LIBRARY Of CONGRESS. 
 
 She! 
 
 f.p3l 
 
 UNITED STATES OF AMERICA. 
 
LOF 
 
 FOR THE LAKES. 
 
 By H, C. PEARSONS. 
 
 " y^ . ^o?^^'''^ 
 
 f" AD 
 
 "sHHw' 
 
 ^c^ 
 
 CLEVELAND : 
 
 BISSELL &. SCRiVENS^ 
 
 1891. 
 

 Copyrighted 1891 
 
 BY 
 BiSSELL & SCRIVENS. 
 
PREFACE. 
 
 It is probably not apprehended bv more than a very few, 
 that in the recent adoption of '^ Standard Time," we have the 
 greatest aid to navigation, on the coast and lakes, that could by 
 any means have been devised. 
 
 The experienced seaman knows that ^^ direction" is the ulti- 
 mate object of all the labored astronomical observations by 
 navigators at sea, and that a knowledge of ''place" is precedent 
 to that of 'direction." 
 
 The determination of place is the office of nautical astronomy, 
 which question alone not infrequently draws largely upon the 
 highest powers of the trained mathematician and astronomer. 
 
 But, fortunately, we are relieved of nautical work on the lakes. 
 ^'The U. S. Lake Survey" has determined the co-ordinates of 
 place for all the light-houses on the whole chain of lakes, and 
 every ship-master on those waters is furnished with the results of 
 this survey, in the ''List of Lights of the Northern Lakes 
 and Rivers," for the asking. And as the '* standard time," now 
 available in nearly every town in the United States, to the 
 fraction of a minute, relieves us of the "chronometer," as used 
 at sea, we have only our "lead" and "log" and "compass," to 
 look after, but which, as we will find, will give us something to 
 do, for the rapid growth in the magnitude of our vessels, result- 
 ing from the enormous growth in the volume of our commerce, 
 make it imperative that those who are to have the care and 
 navigation of them, keep pace in technical and scientific attain- 
 ment with those who design and build them. 
 
IV. PREFACE. 
 
 Accordingly, to help in this matter, I have prepared the 
 following Manual of Xavigation for the Lakes* 
 
 Among those features that may be considered novel, are the 
 articles on the Dumb Compass, with its application to the finding 
 of compass errors; also a method of deducing a table of Time 
 Azimuths of the Sun, from an amplitude when the latitude and 
 the declination are of the same name. This table is available for 
 from one to three hours, requiring but ordinary arithmetical 
 knowledge on the part of the ship-master. 
 
 With the admirable charts in use, and with a precise knowledge 
 of the place of every light, the finding of compass errors is 
 the one problem that suffices for the navigation of the 
 lakes. 
 
 The custom of compensating deviation by means of magnets, 
 is becoming general, the idea being that a needle brought to its 
 place, will remain there indefinitely, and that when it is right on 
 the cardinal points, it is also right on all other points. But this 
 is a sorry mistake. All compasses, are in error, in some parts of 
 the card, even after the most careful adjustment. 
 
 The faces that give position to the needle are so many, and so 
 mutable, that it is not safe to depend on the adjustment for more 
 than a short time, particularly in a new vessel. 
 
 The remedy lies in adjusting the ship-master. 
 
 To attain a confidence in his compass, a living confidence, born 
 of certainty, to meet the growing responsibility laid upon him, 
 the ship-master must be educated up to a full understanding of 
 this question. 
 
 This subject is not, as too often thought, beyond the reach of 
 very moderate attainments, any more than is book-keeping, 
 telegraphy, or photography. The diligent application that 
 masters the one, will master the other. 
 
 Accordingly, I have dwelt on this topic at some length, giving 
 a number of methods of finding compass errors, including that 
 by time azimuths of the sun, which, though costing perhaps a 
 little more time to master, will in the end be found the most 
 efficient and satisfactory of all the methods in use. . 
 
PREFACE. V. 
 
 It is believed that among the several methods given, of finding 
 compass errors, any ship-master of fair capacity will find some 
 method that he can readily master, so as to utilize it. 
 
 The article on the ''correction," or compensation of ship's 
 compasses, is based on the method taught by such scientists as 
 Prof. G. B. Airy, Astronomer Royal; Frederick J. Evans, and 
 Archibald Smith, of the Liverpool Compass Committee. 
 
 While I cannot claim this little work to be free from fault or 
 imperfection, I yet have a confidence that it will be found of 
 sufficient merit to claim respectful consideration from those inter- 
 ested in the maritime affairs of the lakes, and to be particularly 
 helpful to the young student of navigation, in which last event, 
 my highest ambition will have been gratified. 
 
 H. C. PEAESONS. 
 Ferrysburg, Mich., Oct. 27, 1891 
 
CONTENTS. 
 
 PAGE. 
 
 Chapter I. 
 Trigonometry 1 
 
 Chapter II. 
 The Mariner's Compass 22 
 
 Chapter III. 
 The Sailings 40 
 
 Chapter IV. 
 Construction of Charts 64 
 
 Chapter Y. 
 Terrestrial Magnetism, and the Magnetism of Iron in 
 
 Vessels, etc 82 
 
 Chapter VI. 
 The Propeller AVheel 96 
 
 Explanation of the Tables 106 
 
 Table I. 
 Traverse Table 112 
 
 Table II A. 
 Natural Sines and Cosines 113 
 
 Table II B. 
 Natural Tangents and Cotangents 116 
 
 Table II C. 
 Natural Secants and Cosecants 119 
 
Vlll. CONTENTS. 
 
 Table III. 
 Trigonometrical and Conversion Table 122 
 
 Table IY. 
 The Sun's Amplitude 124 
 
 Table \, 
 Sun's Declination, with corresponding Equation of Time. . . . 126 
 
 Table YI. 
 Azimuth and Hour Angle for Latitude and Declination .... 128 
 
 Table YII. 
 For Eeducing Standard to Mean Local Time 142 
 
 Table YIII. 
 Table of Chords to Eadius Unity, for Protracting 144 
 
 Table IX. 
 Meridional Parts 146 
 
 Table X. 
 For Eeducing Departure to Difference of Longitude 153 
 
 Table XI. 
 For the Correction of Middle Latitude, in Middle Latitude 
 
 Sailing 154 
 
 Table XII. 
 Distance of Objects by Two Bearings 155 
 
 Table XIII. 
 For Eeducing Longitude to Time 156 
 
 Plate I. 
 Pearson's Diagram for the Eepresentation of Deviation and 
 
 the Conversion of Compass Errors 158 
 
 Plate II. 
 Illustration of the Components of Magnetic Force 160 
 
 Plate III. 
 Isoo^onic Chart 162 
 
A MANUAL OF NAYIGATION FOR THE LAKES. 
 
 CHAPTER I. 
 
 |Trigonometry. 
 
 Signs and Symbols. — The student not already acquainted with 
 algebraic notation, should make himself familiar with the follow- 
 ing Signs and Symbols, which are among the many used in 
 mathematical language: 
 
 = This is the sign of equality; it implies tJiat the quantities 
 between which it is used are equal, as 12 inches=:l foot, 3 miles 
 =^1 league, etc. 
 
 + This is called the plus sign, implying an increase or addition, 
 as 7+3=10. 
 
 - This is called the minus sign, implying diminution or sub- 
 traction, as 7 — S^=4:, 
 
 The plus and minus signs are also used to imply relative direc- 
 tion, as to the right or left, up or down, heat or cold, forward or 
 backward, etc. 
 
 X This is the sign of multiplication, signifying that the 
 quantities between which it is written are to be multiplied together, 
 as 4Xo=20. 
 
