Introductory Artillery Mathematics and Antiaircraft Mathematics [Reprint 2020 ed.] 9780520346109

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Introductory Artillery Mathematics and Antiaircraft Mathematics

Introductory Artillery Mathematics and Antiaircraft Mathematics

By S O P H I A

H.

LEVY

U N I V E R S I T Y OF C A L I F O R N I A P R E S S B E R K E L E Y AND L O S A N G E L E S • 1 9 4 3

UNIVERSITY O F CALIFORNIA PRESS B E R K E L E Y AND LOS ANGELES CALIFORNIA O CAMBRIDGE UNIVERSITY PRESS LONDON, ENGLAND

COPYRIGHT, 1 9 4 3 ,

BY

SOPHIA H . LEVY

PRINTED I N T H E UNITED STATES OF AMERICA BY T H E U N I V E R S I T Y

OF

CALIFORNIA

PRESS

PREFACE This book has been written for the use of men in the armed services of the United States, primarily for those engaged in the antiaircraft service. I t concerns itself with the parts of mathematics essential to their work; it has been kept brief in the hope that brevity would increase its usefulness. The trial shot problem has received emphasis; descriptions of slide rules have been included; tables, a protractor, and a slide rule will be found at the back of the book. Only type problems have been given and all have been solved completely. The writer has been privileged to instruct about one thousand men of the 101st Coast Artillery Brigade, Fourth Antiaircraft Command (AA). Many of them have subsequently completed their courses at the Antiaircraft Artillery Officer Candidate School, Camp Davis, North Carolina. This book is dedicated to students in my own classes in antiaircraft mathematics, past, present, and future. SOPHIA H . LEVY

Associate Professor of Mathematics University of California Berkeley, California March, 1948

CONTENTS PART

PAGE

I. Algebra, Geometry, Trigonometry, Coordinates in Two Dimensions II. Coordinates in Three Dimensions (Terrestrial and Celestial), Determination of True North . . . .

1

27

III. Logarithmic Scales, Slide Rules, Including the Crichlow Slide Rule, and Solutions of Triangles Using Slide Rules and Using Logarithms 33 IV. Motion of a Projectile, the Trial Shot Problem, Lateral Deviations in Slant and Horizontal Planes . . . .

50

V. Solutions of Triangles of the Trial Shot Problem Using Slide Rules and Using Logarithms VI. Graphical Solutions of the Trial Shot Problem

56 . . .

65

VII. Summary of Directions for Solutions of Triangles and the Trial Shot Problem 71 VIII. Tables—Conversion Tables for Angles and Lengths, Four-Place Tables of Natural Trigonometric Functions, Logarithms of Numbers, and Logarithms of the Trigonometic Functions for Angles in Mils . .

76

Index

99

Introductory Artillery Mathematics and Antiaircraft Mathematics PARTI ALGEBRA, GEOMETRY,

TRIGONOMETRY,

C O O R D I N A T E S IN T W O

DIMENSIONS

Certain algebraic, geometric, and trigonometric developments are basic to the study of introductory artillery mathematics and antiaircraft mathematics. They are presented briefly without classification. Both definitions and theorems are included. 1. Ratio and Proportion The ratio of two numbers is expressed by the fraction having the first number for the numerator and the second for the denominator. 2 For example, the ratio of 2 to 3 is expressed by the fraction 3 Similarly, the ratio of a to & is expressed by the fraction - . b Four quantities a, b, c, and d are in proportion if the ratio of the first to the second equals the ratio of the third to the fourth, that is, if - = - , from which it follows that a-d = b-c and that - = - . b d c d 2. Pythagorean Theorem There is a relation between the three sides of a right-angled triangle given by the Pythagorean theorem: the square on the side opposite the right angle, which is called the hypotenuse, is equal to the sum of the squares on the other two sides. By the square of a number is meant the product of a number by itself, written c • c=c 2 .

c2 = a2 + &2 90>/ L

[i]

2

INTRODUCTORY ARTILLERY

MATHEMATICS

Also, if the sides of a triangle are such that the sum of the squares on two sides is equal to the square on the third side, the angle opposite this third side is a right angle. 3. Concerning Parallel Lines, also the Sum of the Angles in Any Triangle If a straight line crosses two lines which are parallel, it makes the same angles with both of them.

angles 1 are equal angles 2 are equal

1 Also, if a straight line crosses two lines and makes the same angles with both of them, the two lines are parallel. T o find the sum of the angles in any triangle, draw a line through any vertex parallel to the opposite side. The other sides of the triangle become transversals of parallel lines, making equal angles as indicated.

t

3+

The sum of angles 1, 2, and 3 is the sum of the angles within the triangle and is also 180 degrees. If two angles of a triangle are known, the third angle may be found by subtracting their sum'from 180 degrees. Or, since angles •5 and 1 (and 2 and 4) add to 180 degrees, angle 5 = angle 2 + angle 3, and angle 4 = angle 1 + angle 3. Thus angle 3 = angle 5 — angle 2, and angle 3 = angle 4 — angle 1. Angle 3 is determined in this way in antiaircraft mathematics.

A N T I A I R C R A F T MATHEMATICS

3

4. Similar Triangles Similar triangles have corresponding angles equal. Corresponding sides are in proportion. In the figures C

angle A = A', B = B', C = C. The two triangles are similar; — = —,— = — ,— = —. One triangle may be placed on the other, a' b' a' c' b' c' as follows:

A

B

and because of the equality of angles, AB is parallel to A'B'. 5. The Circle and ir The length of the circumference of a circle depends on the diameter and is proportional to it. Q C = ird = 2 i rr, — = ir d The area of a circle is proportional to the square on the radius. A = irr2,

— = x r2

CCA T is the Greek letter used to denote the equal ratios. — ,—, — . Its d 2r V value to six decimals is 3.141593. 6. Angles and their Measurement Any two lines drawn from a point form an angle. Call the two sides of the angle the initial and the terminal sides. An angle is

4

INTRODUCTORY ARTILLERY

MATHEMATICS

swept out by revolving the initial side until it occupies the same position as the terminal side. Revolution in the counterclockwise direction is regarded as + ; clockwise revolution is regarded as —.

