbers of the Brown & Sharpe Twist Drill and Steel Wire Gauge 252 Different standards for Wire Gauge in the United States.. 253 Circumferences and Areas of Circles 254 to 260 ■ N 589#'