 -T- This is the sign of diyision, implying that the number prced- 
 ing it is to be divided by that one foUow^ing, as 9-^3^=3. 
 
 i ) ] h [ ] These are called brackets. Their use is to indicate 
 that the quantities embraced within them are to be considered as 
 one quantity, thus, 5 (5+3)— 6 (3+2)==10. 
 
 This is another form of bracket, and called a yinculum 
 
 or bar, and is used when two or more quantities already connected 
 
A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 by brackets, are to be regarded as one quantity, as 5 (5-|-3)— 6 
 (5^^)-^5 (8— 6)--2=5. 
 
 : :: : These are the signs of proportion. They are a modifica- 
 tion of the signs of division ^-, and equality ^, and may be 
 written thus, -^ = h-. They signify that the first term divided 
 by the second=the third term divided by the fourth. Quantities 
 having this relation are called proportional quantities or numbers, 
 as 4: 8:: 3: 6,, or better, 4-T-8=3-r-6=J. The value in this case, J, 
 is called the ratio of the proportion. 
 
 A power of a number is indicated by a small figure above and 
 to the right of the number, thus, 42=16 indicates that the number 
 4 is used twice as a factor. 4^^64. 
 
 A root is indicated by the symbol ;/, which alone implies the 
 second or square root. . A small figure written over the symbol, 
 thus, ,/, implies the degree of the root, thus, ,/64=4. 
 
 Symbols of CJuantity. — To abreviate mathematical investiga- 
 tions, quantities or numbers are represented by some symbol, as a 
 letter, — usually the initials of the quantity; and the symbols of 
 operation already explained, aj^ply to them the same as to the 
 numerical quantities. 
 
 Lines are separated by letters at different points in their extent, 
 usually at their extremities. Thus, the line between the points 
 A B is called the line AB. 
 
 Angles are defined by giving one letter in each of the lines 
 containing tlie angle, together with the letter at their intersection, 
 thus, the angle between the two lines ^"^^^1^^==^^ AB and BC, is 
 written ABC, the middle letter indicating the angular point. 
 
 Numerals are represented by single letters, the leading letters of 
 the alphabet representing known quantities and the final letters 
 . unknown quantities. 
 
 The use of an equation is, generally, to bring out the value of 
 some unknown quantity, which is the object of inquiry, and 
 which is involved in the equation. Thus, in a proportion, one of 
 the terms is usually an unknown or required quantity represented 
 by one of the terminal letters (usually x) of the alphabet, as 
 
 A:B::C:X. 
 From this, by the principle well known in arithmetic, that the 
 product of the extremes equals the product of the means, we have, 
 
 AX=:BC, 
 in which X is the quantity desired. 
 
TRIGONOMETRY. 6 
 
 Then, frOm the well known axiom, that wc may multiply or 
 divide equal quantities by the same number without destroying 
 the equality between those quantities, we may divide both sides 
 of this equation by the factor, when we have the equation, 
 
 X=BC^A, 
 in which the quantity represented by X is equal to the product of 
 the two quantities B and C divided by the quantity or factor A. 
 This operation is called solving the equation for X, and the 
 expression, X=BC^-A, is called a formula. 
 
 Alg'ebraic Addition is the aggregation into one sum, of quan- 
 tities affected by the opposite signs of -}-j — ? or plus and minus. 
 Thus, if a man has ^100 in bank to his credit, or -{-, and he has 
 bills payable, — , ^90, his estate is only $10 — that is to say, the 
 sum of Dr. and Cr. sides of his % is + f'lOj the two signs can- 
 celling each other to the extent of the smaller number. 
 
 Algebraic Difference is found by changing the sign of the 
 quantity to be subtracted, then combining the quantities as in 
 addition. Example: The difference of latitude between 10° X. 
 and 5° S., or +10° and — 5°=_15°. By changing either sign, 
 both terms become ^^like," i. e.j both -f- or both — , and they 
 come together by addition, so that we may have +15° or — 15° as 
 the difference of latitude of the two places. 
 
 Trigonometry is that branch of geometry that treats of the 
 relations of the sides and the angles of triangles, both plane and 
 spherical. In this work, only plane triangles will be considered. 
 
 In all the wide range of mathematical science, there is, perhaps, 
 no one branch so generally useful or of so wide an application in 
 general affairs of life, as trigonometry. 
 
 To the navigator, the astronomer, surveyor, geographer and 
 the civil engineer, it is simply indispensible. Xatural philosophy, 
 mechanics, optics and geology could not be treated without this 
 branch of mathematics. 
 
 It is hoped the little I may introduce of the subject, and which 
 is as little as we can do with, will induce the student not already 
 familiar with it, to pursue it further in some of the many excel- 
 lent standard works on that science. 
 
 Whatever we may give of trigonometry, we shall treat without 
 the use of logarithms, — the object being to make the subject as 
 elementary as possible. 
 
4 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 There are two kinds of magnitude considered in triangles, viz., 
 au^les and sides or lines. 
 
 For the purpose of measuring angles, a circle is introduced. 
 The circumference of this circle is supposed to be divided into 
 360 equal parts, called degrees; these are further divided into 60 
 equal parts, called minutes; and these, again, into 60 equal parts 
 called seconds. 
 
 The degrees are indicated by a small circle ° over and to the 
 right of the figure; minutes by a dash ^, and seconds by two 
 dashes ^^ above and to the right of the number of minutes or seconds. 
 Thus, 15 degrees, 28 minutes and 46 seconds would be written, 
 15°, 28^, W, etc. 
 
 A degree or a minute, then, is not a magnitude of length, but 
 merely a certain part of the whole circumference, without regard 
 to lineal dimensions; it is merely ratio. 
 
 In measuring an angle, the center of the measuring circle is 
 located at the intersection of the two lines that contain the angle 
 to be measured. The arc intercepted by the two lines is the 
 measure of the angle. 
 
 But, as degrees, etc., and lines are of different kinds of mag- 
 nitude, they cannot be directly compared. Moreover, the mag- 
 nitude of an angle does not vary with the size of the triangle. So, 
 to make these elements comparable, auxilliary lilies have been 
 introduced in and about the arc to be measured, or considered, in 
 such a manner as to make them depend not only on the magnitude 
 of the arc, but on that of the triangle to be considered. 
 
 These auxilliary lines are called fuiictioiis of the arc, and have 
 received different names according to the different positions or 
 relations they have to the arc. 
 
 Names of the Functions. These are eight in number, as 
 
 follows (see Fig, 1): 
 
 In the triangle ADC, let the arc AB be the measure of the angle 
 ACD. Then (1) the Sine is the perpendicular BG drawn from 
 one extremity, B, of the arc, to a diameter or radius drawn 
 through the other extremity of the arc. 
 
 (2). The Tangent, AD, is drawn perpendicular to the diameter 
 through one end of the arc, to meet a line drawn from the center 
 of the circle through the other end of the arc, as at D. 
 
 (3). The Secant, CD, is a right line from the center through 
 one end of the arc, to meet the tangent, as in D. 
 
TRIGONOMETRY. 
 
 (4). The Cosine, CG, is that part of the radius intercepted 
 between the foot of the sine and the angular point C. 
 
 (5). Tlie Tersine, AO, is that part of the radius intercepted 
 between the foot of the sine and the extremity of the arc, or the 
 foot of the tangent. 
 
 The Compliment of an arc or angle is what the angle lacks of 
 being a full quadrant, or 90°. Thus, the angle ACE, being a 
 right angle or 90°, the arc BE is the compliment of the arc AB. 
 The prefix ^'co'^ being used to show that the function before 
 which it is used pertains to the compliment of the arc under 
 consideration. Thus: 
 
 (6). The Co-Tangent, 
 EF, is the tangent of the 
 compliment of the arc AB. 
 
 (7). The Co-Secant, 
 CF, is the secant of the 
 compliment of the arc AB. 
 