The angles between perpendicular lines are each 90°. A degree is —

90'f

90^

Jdo'

of a right angle, a minute is — of a degree, a second is — of a minute. 60 60 An angle of 1 degree (1°) is the same as an angle of 60 minutes (60')An angle of 1 minute (1') is the same as an angle of 60 seconds (60")- Use of this angular unit is almost universal in applications of mathematics. For certain military purposes an entirely different unit has been adopted and used in calibrating some instruments and weapons. This unit is the mil. Equivalents

6400 mils 1600 mils 1 mil

360 degrees 90 degrees 360 , 9 , degrees = degrees 6400 160 = —minutes 8

1 degree

160

9

..

mils

ANTIAIRCRAFT

MATHEMATICS

5

These figures are exact. A table converting angles from either of these units to the other appears on page 78. There is still another measure of angle that is important in mathematics—the radian, which may be defined as follows: take a string equal in length to the radius of any circle and lay it along the circumference. Join the ends of the string to the center of the circle. The angle at the center which is subtended by the string is the angle called the radian and is the unit in the system of radian measure, sometimes called circular measure. No matter what the radius of the circle may be, the angle found in this manner is always the same. The radian is an angle a little over 57 degrees. One radius on a circumference subtends one radian at the center of the circle; thus the circumference subtends 2t radians, and an angle of 2tt radians is the same as an angle of 360 degrees or an angle of 6400 mils. A table converting angles from each of these units to the others appears on page 77. 7. Estimates of the Mil In the preceding section the mil has been defined. The name of this unit was suggested by the fact that an angle of 1 mil is approximately the small angle of a right-angled triangle whose two sides adjacent to the right angle are 1 and 1000 (the Latin word for one thousand is "mille"). l 1000

This angle is too small to be pictured mentally—it is roughly the angle subtended by a 6-foot man 6000 feet away (5280 feet is a mile!). It is better to try to picture an angle of several mils—the diameter of the full moon subtends an angle of about 10 mils at an observer on the earth. It is possible in various ways to estimate the angle subtended by a distant object, and the estimate serves as a range finder if the length of the object is known. Thus, range (distance) may be estimated quickly as follows: multiply the length of the object by 1000 and divide by the number of mils the object subtends. It is assumed, of course, that the length of the object is perpendicular to the line of sight; the object may be vertical or horizontal.

6

INTRODUCTORY ARTILLERY

MATHEMATICS

If the following approximations are made: at at at at

1000 yards, 1 yard subtends 1 mil 1000 yards, 2 yards subtend 2 mils 1000 yards, 3 yards subtend 3 mils 1000 yards, 100 yards subtend 100 mils and so on at 5000 yards, 5 yards subtend 1 mil at 5000 yards, 50 yards subtend 10 mils and so on an error of less than 2 per cent is introduced for an angle as large as 300 mils. Since 1 radian is equivalent to 57?30 and 1000 mils to 56?25, an error of less than 2 per cent is introduced if 1000 mils is taken as equivalent to 1 radian. In the last diagram and in the diagram in the next section, all of the radial lines are sensibly equal. Any or all of them may be given the length 1000.

8. Mil-Gauge To calibrate a mil-gauge use a card of convenient pocket size and insert a string through a hole at its center. Tie two knots, one to be held between the teeth, the other to hold the card at reading distance. View a series of equally spaced marks placed in a direction perpendicular to the line of sight; measure a fixed distance in this line of sight, namely, the distance from these marks. Note on the card, while the knot is held by the teeth, the points in line with the marks. Lengths are to be chosen to make the card read mils. In using the gauge, hold the knot between the teeth and hold the

4

lOOO yards

}

ANTIAIRCRAFT

7

MATHEMATICS

9. The Vernier Measurements of lengths and angles are made with scales subdivided so as to show any unit and fractions of this unit. For example, a ruler may show inches and eighths of an inch, or inches and tenths of an inch. Similarly for angles. With such a scale it is only possible to read measures to the nearest unit and fraction of a unit. With the help of an attachment to the scale called a vernier, it is possible to make more accurate determinations of a length or an angle. The vernier is attached to the scale so that it can slide along it. A typical example of a vernier may be described as follows: let a scale be divided into inches and tenths of an inch. In this case the vernier is subdivided into ten parts while its length is only equal to that of nine of the subdivisions of the scale. Each 9 9 division of the vernier is equal to — of a division of the scale or 10 100 of an inch. Thus, to measure a length, place the zero point of the scale at one end of the length and the zero point of the vernier at the other end of the length. Inches and tenths of inches of the length being measured are now read from the scale. Hundredths of inches are given by the reading on the vernier which most nearly coincides with a subdivision of the scale. In the diagram the fourth mark on the vernier coincides with a mark on the scale. The length indicated is therefore 8.24 inches. 10

123*56789 ' ' •• •• I, I f •• ,• 113+36786 10. Equations—Linear and Quadratic An equation is the statement of the equality between two numbers or expressions. For example: 3 . 5 = 15 ;

- = ?; 2

4

(a + b) (a - b) = a2-

x = 3

b2

The equal numbers or expressions are called the sides or members of the equation.

8

INTRODUCTORY ARTILLERY

MATHEMATICS

If two numbers are equal and the same number is added to each of them the sums are equal; thus the same number may be added to both sides of an equation. Similarly, the same number may be subtracted from both sides of an equation. If two numbers are equal and each is multiplied by the same number the products are equal; thus the two sides of an equation may be multiplied by the same number. Similarly, the two sides of an equation may be divided by the same number. This number cannot be zero—division by zero can never be performed. If desired, an equation may be put into a form in which one side is zero. An equation which is true no matter what values are given to the letters in it, is called an identity. An example: (a+b) (a—b) = o2—b2, no matter what values are given to a and to b. By contrast, the equation 2a; = 4 is true only if x = 2. Also, 2x = 4 may be regarded as a way of asking the question, "What number multiplied by 2 gives 4"? It is in this way that equations are often presented and it is required to find what values the letters must have in order that the equation may be true. Finding these values is called "solving the equation." An example: Solve 13a; — 7 = 5x + 9 First: Add 7 to both sides— 13a; = 5x + 16 Next: Subtract 5x from both sides— 8x = 16 Next: Divide both sides by 8— x = 2 This shows that if there is any number which when substituted for x gives a true equation, that number must be 2. When 2 is put for x, the equation becomes 26 — 7 = 10 + 9 and the equation is satisfied. The value 2 is called the solution or root of the equation. An equation such as this one is called a simple equation or linear equation. It involves a single letter representing an unknown number and involves that letter only to the first power. If an equation involves the second but no higher power of the unknown quantity, it is called a quadratic equation. Examples: x2 — 4 = 0; x2 — 6a; + 8 = 0. If in the first of these equations x is given the value 2, the equation then becomes 22 — 4 = 0, which is true. The equation is satisfied and 2 is called a root of the equation. It is seen that —2 is also a root. The second equation is satisfied if