 (8). The Co-Yersed 
 Sine, E H, is the versed 
 sine of the arc AB. 
 
 The Supplement of an [ 
 arc or angle, is what the 
 arc lacks of 180°. Thus, 
 the arc BCI, embracing 
 the angle DCI, is the sup- 
 plement of the arc AB. 
 And it has the same sine 
 BG as the arc AB. 
 
 The Sine and Cosine 
 connect the two arcs AB and BE, so that when one is known 
 the other is also known, for it will be observed that the sine of 
 one of the arcs is the cosine of the other. The student will observe 
 that the sine and tangent of an angle are always opposite the 
 angle. 
 
 Relations Between the Parts of a Triangle. — It is shown in 
 geometry that the sum of the two acute angles of a right plane 
 triangle is equal to two right angles, so that when one of the 
 two is known the other is also known. 
 
 The Radius is any right line drawn from the center to the cir- 
 cumference of the circle. The radius diminished by cosine is 
 
 Fig. 1. 
 
6 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 yersiue (see AG in Fig. 1), and diminished by sine, is CO- 
 versine, EII. 
 
 In tables of trigonometrical functions, radius being unity, all 
 the internal functions, as sine, cosine, etc., are decimal fractions, 
 as also the tangents of arcs less than 45°, and the co-tangents of 
 arcs greater than 45°. This is more particularly shown in the 
 following table of the 
 
 Limits of Trigonometrical Functions : — Let AB be any arc 
 
 (see Fig. 1) varying from at B to 90° at E; then when 
 
 AB=^0, the sine=0, the versine^O and cosine=:l, 
 
 ^^ tan.=0, " secant=l and cotan.=infinity, 
 AB=90°, the sine^l, the versine=l and cosine=0. 
 
 the tan.=infinity, the secant=infinity and cotan.=:0. 
 AB=45°, the tan.=l and cotan.:=L 
 AB=60°, " cosine=J and versine=^. 
 
 The student should make himself familiar with this table, so 
 that he can tell the limits of any function on hearing it called. 
 He should also observe the effect of any change in the arc AB. 
 Thus, if AB be increased, the radius remaining the same, then 
 all the primitive or direct functions will be increased, while the 
 co-functions will be diminished. Again, if the radius AC be 
 increased, the angle or arc BA remaining the same, then all the 
 functions will be increased in the same proportion. 
 
 Thus it is seen that no change can be made in the magnitude of 
 the angle or in the radius of the measuring circle, without a cor- 
 responding change in the value of all these auxilliary lines. So it 
 is the introduction of these lines that has brought the solution of 
 triangles within the pale of arithmetic. 
 
 The two sides containing the right angle are called the legs. 
 Sometimes base and perpendicular. But any side may be made 
 the radius, i. e., the side by which the others are measured, when 
 the other sides take names according to their relation to the angles 
 of the triangle: Thus, in the following triangle, ABC, if BC is 
 made the unit for measuring the other sides, then AB is sine and 
 AC is the cosine of angle C. 
 
 Again, if in Fig. 3, AC be made the radius, or measuring unit, 
 then AB becomes the tangent and BC the secant of angle C. 
 
 Or if, as in Fig. 4, the perpendicular AB be made radius, then 
 the base AC becomes tangent and BC the secant of the angle B. 
 
 The student should make himself familiar with these changes. 
 
TKIGONOMETRY. 
 
 Fig. 2. 
 
 Fis:. 3. 
 
 The relations existing between the functions of arcs, are those 
 resulting from the comparison of similar triangles, and consist 
 of equality of ratios, which is a fundamental principle of trig- 
 onometry. 
 
 Thus, in Fig. 5, the triangles ABC 
 and A^B^CS having their sides 
 opposite the equal angles A, A ^ ; B, 
 B^, etc., parallel, are similar. 
 
 Also the triangles ABC and A^ 
 B^E (Fig. 6), and ADE of same 
 Fig., having their "like" sideS 
 parallel, are all similar triangles. 
 " Like Sides '' are those that are 
 opposite equal angles. They are 
 also called homologous sides; thus, 
 BC and DE, Fig. 6, are homologous 
 sides, being opposite the angle A, 
 with the sides BC and DE parallel. 
 Again, the sides AD, A^B^, being 
 parallel, and opposite the same angle 
 E, are homologous. 
 
 The ratios of these homologous 
 sides is the same between any two 
 *'like'* sides of similar triangles, i. e., 
 triangles be divided or measured by a 
 triangle, the quotient will be the same 
 as that of any other side in the same 
 triangle, measured by its corresponding 
 ''like'' in the other. 
 
 This relation for a right triangle is 
 expressed thus, for any angle, as ACB 
 
 BC AC AB 
 
 DE AE AD 
 Cosine Sine R 
 
 Fi-. 4. 
 
 if any side of one of the 
 'like" side in the other 
 
 Fig. 5. 
 
 R Tan. Secant C 
 
 These relations are the foundation of 
 
 the several rules for the solution of 
 
 right triangles, and w^hen stated in the 
 
 Fig. 6. 
 
8 
 
 A MANUAL OF NAVIGATION FOIl THP: LAKES. 
 
 form of a proposition, are as follows : 
 
 Cos.: R:: Sin.: Tan. I Sin.: Cos.:: H: Cotan. 
 ::K:Sec. j ::R:Cosec. 
 
 Here, it will be seen, are five quantities besides the right angle, 
 — three sides and two angles — any three of which being given, — 
 one being a side, — the other two may be found. Hence, 
 
 In a right angled triangle, — the two acute angles being compli- 
 ments of each other, — there are only four parts to be con- 
 sidered, viz., three sides and one angle. 
 
 Some two of the six ratios of the preceding article will always 
 contain the two given quantities and the required quantity, And 
 we have the four following cases, viz.: 
 
 (I). Given, the Hypothenuse and An- 
 gles, to find the two sides containing the 
 right angle. 
 
 (II). Given, the Hypothenuse and one 
 Side, to find the two angles and the other 
 side. 
 (III). Given, the two Angles and one 
 Fig. 7. Side, to find the hypothenuse and the 
 
 other side. 
 
 (lY). Given, the two Sides, to find the angles and the hypoth- 
 enuse. 
 
 To Solve a Right Triangle, for any of its parts, it is only 
 necessary to select any two of the six ratios that contain 
 the two given elements or parts and the required part of the 
 problem, and arrange them in the form of a proportion, then 
 Solve the proportion precisely as in arithmetic, by multiplying 
 together the extremes and the means of the proportion in two 
 products, and divide both products by the factor that is found 
 
 connected with the required 
 quantity. 
 
 Example. Suppose we wish to 
 find the hypothenuse of a right 
 triangle, the angles and one side, 
 — say the perpendicular, — being 
 given. 
 
 Solution. Let ABC be the tri- 
 ^^' ' angle whose side b is required, — 
 
 the angles and side AB being given. 
 
TKIGONOMETE, Y . 
 
 It will be found convenient to represent the angles hy the cap- 
 ital letters and the sides by the small letters of the same name as 
 the opposite angle. 
 
 Making either side, say a, the radius, the given side c is the 
 tangent of the angle C. Also, the required side is the secant of 
 angle C. Thus, our two given parts are R and tan., and the 
 required part is see. of angle C. 
 
 Then, as in arithmetic, form a proportion, so that the ratio of 
 the functions will form the first couj^let, and that of given and 
 required side the last couplet, thus: 
 
 Tan. C;Sec. C::c:b. 
 Multiplying extremes and means, as in arithmetic, 
 
 Tan. Cb=Sec. Cc. 
 Dividing both products by first term, as in arithmetic, 
 
 Sec. C c 
 
 b= (1). 
 
 Tan. C. 
 