A N T I A I R C R A F T MATHEMATICS

9

x is given the value 2, for 22 — 6 • 2 + 8 = 0; and if a; is given the value 4 it is satisfied, for 42 — 6 • 4 + 8 = 0. Thus 2 and —2 are roots of the first equation and 2 and 4 are roots of the second equation. The method of solving a quadratic equation may be seen from the following solution of an equation. Example: from x2 — 6x + 7 = 0 it follows that x2—6x = - 7 , then that x2-Qx+9 = 2. But x 2 -6a;+9 = (x - 3) (x - 3) = (x - 3)2. Hence (x - 3)2 = 2, from which it follows that x — 3 = V 2 or a; — 3 = - V2. Thus x = 3 + V 2 orz = 3 - V 2 . The roots of any quadratic equation ax2 + bx + c = 0 are x =

and s = usually written

-b

+ V&2 - 4ac 2a

-b

-

Vb2

- 4ac

o 2a

2

-b ± V& - 4oc x = —

2a

11. Exponents A product of equal numbers is called a power of that number; thus 2 • 2 • 2 = 8 is the third power of 2. The number of factors in the product is called the exponent and 2 • 2 • 2 is written 23. If two powers of the same number are multiplied together the result is a power whose exponent is the sum of the exponents of the two powers which are multiplied, thus 22 • 23 = 2 • 2 • 2 • 2 • 2 = 25, or 22 • 23 = 26 = 32. In general am • an = am+n. If the factors of a product are the reciprocals of a number, the product is called a negative power or a power with a negative exponent, thus ! • - • - = — = 2 -3 . 2 2 2 23 The exponent of the product of two powers of the same number is the sum of the exponents of the factors; thus 23 • 22 = 25 and 23 • 2"2 = 21 = 2 (when the exponent is 1 it may be omitted). In general am • an = am+n for m and n either positive or negative.

10

INTRODUCTORY ARTILLERY

MATHEMATICS

am Since am • a~m = — = 1, and also am • cr™ = a m_m = a 0 , it follows am that a0 = 1. (a cannot be equal to zero). If the factors of a product are equal powers of the same number, the product has an exponent equal to the exponent of the factor multiplied by the number of factors, thus (22)3 = 26 or (a m ) n = amn. A fractional exponent is used to express the extraction of a root; thus 2i means the cube root of 2 or (-?2 = 1.260). This use of an exponent is consistent with the laws of exponents stated above. 12. Rectangular Coordinates or Grid Coordinates are numbers which serve to locate the position of a point. The simplest kind, called by any of the names Cartesian, rectangular, or grid, may be described as follows: two perpendicular lines, called the X and F axes, intersect at 0, called the origin.

Y 5 H

800 = R

a < •

tan

e

H —

R

" L " onD, scale E " S " on €, scale B " S " o n index Read R on "L," scale E

R

cot e " S " o n index " L " on e, scale C " L " on Ì2, scale E Read i f on " S , " scale E

" S " o n index " L " on e, scale C " S " on R, scale E Read H on " L , " scale E

C a s e 5 :

800

= D

+

OR

[71]

-±COS €

" S " on index " L " on e, scale B " L " on D, scale E Read R on " S , " scale E

72

INTRODUCTORY ARTILLERY

MATHEMATICS

Case 6 : « > 800 R —

H

tan e "S" on index "L" on €j scale C "L" on H, scale E Read R on "S," scale E

Case 7:

€ < 800 R = H • cot € "S" on index "L" on e, scale C "S" on H, scale E Read R on "L," scale E

Case 8:

1 _ D sine H

1 _D cos € R

"L" on D, scale E "S" on H, scale E "S" on index Read e on "L," scale D

"L"on D, scale E "S" on R, scale E "S" on index Read e on "L," scale B

Case 9 : H > R;

tan e = — R "L" on H, scale E "S" on R, scale E "S" on index Read e on "L," scale C

H < R;

cot e = — H "L" on R, scale E "S" on if, scale E "S" on index Read e on "L," scale C

Case 10: Lat. Dev. (horiz.) = Lat. Dev. (slant) • —!— cos e "S" on index "L" on e, scale B "S" on lateral deviation in slant plane, scale E Read corresponding lateral deviation in horizontal plane under "L," scale E

ANTIAIRCRAFT

MATHEMATICS

73

48. Oblique Triangles Case 1 : 1 _ R, 1 sin T base-line sin 0 2 "L" on Rh scale E "S" on base-line length, scale E "S" on 02, scale D Read Ton "L," scale D

Case 2 : Ri = base-line • — ; — sin T sin Oi "L" on T, scale D "S" on Oh scale D "S" on base-line length, scale E Read R2 on "L," scale E

49. Trial Shot Problem 1. Select azimuth of TSP from 0 2 2. Draw position sketch to scale showing Oi, 02, TSP, north lines, azimuths 3. Determine angle 0 2 (A) If Oi is on left of 02, subtract back azimuth of base-line from azimuth of TSP from 0 2 (add 6400 mils if necessary to keep angle positive) (B) If Oi is on right of 02, subtract azimuth of base-line from azimuth of TSP from 0 2 (add 6400 mils if necessary to keep angle positive)

74

INTRODUCTORY ARTILLERY

MATHEMATICS

4. Determine angle T .Ri 1 _ sin T base-line

1 sin 0 2

"L" on scale E "S" on base-line length, scale E "S" on 0 2 , scale D Read T on "L," scale D 5. Determine angle Oi (A) If Oi is on left of 0 2 , exterior angle Oi = angle 0 2 + angle T (B) If Oi is on right of 0 2 , interior angle Oi = angle 0 2 - angle T 6. Find azimuth of TSP from Ox (A) If Oi is on left of 02, azimuth TSP from Oi = back azimuth base-line + angle Oi (B) If Oi is on right of 0 2 , azimuth TSP from Oi = azimuth base-line + angle Oi If azimuth is greater than 6400 mils, subtract 6400 from it. 7. Determine Ri R 2 = base-line • — ; — sin T sin 0\ "L" on T, scale D "S" on Oi, scale D "S" on base-line length, scale E Read Ri on "L," scale E 8. Find eg using R 2 and H , H tan «2 = — Ri H > Ri "L"onH "S" on Ri "S" on index Read a on "L," scale C

, Ri cot ti — — H H < Ri " L " on Ri "S" on H "S" on index Read a on "L," scale C

TRIAL SHOT PROBLEM—COMPUTING FORM Data for Position Sketch Position Sketch #1. Azimuth of base-line mils #2. Length of base-line yards #3. Ri, horiz. range to TSP yards — — #4. Approximate azimuth of TSP from Oi mils Additional Data #5. H, altitude of TSP #6. €i, angular height of TSP from Oi #7. Azimuth of TSP from 0 2 (taken from sketch) Oileft

yards ~~ mils mils

I

I

I

I

I

Directions for Computations

I

I

Oi right

# 8. Azimuth of TSP from 0 2 (same as #7) # 9. Add 6400 if necessary #10. Azimuth of base-line (same as #1) (-)

(-)

# 11. Back azimuth of base-line (add 3200 to #1) #12. Oi angle [Oi left, #8 - #11 ; Oi right, #8 - #10] #13. To find T; set "L" on #3 on scale E and "S" on #2 on scale E; move "S" to #12 on scale D and read T on "L" on scale D, selecting value which conforms to position sketch #14. Oi angle [Oi left, #12 + #13; Oi right, # 1 2 - # 1 3 ] #15. Azimuth of base-line (same as #10)

(+)

#16. Back azimuth of base-line (same as #11) #17. Subtract 6400 if necessary #18. Azimuth of TSP from Oi [Oi left, #14 H- #16; Oi right, #14 + #15] #19. To find Ä2; set "L" on #13 on scale D and "S" on #14 on scale D; move "S" to #2 on scale E and read Äs on "L" on scale E #20. Tofind€2« If #5 is larger than #19: set "L" on #5 on scale E and "S" on #19 on scale E; move "S" to index and read «2 on "L" under tan on scale C. «2 > 800 mils. If #5 is smaller than #19: set "L" on #19 on scale E and "S" on #5 on scale E; move "S" to index and read a on "L" under cot on scale C. 12 < 800 mils.

I

(+)

PART V I I I TABLES CONVERSION TABLES FOR ANGLES A N D LENGTHS, F O U R - P L A C E T A B L E S OF N A T U R A L T R I G O N O M E T R I C F U N C T I O N S , L O G A R I T H M S OF N U M B E R S , A N D L O G A R I T H M S OF T H E TRIGONOMETRIC FUNCTIONS FOR ANGLES IN MILS 60. Some Formulas for Reference Logarithms

,

,

, ,

log pq = log p + log q log-

= log p — log q

log p"

= q • log p

log y/p = - • log p Trigonometric Functions . _ y _ opposite side t -— — J r hypotenuse

cosec A —

. x adjacent side cos A = - = —' ; r hypotenuse

sec A =

.

sin A

w

=

^

=

x

oppositeside. adjacent side

sin A 1 cos A

cot A =

1 tan A

Law of Sines If sides of any triangle are a, b, c, and angles opposite those sides A, B,C, respectively, a _ sin A b

sin B'

t

a _ sin A C

sin C '

b _ sin B c

Law of Cosines a2 = 62 + c2 — 26c • cos A b2 = a2 + c2 — 2ac • cos B & = a2 + ¥ - 2ah • cos C [76]

sin C

A N T I A I R C R A F T

M A T H E M A T I C S

77

51. Equivalents : Measures of Angles Mils

Degrees

IOO 200 300 400

5° II 16 22

500 600 700 800

28

Mils

Degrees

37' 15 52 3°

30-0 00.0 30.0 00.0

0.0982 0.1964 0.2945 O.3927

33°° 3400 3500 3600

185° 191 196 202

37' 15 52

0.4909 O.589I 0.6872 0.7854

3700 3800 3900 4000

208

07

45

°7 45 22 OO

30.0 00.0 30.0 00.0

213 219 225

45 22 00

30.0 00.0 30.0 00.0

O.8836 0.9818 I.0799 I.1781

4100 4200 4300 4400

230 236 24I

37

247

3°

30.0 00.0 30.0 00.0

I.2763

4500 4600 4700 4800

253 258 264 270

07

4900 5000 5100 5200

275 281 286 292

33

39

900 IOOO ncra 1200

I6 16

37 !5 52

67

3°

1300 1400 1500 1600

73 78 84 90

07

1700 1800 1 goo 2000

95 IOI 106 112

2100 2200 2300 2400

118 123 129

2500 2600 2700 2800

140 146

2900 3000 3100 3200

163 168

m

45 22 OO 37 J5 52 3° 07 45 22 00

!5I

37 !5 52

57

3°

I

Radians

!74 180°

07 45 22 00'

30.0 00.0 30.0 00.0

r-3745 1.4726 1.5708

(= r/2) 1.6690 1.7671 1.8653

1

•9635

30.0 00.0 30.0 00.0

2.0617 2.1598 2.2580 2.3562

5300 5400

30.0 00.0 30.0 00.0

2.4544 2.5525 2.6507 2.7489

57°° 5800 5900 6000

30.0 00.0 30 0

2.8471 2.9452

6100 6200 6300 6400

ooTo

3-0434 3.1416

(=*)

55°° 5600

3°

52

45 22 00 37 15 52 3°

298

07

3°3 3°9 315

45 22 00

320 326

37 *5 52

33 337 1

343 348 36°°

3° 07 45 22 00'

Radians

3°-° 00.0 30.0 00.0

3 3 3 3

3379 4361

30.0 00.0 30.0 00.0

3 3 3 3

6325 7306 8288 9270

30.0 00.0 30.0 00.0

4 0252 4 !233 4 2215 4 3197

30.0 00.0 30.0 00.0

4 4179 4 5160 4 6142 4 7124 ( = 3 « "/ 2 ) 4 8106 4 9087

30.0 00.0 30.0 00.0

2398

5343

5 0069 5 1051

30.0 00.0 30.0 00.0

5 5 5 5

30.0 00.0 30.0 00.0

5 5 5 5

30.0 00.0 30 0

5 9886

oo'.o

2032 3°i4 3996 4978 5959 6941 7923 89°5

6 0868 6 1850 6 2832 = 271")