 Again, making b the radius, then, 
 
 Sin. C : K : : c : b, and, as before. 
 Sin. C b=:Rc, then dividing and K being unity, 
 C 
 
 b=-— (2). 
 
 Sin. C. 
 Again, making AB or c the radius, 
 
 R : Cosec. C : : c : b, or 
 Eb=Cosec. Cc. 
 AVhence, by dividing, and remem- 
 bering that R is unity, wc have, 
 
 b=Cosec. Cc /o\ 
 
 =3Sec. A. ^''''* 
 
 From the above, it will be seen 
 that we may make any side of the 
 triangle the measuring side, but to 
 make the given side, as c, or the re- 
 quired side, as b, in the above ex- Fig. 9. 
 ample, the measuring unit will give the simple solutions, as seen 
 in equations (2) and (3), above. 
 
 Thus, to find the side BC or a, we would in the same manner 
 have, 
 
 a==c tan. A, 
 
10 A MANUAT^ OF NAVIGATION FOR THE LAKES. 
 
 or, if we make a the radius, then, 
 
 c 
 Si= ■ 
 
 tan. c. 
 
 In this manner many formulas have been constructed for the 
 solution of right jDlane triangles. From them I have selected a 
 few of the most useful, and which are sufficient for the solution 
 of such questions as are likely to come before the navigator in 
 *' dead reckoning." 
 
 Preparatory to the solution of practical questions, the student 
 must be informed of some of the Trigonometrical Tables and 
 their uses. 
 
 Table of the Natural Functions. — These are the Sines, Co- 
 sines, Co-tangents, the Secants and Co-secants, Yersines and Co- 
 versines of arcs varying by one minute, for a whole quadrant. 
 
 They are called natural functions to distinguish them from the 
 same functions when given by their logarithms. 
 
 This table gives all the above elements to radius unity, — thus, 
 we have all the three sides and their corresponding angles for 
 5400 right triangles; and by considering the angles to vary by 
 parts of a minute, as by 10^^ or 20^^ we could readily deduce all 
 the elements for many thousand more triangles. 
 
 So, having the elements of any proportioned triangle to radius 
 unity, that can come before us in practice, we have only to insti- 
 tute a proportion between the known parts of one triangle and 
 the corresponding parts of the tabular triangle, and reduce the 
 proportion to find the parts desired. 
 
 These sines, tangents, secants, etc., are arranged in columns, 
 under their rbspective names. 
 
 The degrees for arcs of less than 45° are found at the top of the 
 page, with the minutes in a column at the left. For arcs greater 
 than 45°, the degrees are found at the foot of the page, with the 
 minutes in a column at the right. (See table II). 
 
 Let the student look out the following functions. (Our table 
 gives them only to intervals of 5°) : 
 
 The natural sine of 25°, 40^= .4331 
 " '' cosine '' " = .9013 
 
 '' " tangent '' '' = .4805 
 
 " '' cotangent '' '' =2.0809 
 
 '' '' secant '' '' =1.1095 
 
TRIGONOMETRY. 11 
 
 The natural cosecant af 25°,40^=2.30S7 
 '' '' sine of 74°, 25^= .9632 
 
 " " cosine '' '' == .2686 
 
 '« " tangent " " =3.5856 
 
 " '' cotangent '' '' = .2789 
 
 '« " secant *' '^ =3.7224 
 
 '' '' cosecant '' " =1.0382 
 
 The equations spoken of in a former article, as being selected 
 for this work, are the following: 
 
 Formulas for the Angles of a Right Triangle. 
 
 -, J Sin. C=c^-b^==perpen(licular-i-hypothenuse. 
 
 ' \ Cos. C::=:a^-b^base-^hypothenuse. 
 o r Tan. C^=:c-7-a=perpendicular-T-base. 
 
 *\Cotan. C=a-4-c=zrbase .-perpendicular. 
 J Sec. C=bH-a=hypothenuse-T-base. 
 
 * I Cosec. C=b-i-c=hypothenuse-i-perpendicular. 
 
 It will be observed that the preceding equations are in pairs, 
 €ach pair finding an angle by a function with its co-function. 
 
 It will be observed, too, that each angle is found indirectly by 
 i3ome function, as sine, cosine, etc., so that the angle correspoding 
 lo the function must be found from the table of sines, cosines, etc. 
 
 It will be observed, also, that any function is found by dividing 
 one side by some other side of the triangle, which is for the pur- 
 pose of reducing the sides of the triangle in question, to radius 
 unity, for the purpose of making them comparable with a tabular 
 triangle. 
 
 It will also be observed, in the second member of each of the 
 equations in the preceding article, that some one side of the tri- 
 4ingle is divided by some one of the other sides. Then, turning 
 this divisor over to the other member of the equation, as a 
 multiplier, we find a side of the triangle. In this manner we 
 derive the following: 
 
 Formulas for the Sides of Right Triangles. 
 
 r at=b cosine C=hypothenuseXcos. C, or sin. A. 
 
 • \ =c cotan. C=perpendicularXcotan. C, or tan. A. 
 rb=ra sec. C=baseXsec. C, or cosec. A. 
 
 • \ =0 cosec. C=perpendicularXcosec. C, or sec. A. 
 
 r c=b sin. C=hypothenuseXsin. C, or cosine A. 
 
 * \ =a tan. C==baseXtan. C, or cotan. A. 
 
 (See Fig. 10). 
 
12 
 
 A MANUAL OF NAVKxATION FOR THE LAKES, 
 
 Geometry gives the following equations for the sides 
 triangle, viz.: 
 
 -(1). 
 
 -(2). 
 -(3). 
 
 )f a 
 
 a^:r^b- 
 b 
 
 r|/rad.- — sin. 2 C 
 
 y^a--t-c2=r|/sin.- C+C08.- C- 
 
 c^=/b2 — a2=-|/rad.2— oos.2 C 
 
 The student should make himself so thoroughly familiar with 
 the three sets of equations in each of the three preceding articles, 
 ^^ that he can readily select any one 
 wanted for the occasion. 
 
 With regard to numerical work, the 
 student, having made himself familiar 
 with the preceding articles, is prepared 
 for the solution of plane riglU triangles, 
 but preparatory to a numerical solution, 
 he should be able to solve them by coii- 
 Jd struction, which involves the use of a 
 
 Tal)le of Chords of Arcs, or a 
 ^^S' 10« Protractor. — A scale of chords is 
 
 sometimes engraved on the ordinary drafting scales. But it is 
 to a particular radius given on the scale, — and therefore of but 
 limited use, — moreover, it is not sufficiently precise for good 
 work, though a good protractor w^ill do. 
 
 By a principle of geometry, the chord of an arc is twice the 
 sine of half the arc. Whence, a table of chords can be con- 
 structed directly from a table of natural sines. Thus, 
 
 Eequired to find the chord of 26°, 28^. The sine of the half of 
 
 this arc, or 13°, 14^, is .22892. Twice this decimal, retaining four 
 
 places of figures, is .4578, which is the chord of 26°, 28"^ to radius 
 
 unity. (Our tables, varying by 10^, will give the chord for 26°, 
 
 '30^=.4584). 
 
 In this manner was the table of chords computed. It is arranged 
 with the degrees at the head of the columns and the minutes in 
 columns at the margin of the page. And it is computed for 90°, 
 varying by 10^ (See table YIII). 
 
 Thus, the chord of 43°, 10^= .7357 
 '' '' 60°, 00^=1.0000 
 
 '' '' 84°, 40^=1.3469 
 
 '' " 4°, 05^= .0713 
 
 " " 15^= .0044 
 
TRIGONOMETRY. 
 
 13 
 
 In using a table of chords for the construction or the measure- 
 ment of angles, the student should have a scale divided decim- 
 ally. Then, by removing the decimal point one or two places to 
 the right, he may have a large scale to set off his decimals with, 
 and thus attain great precision. 
 