(=

78

I N T R O D U C T O R Y

A R T I L L E R Y

M A T H E M A T I C S

52. Conversion Table: Degrees—Mils—Degrees I degree = 1 minute =

Degrees I

1 mil = 0 . 0 5 6 2 5 degrees 1 mil = 3 . 3 7 5 0 0 minutes

1 7 . 7 7 7 7 8 mils 0 . 2 9 6 3 0 mils

Degrees

Mils

Degrees

Mils

Minutes

Mils

Minutes

I 2

0-3

31

31 32 33 34 35

551.1 568.9 586.7 604.4 622. 2

6L 62

1084.4 1102.2

63

1120.0 "37-8

65

1155.6

36 37 38 39

64O.O 657.8 675.6

66

"73-3

10

106.7 124.4 142.2 160.0 177.8

40

693-3 711 . I

69 70

11

195.6

1

728.9

12

213-3 231-1

2

3 4 5 6

7

8

9

17.8

Mils

35-6 53-3

71.1 88.9

4 42 43 44 45

817.8

835-6 853-3

337-8 355-6

49 5°

21 22

23 24 25 26

27 '

28

29

30

48

1

373-3 391 •1

5 52 53 54 55

408.9 426.7

444-5 462. 2 480.0 497.8 515.6

56 57 58

P 60

533-3

72

47

284.4 302.2 320.0

19

7

1

46

16

20

68

800.0

248.9 266.7

18

764.4

^

73 74 75

H 15 17

746.7

64

1191.1 1208.9 1226.7 1244.5

1262.2 1280.0 1297.8 1315.6

1.8 2.1 2.4

36

10.7 II.O

9

2-7 3-°

39

11.6 11.9

11

3-3 3-6 3-9 4-1 4-4

41 42

12.1 12.4 12.7 13.0

6

7

8

10

12

76

I

1

16

77 78 79

1368.9 1386.7 1404.5 1422.2

!7

906.7 924.4

81 82

960.0

83 84

1440.0 1457.8 1475.6

977-8

85

995.6

86

1013.3 1031.1 1048.9 1066.7

87 88

87I.I 888.9

942.2

80

89 90

35 -

H93-3 1511.1

1528.9

1546.7 1564.5 1582.2 1600.0

9-5

9.8 IO.I 10.4

1333-3 1

9.2

32 33 34 35

3 4 5

13 14 !5

782.2

0.6 0.9

Mils

18

•9

1.2 M

4-7 5-o 5-3 5.6

20

5-9

21 22

6.2

25

H 6.8 7-i 7-4

26

7-7

23

24

27 28 29

3°

8.0

8-3 8.6 8.9

37

38 40

43

44

11

-3

45

13-3

46

13.6

47

48

T

3-9

5°

14.2 14.5 14.8

51 52

15.1 15.4

53 54 55

15-7

56

16.6 16.9 17.2

49

57

58 59

60

16.0 16.3

r

7-5

17.8

EXAMPLES

I. T o convert 3 7 ° 1 7 ' to mils add two numbers taken from table, one for 3 7 degrees, the other for 1 7 minutes: 37°

= 6 5 7 . 8 mils

17' = 5.0 mils 3 7 ° 1 7 ' = 662.8 mils II. T o convert 3 7 ? 2 8 to mils, add two numbers taken from table, one for 3 7 degrees, the other for 28 degrees, moving decimal for latter number two places to left: 370 = 6 5 7 . 8 mils o?28 =

5 . 0 mils

3 7 ° 2 8 = 662.8 mils

I I I . T o convert a number such as 37^96 to mils, add taken with the decimal. Different solutions are open, two 37° directly from table OR 37° o?9 use 9 0 and move decimal 0?90 one place to left o?o6 use 6° and move decimal o?o6 two places to left

three numbers, care being being given here: directly from table use 90° and move decimal two places to left use 6° and move decimal two places to left

ANTIAIRCRAFT MATHEMATICS

79

53. Natural Trigonometric Functions Mils

Sine

Cosine

Tangent

Cotangent

0

0.0000

1.0000

0.0000

00

5°

150

0.0491 .0980 .1467

0.9988 •995 2 .9892

0.0491 .0985 .1483

20.355 10.153 6.741

200

.1951

.9808

.1989

5.027

250 300 35°

.2430 .2903 •33 6 9

.9700 •9569 •9415

.2505 •3033 •3578

3-99 2 3-297 2.795

400

0.3827

0.9239

0.4142

2.414

450 500 55°

.4276 • 47*4 .£141

.9040 .8819 •8577

•4730 •5345 •5994

2.114 1.871 1.668

600

•5556

•8315

.6682

!-497

•5957 • 6344 .6716

.8032 •773° .7410

.7416 .8207 0.9063

1.348 1.219 1-103

IOO

650 700 750 800

0.7071

0.7071

1.0000

1.0000

850 900 950

.7410 •773° .8032

.6716 • 6 344 •5957

1.103 1.219 1-348'

0.9063 . 8207 .7416

1000

•8315

•555 6

1.497

.6682

1050 1150

•8577 .8819 .9040

•5I4I • 47H .4276

1.668 1.871 2.114

•5994 •5345 •473°

1200

0.9239

0.3827

2.414

0.4142

1250 I3°° 135°

•9415 .9569 .9700

•3369 • 2903

.2430

2.795 3-297 3-992

•357« • 3033 .2505

1400

.9808

.1951

5.027

.1989

1450 1500 1550

.9892 •995 2 0.9988

.1467 .0980 0.0491

6.741 10.153 20-355

• 1483 .0985 0.0491

1600

1.0000

0.0000

00

0.0000

I IOO

80

INTRODUCTORY ARTILLERY

MATHEMATICS

64. How to Use Tables of Logarithms Tables of logarithms of numbers and logarithms of the trigonometric functions of angles follow. The mantissas are given to four decimal places. The characteristic is printed in the table of logarithms of the trigonometric functions. Minus 10 (—10) must be affixed to all values of log sin and log cos and to all values of log tan for angles between 0 and 800 mils and to all values of log cot for angles between 800 and 1600 mils. Minus 10 (—10) is therefore affixed to all numbers taken from three of the four columns in the table of logarithms of the trigonometric functions of angles. The characteristic is determined as follows when finding the logarithm of a number: For numbers 1 or greater, count the number of digits preceding the decimal point. The characteristic is one less than this number. For numbers less than 1, count the number of zeros between the decimal point and the first significant digit. The characteristic is the number obtained when this number of zeros is subtracted from 9. Minus 10 ( — 10) appears in these cases. The use of the table will be illustrated by the following examples: I. To find log 51.4. Read the number in the table located by 51 in the column at the left and under the heading 4 in the line at the top of the page. The characteristic is 1. Thus log 51.4 = 1.7110. II. To find log 51.43. The result will be found by increasing log 51.4 by 0.3 of the difference between log 51.4 and log 51.5. This difference is 8 (in units of the fourth decimal place). Multiply 8 by 0.3 and the result is 2.4, which shows that 2 is the amount to add to log 51.4 to get log 51.43. Thus log 51.43 = 1.7112. Tables of Proportional Parts (P.P.) perform the indicated multiplications. III. To find the number N having log N = 2.3655. The characteristic being 2, the number will have three digits preceding the decimal point. The number 3655, which is the mantissa of the logarithm, is located in the table and it is found that it corresponds to the reading 23 at the left and 2 at the top of the column in which it is located. The number N is therefore 232, or 232.0, if an additional digit is desired. IV. To find the number N having log N = 2.3667. Again, there will be three digits preceding the decimal point. Also, 3667 will be

A N T I A I R C R A F T MATHEMATICS

81

found between the mantissas of log 232 and log 233; 12 is the amount which has been added to log 232 to produce log N, and 19 is the difference between log 232 and log 233. In the table of proportional parts look under 19 and find the number nearest to 12. It is 11.4. The table shows that 11.4 is 0.6 of 19. N is therefore 232.6. In the table of logarithms of the trigonometric functions, the mil is taken as the unit of measure, and the logarithms of the functions are given at intervals of 5 mils. To avoid making it necessary to take proportional parts both in the mil column and in the log column, the multiplications and divisions indicated above have all been performed and are given in an extended table of proportional parts. Use of these tables of proportional parts is illustrated by the following examples. I. Tofindlog sin 782 mils. The table gives log sin 780 mils and log sin 785 mils, and the difference between them is 22 (in units of the fourth decimal place). The log sin 782 mils will therefore be found by increasing log sin 780 2 mils by - , or 0.4, of 22, which is 9. The table gives log sin 780 mils 5 = 9.8408. Thus log sin 782 mils = 9.8417 - 10. To use the table of proportional parts, find the number having the difference between the tabular logarithms at the top of the column and the difference in mils at the left. In this case, look for 22 at the top and 2 at the left. The correction is 9, and 9 added to log sin 780 mils again gives log sin 782 mils = 9.8417 — 10. II. TofindA when log tan A = 9.9846 - 10. From the table it is seen that 9.9846 is between log tan 780 mils and log tan 785 mils, the difference between these tabular logarithms being 43 (in units of the fourth decimal place). The difference between log tan 780 mils and log tan A is 17. In the table of proportional parts, it is seen that with the difference of 43 read at the top of the column and the difference 17 read in the table, the number 2 appears at the left. This is the number of mils to be added to 780 to get A. Thus A = 782 mils if log tan A = 9.9846 - 10.

82

INTRODUCTORY ARTILLERY

MATHEMATICS

Use of tables of proportional parts is based on the assumption that changes in the tabulated values are proportional to the changes in the numbers. Thus they can be used when variations are approximately uniform, and not under other conditions. The operations described in this section are often referred to as interpolation. For small angles the differences are such that proportional parts cannot always be used. Values of sines and tangents cannot be interpolated for small angles. From the table of the logarithms of the trigonometric functions, values could be interpolated by proportional parts to four places of decimals with an inaccuracy of but 1 unit in the fourth decimal for sines and tangents of angles as small as 100 mils and for cosines and cotangents of angles as large as 1500 mils. Such angles do not arise in the antiaircraft problems. Because of this, the table of proportional parts has been somewhat restricted. Since cosec A = —-—, log cosec A = log 1 — log sin A and sin A thus log cosec A = — log sin A. If log sin A = 9.6734 — 10, log cosec A = 10 - 9.6734 = 0.3266. In this manner the values of log cosec A and log sec A may be found from the tabulated values of log sin A and log cos A. Similar relations exist between log tan A and log cot A, but both of these functions have been tabulated for convenience. For an angle tabulated in mils in the column at the left hand side of the page, the function is read at the top of the column; when the angle is read at the right hand side of the page, the function is read at the bottom of the column. Thus with angles from 0 to 800 mils at the left and from 800 to 1600 mils at the right, the table contains the logarithms of the trigonometric functions of all angles from 0 to 1600 mils, at intervals of 5 mils. It makes use of the relations sin (1600 - A) = cos A cos (1600 - A) = sin A tan (1600 - A) = cot A cot (1600 - A) = tan A

A N T I A I R C R A F T MATHEMATICS

83

If functions of angles between 1600 and 6400 mils are desired, they may be taken from the table by suitable use of the following relations: sin (3200

—

A)

=

+sin A

cos (3200

—

A)

=

—cos A

tan (3200

—

A)

=

—tan A

cot (3200

—

A)

=

sin (3200 +

-

cos (3200 + A) cot (3200 +

—sin A —cos A

=

tan (3200 + A)

—cot A

-

+tan A

=

+cot A

sin (6400

—

A)

=

—sin A

cos (6400

—

A)

=

+cos A

tan (6400

—

A)

=

—tan A

cot (6400

—

A)

=

— cot A

84

INTRODUCTORY ARTILLERY

MATHEMATICS

55. Logarithms of Numbers N

0

100

1

2

3

4

5

6

7

8

9

0000 0004 0009 0013 0017 0022 0026 0030 0035 0039

101 0043 0048 0052 0056 0060 0065 0069 0073 0077 0082 102 0086 0090 0095 0099 0103 0107 OIII 0116 0120 0124 103 0128 oi33 0137 0141 0145 0149 0154 0158 0162 0166 104 0170 017c 0179 0183 0187 0191 0195 0199 0204 0208 0212 0216 0220 0224 0228 0233 0237 0241 0245 0249 106 0253 02J7 0261 0265 0269 0273 0278 0282 0286 0290 107 0294 0298 0302 0306 0310 ° 3 H 0318 0322 0326 033° 108 °334 0338 0342 0346 0350 u 354 0358 0362 0366 0370 109 0374 0378 0382 0386 0390 0394 0398 0402 0406 0410 110