 He will also want compasses for ink and pencil, a T square and 
 triangle or parallel rule. If not already provided with such instru- 
 ments, he should send twenty or thirty cents to some dealer in 
 drawing instruments, for an Illustrated Catalogue. 
 
 Solution of Plane Triangles.— Examples : 
 
 1. Sailed S. 48° W., 126 miles. 
 How far did I sail south, and 
 how far west ? 
 
 By Construction. Draw a 
 vertical line AB (see Fig. 11) to 
 represent the meridian from 
 which the course is reckoned, A 
 representing the starting point. 
 
 Look in the table for the chord 
 of 48°, which we find to be .8135 
 to radius unity. Removing the 
 decimal point one place to the 
 right, we have 8.135 as the chord 
 of 48° to radius 10. 
 
 Take 10 units to any convenient scale and set them off from A 
 to m. Then take the chord 8.135 to the same scale and set them 
 off from m to n. 
 
 Through the points A and n, draw an indefinite line, and on it 
 set off the distance AC, 126 miles, to any convenient scale. 
 
 Through C, draw a line at right angles to the line AB, meeting 
 that line in B. 
 
 Measure the lines AB and BC by the same scale as that by 
 which AC was set off, and we have the measure desired of the 
 two sides. 
 
 AB or c=southings=84.31 miles. 
 BC or a==westings==93.63 miles. 
 
 By Computation. In this problem, it is seen that we have 
 the angles and hypothenuse from which to find the sides a and c, 
 containing the right angle. 
 
 Fig. 11. 
 
14 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 By reference to the first equation of the third pair of equations, 
 (page 9), we see that the perpendicular c is found by multiplying 
 the hypothenuse b into the cosine of the angle A, adjacent to c, 
 or into the sine of C, the angle opposite, or 
 c=bXcos. (A=48°) 
 =126 X .6691=84.31 miles southings. 
 In the same manner from the first equation of the first pair, we 
 
 have, 
 
 a=bXsine (A=48°) 
 
 =126 X .7431=93.63 miles westings. 
 It will be seen that the numerical work is only indicated here. 
 The student should look the functions from the tables and per- 
 form the multiplication; and he may also construct the angles 
 with a protractor, though the method by ^'chords" should be 
 learned. 
 
 2. Example: Sailed due north 146 miles, then due east 76 
 miles. Required the course and distance back to the place sailed 
 from. 
 
 By Construction. Draw vertical line on the page to represent 
 the meridian, and set off on it, to 
 any convenient scale, the distance 
 sailed north, 146 miles, C to B. 
 
 Then turn off a line at right an- 
 gles to the right, and set off on it 
 the distance sailed east, 76 miles, 
 B to A, then join A with C. 
 
 AC, measured on the same scale, 
 
 will be the distance required. We 
 
 are now to find the angles at A and 
 
 C. We could apply a protractor 
 
 and read the angle directly from it, 
 
 but it is better to measure it by 
 
 means of its chord to some radius — 
 
 Fig. 12. say 100. Thus, 
 
 Take 100 in the compass, to any convenient scale, and sweep an 
 
 arc from the side a to the side b, with C as the center. Also from 
 
 the side b to the side c produced, from A as a center, sweep the 
 
 arc m, n, for the purpose of measuring the angle A. 
 
 The object of measuring or computing both angles is to have a 
 check on our work. The student will soon see abundant reason 
 
TRIGONOMETRY. 15 
 
 for this in the non-pr.ecision of his first efforts. A and C together 
 should=90°. 
 
 The chord measuring the angle C to radius 100, will be found 
 =47.5, and that for A=103.7. 
 
 Kemoving the decimal point, in each case, two places to the 
 left, for the purpose of finding the chord for radius unity, we 
 have for C, .405, and for A, 1.098. Taking these chords to the 
 table for the angles, we find A=62°, 30^ and C=27°, 30^ and A 
 C=164.6. 
 
 As courses are always measured from the meridian, we must 
 take the compliment of the angle A for our return course, which 
 is S. 27°, 30" W. and distance=164.6 miles. 
 
 By Computation. In this case we have the two sides contain- 
 ing the right angle to find the angles and the hypothenuse. By 
 equation 2 (page 9), we have. 
 
 Tan. C^^perpendicular-r-base, 
 
 =76^146=.5205=tan. (27°, 30^=C). 
 Thus, dividing the perpendicular by the base, we get the tangent 
 of the angle opposite the perpendicular, — in this case, the decimal 
 .5205. Then, looking in the table of natural sines, tangents, etc., 
 we find this decimal in column of tangents, under 27° at the head, 
 and in line of 30^ on the left. 
 
 One of the two angles being found, we may subtract it from 90° 
 for the other-=62°, 30^ 
 
 But, as a check, it is better to compute it in the same manner, 
 by the same rule, thus: 
 
 146-^-76=31. 92105=tan. (A-=62°, 300- 
 
 From equation (2) of (page 9), we have, 
 b=a X secant (C-=27°, 30^ 
 =146 X 1.1274=164.6=distance. 
 Examples: 
 
 3. Given the hypothenuse 26. The angle at the base=26°, 
 20''. Bequired base and perpendicular. (See eq. 3 of page 9). 
 Ans. Base=23.37. 
 Perp.=11.38. 
 
 Note. — Let the student construct graphically, with care. He 
 should use a sharp lead pencil, making a line as small as can be 
 clearly seen, and using a needle point for pricking off distances. 
 
16 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 4. Given the hypothenuse 146. Angle at the base, 72°, 30''. 
 Find perpendicular and base. 
 
 Ans. Base=43.95. 
 Perp.=89.28. 
 
 5. Given hypothenuse 84, base 46. Required the perpendic- 
 ular and the angles. (See pages 8-9 for angles). 
 
 Ans. Angle at base==o6°, 50'' 
 
 ^' '' vertex=33°, 10^ 
 Perpendicular=70.29. 
 
 6. Given hypothenuse 218, one of the sides contain the right 
 angle=46. Required the angles and other side. 
 
 Ans. Angle opposite the greater leg=77°, 50^ 
 '' '' '' smaller '' =12°, 10^ 
 
 The other leg=213.13. 
 
 7. One leg or side=76, angle opposite=43°, 28^. Required 
 the hypothenuse and other side. 
 
 Ans. Hypothenuse=110.5. 
 Other Side, =80.18. 
 
 8. One side=243, adjacent angle=18°, 40^. Required the 
 hypothenuse and other side of the triangle. 
 
 Ans. Hypothenuse=256.49. 
 Other Side, = 82.08. 
 
 9. Given hypothenuse 180.3, legs 176, and 39.04. Required 
 the angles of the triangle. See group of equations for angles, 
 (page 8). 
 
 Ans. Angles, 12°, 37^ and 77°, 23^ 
 Let the student find an angle, say the smaller one, from each of 
 the three sides, thus: 
 
 Hypoth.H-perp.=cot. angle at base=12°, 37^. 
 Perp.H-base=tan. angle at base=:12°, 37^. 
 Hypoth.^-base=sec. angle at base=12°, 37^. 
 Note. — As our tables vary by 5^, the nearest minute given by 
 them for this case will be 12°, 35^, and as our compasses are not 
 graduated to appreciate angles smaller than 1°, the practice of 
 computing, using 5 or 6 place decimals with intervals of V and 
 finding all angles to the nearest V , is simply a waste of time, — 
 accordingly, we have abridged the tables, and thereby the work. 
 Oblique Plane Triang-les. — The solution of oblique plane 
 triangles will require the use of a few formulas different from 
 those required for right plane triangles. 
 
TRIGONOMETRY . 
 