0414 0418 0422 0426 0430 0434 0438 0441 0445 0449

1 1 1 °453 0457 0461 0465 0469 0473 0477 0481 0484 0488 1 1 2 0492 0496 0500 0504 0508 0512 0515 0519 0523 0527 " 3 ° í 3 I ° 5 3 i 0538 0542 0546 0550 0554 0558 0561 0565 1 1 4 0569 °573 0577 0580 0584 0588 0592 0596 0599 0603 0607 0611 0615 0618 0622 0626 0630 0633 0637 0641 1 1 6 0645 0648 0652 0656 0660 0663 0667 0671 0674 0678 1 1 7 0682 0686 0689 0693 0697 0700 0704 0708 0711 0715 118 0719 0722 0726 0730 0734 0737 0741 0745 0748 0752 1 1 9 0755 ° 7 i 9 0763 0766 0770 0774 0777 0781 0785 0788 120

07 92 0795 0799 0803 0806 0810 0813 0817 0821 0824

121 0828 0831 ° » 3 Í 0839 0842 0846 0849 0853 0856 0860 122 0864 0867 0871 0874 0878 0881 0885 0888 0892 0896 123 0899 0903 0906 0910 0913 0917 0920 0924 0927 0931 124 °934 0938 0941 0945 0948 0952 °955 0959 0962 0966 0969 0973 0976 0980 0983 0986 0990 °993 0997 1000 1004 1007 i o n 1014 1017 1021 1024 1028 1031 i ° 3 S lib 127 1038 1041 128 1072 1075 129 1106 1109 130

p.p.

"39

"43

1045 1048 1052 1055 1059 1062 1065 1069 1079 1082 1086 1089 1093 1096 1099 1103 1 1 1 3 1 1 1 6 1 1 1 9 1 1 2 3 1126 1129 " 3 3 1136 1146

1149

"53

1156

"59

1163

1166

1169

'3 1 1 7 3 1176 1179 1183 132 1206 1209 1 2 1 2 1216 133 1239 1242 1245 1248

1186 1189 " 9 3 1196 " 9 9 1202 1219 1222 1225 1229 1232 1235 1252 1 2 55 1258 1261 1265 1268

134 1271 I r 35 3 ° 3

1284 1287 1290 1294 1297 1300 1 3 1 6 r 3 J 9 r 323 1326 1329 !332

1

1274 1278 1281 1307 1 3 1 0 ! 3 ! 3

5 1 2 3

4 5 6 7

8 9

o.S 1.0 1 -5 2.0 2.5 30 3-5 4.0 4-5 4

1 2 3 4 5 6 7 8 9

0.4 0.8 1.6 2.0 2.4 2.8 3.6

3 2 3 4 5 6 7 8 9

0.3 0.6 0.9

1.2 i-5 1.8 2.1

2.7

ANTIAIRCRAFT MATHEMATICS

Logarithms of Numbers N 136 '37 138 J 39 140

0

1

i3°3 1307 '335 1339 1367 1370 '399 1402 1430 1433 1461 1464

141 1492 142 '523 H3 1553 144 1584 1614 146 1644

2

3

1310 3 3 1342 '345 '374 1377 1405 1408 1436 1440 1467 1471 I I

4

1477 1508 1538 1569

A 1599 1629 1658 1688 1717 1746

1772 1775 1801 1804 1830 1833 1858 1861 1886 1889 1915 1917 1942 1945

57 '959 1962 1965 1967 1970 1973 158 1987 1989 1992 '995 1998 2000 '59 2014 2017 2019 2022 2025 2028 160 2041 2044 2047 2049 2052 2055 J

161 162 163 164 165 166 167 168 169

2068 2071 2074 2076 2095 2098 2101 2103 2122 2125 2127 2130 2148 2151 2154 2156 2175 2177 2180 2183 2201 2204 2206 2209

6

1316 1323 1348 3S 1355 1380 1383 1386 1411 1414 1418 1443 1446 1449

1474 1495 1498 1501 1504 1532 '535 1526 1529 1556 1559 1562 1565 1587 1590 *593 I5 1617 1620 1623 1647 1649 1652 1626 1655 1676 1679 1682 1685 1706 1708 1711 1714 1735 1738 1741 1744

H7 I673 148 :7°3 149 1732 150 1761 1764 1767 1770 151 1790 1793 1796 1798 152 1818 1821 1824 1827 1 S3 1847 1850 1853 1855 J 54 1875 1878 1881 1884 I9°3 1906 1909 1912 156 1931 1934 1937 1940

5 I 1 I3 9I

2079 2106 2133 2159 2185 2212

2082 2109 2135 2162 2188 2214

7

8

9

1326 1329 1332 1358 1361 i364 1389 1392 1396 1421 1424 1427 1452 1455 1458

1480 1483 1511 1514 1541 1544 1572 1575 1602 1605 1632 1635 1661 1664 1691 1694 1720 1723 1749 1752 1778 1781

i486 1489 1517 1520 1547 '550 1578 1581 1608 1611 1638 1641 1667 1670 1697 1726 1755 1784

1700 1729 1758 1787

1807 1810 1813 1816 1836 1838 1841 1844 1864 1867 1870 1872 1892 1895 1898 1901 1920 1923 1926 1928 1948 1951 r953 1956 1976 1978 1981 1984 2003 2006 2009 2011 2030 2033 2036 2038 2057 2060 2063 2066 2084 2111 2138 2164 2191 2217

2227 2230 2232 223j 2238 2240 2243 2253 2256 2258 2261 2263 2266 2269 2279 2281 2284 2287 2289 2292 2294 170 2304 2307 2310 2312 23!5 2317 2320

2087 2090 2092 2114 2117 2119 2140 2143 2146 2167 2170 2172 2193 2196 2198 2219 2222 2225 2245 2271 2297 2322

2248 2251 2274 2276 2299 2302 2325 2327

85

INTRODUCTORY ARTILLERY

86

MATHEMATICS

Logarithms of Numbers N

0

1

2

3

170

2304

2307

2310

2312

4

5

2315 2317

6

7

2320 2322

8

9

2325

2327

P.P.

233° 2333 2335 2338 2340 2343 2345 2348 2350 2353 2355 2358 2360 2363 2365 2368 2370 2373 2375 2378 173 2380 2383 2385 2388 2390 2393 2395 2398 2400 2403

0.3

174

2. i 2-4 2.7

171 172

176 177 178 179

2405 2408 2410 2413 2415 2418 2420 2423 2 4 2 5 2428 2430 2433 2435 2438 2440 2443 2445 2448 2450 2453 2455 2458 2460 2463 2465 2467 2470 2472 2477

2475

2480 2482 2485 2487 2490 2 4 9 2 2494 2497 2499 2502 2504 2507 2509 2 5 1 2 2 5 1 4 2 5 1 6 2519 2521 2 5 2 4 2526 2529 2531 2533 2536 2538 2541 2543 2545 2548 2550

180 2553 2555 2558

2560 2562

2565 2567

2570 2572

2574

00 00 CO

2577 2579

00 00 00

2582 2584 2586 2589 2591 2594 2596 2598 2605 2608 2610 2613 2615 2617 2620 2622 2629 2632 2634 2636 2639 2641 2643 2646

2648 2651 2653 2655 2658 2660 2662 2665 2667 2669 2672 2674 2676 2679 2681 2683 2686 2688 2690 2693 2695 2697 2700 2702 2704 2707 2709 2 7 1 1 2714 2716

187

2718 2721 2723 2725 2728 2730 2742 2744 2746 2749 2751 2753 2765 2767 2769 2772 2774 2776

2732 2735 2737 2739 2755 2758 2760 2762 2778 2781 2783 2785

190

2788

2801

191 192

2810

194

2878 2880 2883 2885 2887 2889 2891 2894 2896 2898 2900 2903 2905 2907 2909 2911 2914 2916 2918 2920 2923 2925 2927 2929 2931 2934 2936 2938 2940 2 9 4 2

188 189

193 196

2601 2625

2603 2627

2790 2792 2794

2813 2835 2856 2858

2833

197 198 199

2945

200

3010

2797 2799

2804 2806 2808

2815 2 8 1 7 2819 2822 2824 2826 2828 2831 2838 2840 2842 2844 2847 2849 2851 2853 2860 2862 2865 2867 2869 2871 2874 2876

2947 2949 2951 2967 2969 2971 2973 2989 2991 2993 2995

3012 3OI5

2953 2975 2997

3017 3019

2956 2958 2960 2962 2964 2978 2980 2982 2984 2986 2999 3002 3004 3006 3008 3021

3023

3025 3028

3°3°

0.6

0.2 0.4 0.6 0.8

1.0 1.2 1.4 1.6 1.8

Logarithms of Numbers N

0

1

20

3010

3°32

2

3

3 ° 5 4 3075

4

5

3096 3 1 1 8

6 3139

7

8

3160 3 : 8 1

21 22 23

3222 3243 3263 3424 3444 3 4 6 4 3617 3636 3655

24

26

3802 3820 3838 3856 3874 3892 39°9 3927 3945 3979 3997 4014 4031 4048 4065 4082 4099 4 1 1 6 4150 4166 4183 4200 4216 4232 4249 4265 4281

27 28 29

43*4 4 3 3 ° 4472 4487 4624 4 6 3 9

3284 33°4 3324 3345 2 3522 3 5 4 i 3483 3 5 ° 3692 3 7 " 3729 3674

4346 4362 4378 4533 4502 4654 4669 4683

3365 356° 3747

4393 4409 4425 4548 4564 4579 4698 4 7 ' 3 4728

3385 3404 3598 3766 3784 3962 4133 4298

4440 4456 4594 4609 4742 4757

4771

4786

4800 4814

4857

4871

4886 4900

31 32 33

4914 5051 5185

4928 5065 5198

4942 4955 4969 4983 4997 5079 5092 5105 5 1 1 9 5132 5211 5224 5237 5250 5263

5011

5024 5038 5 1 5 9 5172 5289 5302

34

53 T 5 544i 55 6 3

5328 5453 5575

534° 5465 5587

36

5353 5478 5599

4843

5366 5378 5490 5502 5611 5 6 23

6o53

6064 6075

41 42 43

6128 6232

6138 6243 6345

6149 6253 6355

6160 6263 6365

50 51 52 53

7007

7076 7160

7°93

7243

7084 7168 7251

5428

5539 5658

555i 5670

5740 575J 5866 5966 5977

57 6 3 5877 5988

5775 5888

6222

7016

7101 7l77 718J 7259 7267

6857 6946

7024 7°33 7110 7*93 7275

7348 7356 7 3 6 4 7372 7427 7435 7443 7 4 5 i 7 5 ° 5 7 5 ' 3 7520 7528

7380 745? 7466 7 5 3 6 7543

7559 7566 7 5 7 4 7582 7597 7604 7 6 1 2 7619 7627 7 6 3 4 7642 7649 7657 7664 7672 7679 7686 7694 7701 7709 7 7 1 6 7723 7731 7738 7745 7752 7760 7767 7 7 7 4

60

7782

7803

2

3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4

1-4 2.8 4.2

7810

7818

,

7825

7832 7839

7396 7474 7551

7846

3 4 5 6 7 8 9

4-5 6.0 7-5 9.0 10.5 12.0

5-6 7.0

13.5

8.4 9.8 11.2 12.6

13

12

11

1.3 2.6

1.2

7042 7050 7059 7067

57 58 59

7796

14

1.5 3.0

1 2

7324 7332 7 3 4 ° 7404 7 4 1 2 7 4 i 9 7482 7490 7497

7789

15

6794 6803 6884 6893 6964 6972 6981

7 1 1 8 7126 7 I 3 5 7 H 3 7 1 5 2 7202 7210 7218 7226 7235 7284 7292 7300 73o8 7 3 i 6

15-3

1.6

11.7

6|85

144 16.2

16

9

6776 6866 6955

3.6 S -4 7.2 9.0 10.8 12.6

1

3.9 s•2 6.5 7.8 9.1 10.4.

54

Si

13-3 15.2 17.1

2 3 4 S 6 7 8

6464 6474 6 4 8 4 6493 6503 6 5 ^ 65 22 6