 17 
 
 Fig. 13. 
 
 Proposition 1. In any plane triangle, the sides are propor- 
 tioned to the sines of the opposite angles. 
 
 In the oblique triangle ABC, if we make the two sides AC and 
 BC radius, the perpendicular CD is the sine of each of the angles 
 A and B. Then, 
 
 AC : CD : : K : sin. A, or AC sin. A=E CD. 
 And BC : CD : : K : sin. B, or BC sin. B=E CD. 
 But the second members in each 
 of the above equations are the same, 
 whence the other two are propor- 
 tioned, and we have, 
 
 AC sin. B=BC sin. B, or 
 AC : BC : : sin. A : sin. B, 
 according to proposition. 
 
 Solution by the Properties of Ri^ht Triangles,— three 
 €ases. — As all cases of right triangles may be solved by the prop- 
 erties of right triangles, we give no more rules for their direct 
 solution, though a number are known. 
 
 I. The three parts given may all be adjacent, as when two sides 
 with their included angle are given. 
 
 In this case, demit a perpendicular from the extremity of the 
 smaller side onto the other side, as from C to D in the triangle 
 ABC. 
 
 In the right triangle A C D, we 
 have hypothenuse and one side, with 
 the angles from which to find the 
 perpendicular CD, which may be 
 found by equations on page 9, as 
 also may the base A D. 
 
 In the triangle BCD, subtracting ^^^' -*^^* 
 
 AD from AB, we have BD. Then, CD having been computed, 
 we have the two sides containing the right angle from which to 
 find the angles. 
 
 The angles A and B, being now known, we have only to take 
 their sum from 180°. when we have the angle at C, on the principle 
 that the sum of the three angles of any plane triangle is 180°. 
 
 The side BC remains to be found. We may find it by Prop. 1, 
 just explained, and called the sine proportion, or we may find it 
 
18 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 by means of the equations for the sides of aright triangle, page 9» 
 By the ''sine proportion," the solution is, 
 
 Sin. B : AC : : Sin. A : BC. (Page 12). 
 
 II. When the Angles and One Side are Giren. — From one 
 extremity of the given side, demit a perpendicular onto one of the 
 other sides, or onto one of them produced, as from C to D on the 
 side AB produced, of the triangle ABC. (Mark with a dash, — , 
 the given parts). 
 
 In the right triangle ACD, we have all the angles and the 
 hypothenuse AC given, from which we may find AD and DC^ 
 from rules already explained. 
 
 Then, in the right triangle DBC, the angle at B is known,, 
 because it is the supplement of B in the 
 oblique triangle ABC, whence both angles 
 of the triangle BDC are known, and DC 
 having been computed, BC can be found 
 from equations, page 9. 
 ,_. Now, AB may be found from the ''sine 
 
 proportion," Prop. 1, page 12, when the 
 triangle is determined. 
 The proportion for BC is. 
 Sin. B : AC : : Sin. A : BC, 
 and for AB it is. 
 
 Sin. B : AC : : Sin. C : AB. 
 Thus the case may be solved either by the rules for right triangles 
 or by the sine proportion. 
 
 III. When the Three Sides are given, to find the Angles* 
 
 The solution of this will depend on the following: 
 
 Proposition 2. In any plane triangle, as the greater side 
 : The sum of the other two sides, 
 : : The difference of the other two sides, 
 : The difference of the segments of the greater 
 side made by a perpendicular from the 
 opposite angle. 
 In the triangle ABC, let AB be the greater side. With AC as 
 a radius, and from C as a center, sweep the arc EAF and produce 
 BC to E. Then BE is the sum of the two smaller sides, and BF 
 is their difference. 
 
 AB is the sum of the two segments of the greater side, made by 
 the perpendicular CD, and BG is their difference. 
 
TRIGONOMETRY. 
 
 19 
 
 Then, because the two lines BA and BE are two secants, cutting 
 the circle and meeting outside the circle, the whole secants and 
 their external segments are inversely proportional, and w^e have 
 the proportion, 
 
 AB : BE : : BF : BG, 
 which is the proposition. 
 
 Then, half of this difference added to half the base, or greater 
 side=tlie greater segment. And half this difference sub- 
 tracted from half the base, gives the smaller segment. 
 
 This being done, the triangle 
 is divided into two right triangles, 
 in each of which the base and 
 hypothenuse are known, from 
 which the angles are determined 
 by the equations on pages 8-9. 
 
 Example: In the oblique tri- 
 angle ABC, given the three sides, 
 64, 56, 34, to find the angles. 
 
 Solution: Demit a perpendic- 
 ular from C onto the greater side ^ ^S- ^^* 
 at D, dividing that into the two segments m and n, then by the 
 last proposition we have, 
 
 As AB (=m+n)=64 
 :AC hCB=90 
 ::AC— CB=22 
 : m— n=::90X22^64=30.9416. 
 
 Then, J (m— n)=15.4708, 
 
 1 (m+n)=32.0000. 
 
 Whence, m =47.4708, 
 
 and n =16.5292. 
 
 The oblique triangle is now divided 
 into two right triangles, in each of 
 Avhich the base and hypothenuse are 
 given to find the angles, which can be done by means of the equa- 
 tions on page 8. 
 
 Then, the sum of the angles A and B being taken from 180° 
 before explained, we have angle C. 
 
 as 
 
20 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Examples of the three cases of oblique triangles for solution. 
 1. Given the two sides, 91 and 60, with their included angle, 
 42°, 30^. Required the other side and the other two angles. 
 
 Solution: First, by Construction. In the triangle ABC, lay 
 off one side, preferably the longer side AB, to any convenient 
 scale=to the given side, 91. From each point, A and B, sweep 
 an arc on a radius of 10 or 100 units, — say 100, to any convenient 
 scale, for the purpose of measuring the angles at B and C. 
 
 On the arc opposite A, set off the chord of 42°, 30^, from n to 
 c, and draw the line A c. On this line, set off from A, the dis- 
 tance AC:=60, and join 
 CwithB. Then is the 
 triangle constructed. 
 
 Measure BC by the 
 same scale, and that side 
 is determined=(61.8). 
 
 Produce BC to b, and 
 measure the chord mb=^ 
 70, w^hich corresponds 
 to (40°, 560 for the an- 
 gle at B. Then 180° — 
 (BTC=383°, 260-^(96° 
 34^)=angle at C. 
 
 Verify by construct- 
 ing C thus: With the 
 same radius as for the 
 other angles, sweep an arc from C as center and produce, if neces- 
 sary, the CB and CA to it, as at o and p. We find the chord o p 
 greater than the chord of 90, the limit of the table of chords, so 
 we measure the arc in two parts, and find it to be (96°, 34^), as 
 before, or verify with a protractor. 
 
 By Computation. In the triangle 
 ACD, Ave have the angles and one 
 side (the hypothenuse) to find the 
 other sides. 
 
 Then multiplying AC by sine of 
 
 A, we have DC or .6756X60=.(40.54 
 
 Fig. 19. =DC). Eq. 3, page 9. And ACX 
 
 cos. 42°, 300=AD; or .7373X60=(44.24=AD). Eq. 1, page 9. 
 
 And AB-AD=91— 44.54=(46.76=BD). 
 
 U^.2^ 
 
 ^6.7^ 
 
TRIGONOMETRY. 21 
 
 The angle at B is found by its tangent, by dividing DC by DB, 
 or 40..j4^46.76=(.S670=tan. 40°, oij'). Eq. 2, page 8. 
 
 The side BC is found by multiplying the side or base BD by 
 sec. of angle at B, or 46.76X1.324=(61.85=BC). Eq. 2, page 9. 
 
 The angle at C is found by taking the supplement of the sum of 
 the angles A and B, which gives (96°, 34^=C). 
 
 The student should complete the multiplications indicated in 
 the above computations, looking the functions out of the tables, 
 and he should make careful constructions to scale, of all his prob- 
 lems of triangles, as the geometrical figure will always help to a 
 clear conception of the numerical work required. 
 
 There are many other theorems for the solution of oblique 
 triangles, but those given are deemed sufficient for all of that class 
 of questions likely to occur. 
 
 Examples of Oblique Triangles. — 
 
 1. Given one side b of a triangle=32 rods. The angle A 56°, 
 20^, and the angle at C=49°, 10^ to find the sides a and b. 
 
 Ans. a=27.5 rods, 
 b=25.0 " 
 
 2. Given the angle A=63°, 35^, the side b=32, and the side a 
 =36. Required the other side and the other two angles. 
 
 Ans. Angle B=o2°,45J% 
 " C=63°, 30.}^, 
 Side AB=36, nearly, 
 
 3. Given angle A=26°, 14% the side b=78, the side c=106. 
 Eequired the angles B and C, and the side a. 
 
 Ans. B=43°, 14% C=110°, 2% side a=oO. 
 
 4. Given the side a^o4, c=78 and b=70. Required the 
 angles. 
 
 Ans. A=42°, 22% B=60°, 52t% C=76°, 45V. 
 
CH.APTER II. 
 
 THE MAHINEK S COMPASS. 
 
 Concerning the early history of this instrument, we have but 
 Jittle reliable information. 
 
 We first hear of it in China, where the needle or 'Moadstone'' 
 was used to give direction, by floating it on a piece of cork, on 
 water. 
 
 Then we hear of it sus2:>ended at the middle by a string, when 
 the magnetic force of the earth would give it direction. 
 
 Early in the 14th century, the compass was improved by placing 
 the needle on a '^pivot post;" and in 1608, the Rev. Wm. Barlow 
 further improved it by applying the ^'gimbal joint" to the bowl 
 carrying the needle. But we must consider it as at present con- 
 structed. 
 
 The Needle is made of steel, hardened and magnetized. It is 
 then surmounted by a card graduated into 32 points, and some- 
 times into quadrants of 90° each, and sometimes into degrees, con- 
 tinuously, from at the north, around by the east, south, west, 
 to 360° at the north. 
 
 The better class of compasses have two to four needles under 
 the card. 
 
 Names of the Points. — The line passing through the point on 
 which the needle swings, and the north point of the card, is called 
 the Zero, or Meridian Line, — the intersection of this line with 
 the north part of the card being marked with a ^ 'fleur-de-lis," by 
 way of distinction; the south side being marked by the letter S. 
 
 The line at right angles to this meridian, is called the '^ Equa- 
 torial," from its analogy to the equator, — its extremities being 
 marked with the initials of the directions they represent, — E for 
 east, and W. for west. 
 
 The four letters, N., S., E., W., are called the cardinal letters, 
 and the points they represent are called the four Cardinal Points., 
 Sometimes the Equatorial is called the Prime Yertical. 
 
THE COMPASS. 23 
 
 SubdiTiding" the Card. — The points midway between the car- 
 dinal points are called the Inter-Cardinal Points, sometimes 
 the Quadrantal Points. 
 
 A quadrantal point takes the name of the two cardinal points 
 between which it lies. Thus, the point midway between the north 
 and east, is called the N. E. point, and so of the other quadrants. 
 
 In the same manner, the point midway betw^een the cardinal 
 point and a quadrantal point, is indicated by calling the letters 
 between which it lies, — as N., N. E., indicates the point midway 
 between the north and the northeast, and so for the other octants. 
 
 The remaining sixteen points are found adjacent to the cardinal 
 letters and the quadrantal letters, one on each side of each letter, 
 the adjacency being indicated by the word *^ by." 
 
 In defining one of those points, the reference letter is first 
 named, then the point immediately to the right or left of the 
 reference letter, as the case may be. Thus, the first point imme- 
 diately to the right of east, is called E. by S.; that to the left of 
 north, N. by W., etc. Thus, the plan of naming the points being 
 understood, the name of any point of the compass is easily 
 brought to mind. 
 
 The naming of the points of the compass in consecutive order 
 around the card, is called "Boxing the Compass," and the stu- 
 dent should make himself so familiar wath it that he can name 
 them readily in either direction. 
 
 Reading the Compass Card by Numerals. — In giving the 
 course on which the ship is to be steered, the officer of the deck 
 usually refers the same to the nearest cardinal or quadrantal letter. 
 But in making computations for bearings, or for reduction of 
 course, or the finding of compass errors, reference is always made 
 to the nearest meridian letter, N. or S., and the bearing is reck- 
 vpned in points and parts of a point, or in degrees, toward the east 
 or west, as the case may be. 
 
 This is made necessary by the fact that all trigonometrical tables 
 used in calculating courses, are arranged in that manner. 
 
 The student must therefore make himself familiar with reading 
 the card in that manner. Thus, 
 
 E. by N, would be read, N. 1 p. E. 
 
 N. E. '' '^ N. 4 p. E. 
 
 E. JN. " '^ ^r. 7f p. E.,etc. 
 
24 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 The student should also be able to change a course given by its 
 numerical value to its cardinal name. Thus, 
 
 N. 3 p. E. would be called X, E. by X\ 
 
 s. 6 p. ^y. '^ '' w. s. \v. 
 
 N. 5p. W. '' '' X. W, byW. 
 
 X. 45° E. '' '' X. E. 
 
 S. 33f° E. '' '' S. E. by S., etc. 
 
 Table III will facilitate this reduction of courses, and it is 
 important that the student make himself expert in these reduc- 
 tions. Some compass cards are graduated both in degrees, and in 
 points and \ points. Such a card is then, in itself, a table for 
 the reduction of course from one denomination to the other. 
 
 The Use of the Compass, is to give the bearing of a line, or 
 to point out the direction in which a ship is to be steered. All 
 courses are ultimately referred to the astronomical meridian, when 
 they are called True or Astronomical Courses, sometimes they 
 are called Chart Courses. 
 
 But the compass seldom points to the true north. In most 
 places it is turned away from that direction by the earth's mag- 
 netism, more or less. 
 
 It is also disturbed from the position which it assumes under 
 the earth's magnetism, by the magnetism of the iron in the ship. 
 This is the most serious disturbance that can come to the compass 
 needle, — for as the ship heads to different courses, the needle also 
 takes different directions, — so that the change in the readings of 
 the card is no indication of the change made in the course of the 
 ship. Yet, at sea, in dark weather, the navigator is obliged to 
 refer his course to the disturbed needle, — whence it is of the first 
 importance that the navigator know the error of his compass. 
 
 The Ag'OUic Liue, or line of no variation, is a line on which 
 the magnetic needle, when not disturbed by local causes, points 
 to the true north. 
 
 This line, in the United States, commencing in South Carolina, 
 takes a course a little to the west of north, crossing the head of 
 Lake Erie near the mouth of the Detroit Kiver, thence through 
 the eastern part of the State of Michigan, crossing Lake Huron a 
 little west of Mackinac Island, thence across the eastern part of 
 Lake Superior to a little east of the Copper Islands. 
 
THE COMPASS. 25 
 
 From this chart it will be seen that east of the agonic line, the 
 north end of the needle stands to the left, or west of the astron- 
 omical north; and to the west of this line, the north end of the 
 needle points to the east of the true meridian. Thus, in the Gulf 
 of St. Lawrence, the needle points 20° to the left of north, while 
 on the Pacific coast, near the mouth of Columbia Kiver, the north 
 end of the needle stands to the right of the true north, 22°. 
 Thus, in crossing the continent, the north end of the needle 
 swings toward the east nearly four points. 
 
 Preparatory to finding the error of the needle, Ave give a few 
 definitions: 
 
 Astronomical Meridian. — This is the north and south line 
 given on all charts, and to which all courses are eventually 
 referred. The meridian of any place is in the plane containing 
 the earth's axis, and that place. 
 
 Mag'netic Meridian. — This is the line pointed out by the mag- 
 netic needle, when acted upon by the earth's magnetism alone. 
 
 Compass Meridian. — This is ;he line pointed out by the mag- 
 netic needle, acting under the combined influence of the earth's 
 magnetism and that of the iron in the ship. 
 
 Variation, is the difference between the directions of the 
 astronomical meridian and the magnetic meridian. The north end 
 of the needle may stand either to the right (east) or to the left 
 (west) of the true meridian, in which case the variation is, 
 called east or Avest, as the case may be. Easterly variation is con- 
 sidered + (plus) and westerly — (minus), and is usually given in 
 degrees. 
 
 Deviation, is the difference between the directions pointed 
 out by the magnetic and the compass meridians. The compass, 
 when aboard ship, may be disturbed by the ship's magnetism, so 
 as to take position to the right or left of the magnetic meridian, 
 precisely as the magnetic meridian takes position to the right or 
 left of the true meridian. Whence, 
 
 Deviation is east or west, + or — ? precisely as in variation 
 
 Variation and deviation are thus seen to be strictly analogous,' 
 but with this difference, viz.: Variation is constant for all head- 
 ings of ship, but deviation is different for all headings of ship. 
 
 Three Kinds of Bearing. — Bearing being the direction of an 
 object or place, with regard to some line of reference, as a mer- 
 
26 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 idian line, it follows from the above definition thai we have three 
 kinds of bearing, viz.: 
 
 Astronomical, or True Bearing*, which is the direction of an 
 object, with regard to the true meridian. 
 
 Magnetic Bearing is the direction of an object or place, with 
 regard to the magnetic meridian; and 
 
 Compass Bearing" is the direction of an object or place, as 
 given with the deviated needle. Whence, we have at sea: 
 
 Variation and Deviation combined into one error called Cor- 
 rection or Total Variation, and which is the algebraic sum of 
 variation and deviation, for all compasses at sea are under the 
 influence of both, — the earth's magnetism and that of the ship. 
 
 That part of compass error due to variation, is found at the 
 time of making a survey of the coast, or of the lake, or of any 
 locality, and recorded in the charts of the same; and is regarded, 
 for the time, as constant, though it changes slowly by the slow 
 change in the ' 'agonic" line. Thus, in 1840, Prof. Elias Loomis 
 informed us, in the American Journal of Science, that the agonic 
 line, commencing at a point near Wilmington, N. C, went north- 
 westerly, crossing Lake Erie east of Cleveland; thence, through 
 the middle of Lake Huron and entirely east of Lake Superior. 
 The variation in 1890, or rather the agonic line, was from 1J° to 
 2° to the east of what it was in 1840, i. e., places that then had 
 variation, have variation of 1J° to 2° west, now. As a conse- 
 quence, the variation that was given on charts that were made 
 from early surveys, are in error, according to the age of their 
 surveys — say 1° to 1J° — westerly variation increasing and easterly 
 variation decreasing. 
 
 But not so with that part of compass error due to deviation. 
 This must be found for the compass of each individual ship, and 
 for compasses in different parts of the same ship. Nor can we tell 
 from the behavior of a compass in one ship, what may be looked 
 for in the compass of another ship. 
 
 Azimutli, is the angle or course by which an object is referred 
 to a meridian. In geodesy, it is usually reckoned from the south 
 part of the meridian (in north latitude), and from to the right, 
 by the west to the north, and east 360° back to the south. In 
 navigation and surveying, it is reckoned from both parts of the 
 meridian, 90° each way to the east and west. 
 
THE COMPASS. 27 
 
 Amplitude, is the bearing of an object when referred to the 
 east or west point of the compass. 
 
 Azimuth and Amplitude are compliments each to the other, i. 
 €., each is what the other lacks of eight points, or 90°. 
 
 Swinging Ship for Compass Errors. — That part of the com- 
 pass error due to deviation, is found by "swinging ship,'' and 
 <jomparing the reading of the ship's compass with the simultaneous 
 readings of a "shore compass" on successive headings, as the ship 
 turns around in azimuth. 
 
 The "shore compass" is the ordinary compass used by surveyors, 
 but sent on shore to get it away from the influence of the ship's 
 magnetism to Avhere it is acted on only by the earth's magnetism. 
 The line between the ship's compass and the "shore" compass is 
 a reference line. The bearing of it, for successive headings of 
 ship, compared with the bearing given by the "shore" compass, 
 gives the error (deviation) of the ship's compass on the several 
 headings. 
 
 Standard Compass. — The above method of finding the error 
 of ship's compass, involves the use of some appliance on board the 
 ship for measuring the angle between the ship's heading and the 
 Teference line. 
 
 At sea, this appliance is found in the movable ring and movable 
 sights attached to the standard compass, which is placed on the 
 open deck, and high enough so that bearings can be taken over 
 the rail, even when the ship is heeled over a few degrees. 
 
 But on our lake vessels there are no such appliances. The com- 
 passes are not only not provided with the ring and sights, but 
 they are boxed up in the pilot house where such things could not 
 he of any avail if we had them. As a consequence, our lake ves- 
 sels, as a class, are utterly without the means of finding their 
 compass error. 
 
 The Dumb Compass has been prepared to meet this deficiency. 
 It is the movable ring and sights of the standard compass, as used 
 at sea, but in another form. It is not necessarily expensive, — a 
 good one may be made by the ship's carpenter, — though better by 
 the instrument maker. 
 
 The essential feature of the dumb compass is that it may be 
 read by two indexes. It has no needle. The card is graduated 
 like that of the ship's compass. 
 
28 A MANUAL OF NAVIGATION FOR THE I.AKES. 
 
 The lower index corresponds precisely to the ^^lubber's" mark 
 of the ordinary compass bowl, — the line joining it with the center 
 of the compass to be fixed parallel with the center line of ship, as 
 is that of ship's compass. 
 
 The plate on which the card is fixed is movable to any position 
 about a vertical spindle, and may be clamped in any position. 
 
 The sights are on a bar that turns in azimuth about the same 
 spindle as that carrying the card, and which can be clamped to 
 the card in any position. 
 
 It should be mounted on a tripod and be provided with a socket 
 joint, and a level vial should be fixed in the plate carrying the 
 card, for adjusting the card to a horozontal position; and it should 
 be set up high enough so that a clear view can be had from it of 
 the whole horizon. 
 
 (XoTE. — The ideal instrument would be on ^^ gimbals,' ' but 
 this is not necessary, or even important, for, as the instrument is 
 not used for a steering compass, but merely as an auxilliary com- 
 pass, it is quite sufiicient if it be mounted on a tripod provided 
 with a ball and socket joint, and with cross levels in the card). 
 
 The Use of the Dumb Compass, is to find the angle between 
 the heading of the ship and any line that is used as a reference 
 line in finding compass errors. Thus, in port, w^hen it is desired 
 to find compass errors by reference to a line of known magnetic 
 bearing, the process is as follows: 
 
 1. Send the shore compas-s ashore and set it up at any con- 
 venient point where it is away from any local disturbance of the 
 needle, and take the bearing to the ''dumb compass" on board 
 ship. 
 
 2. The dumb compass being set up on deck in a suitable place 
 for observation, the zero line and lubber line being parallel with 
 ship's center line, and '' looking forward," the same as the ship's 
 compass. 
 
 3. Set the upper index, or sights, to the same reading as that 
 given by the shore compass, the north side of the card being to 
 the north. 
 
 4. The card and sights being clamped together, turn them 
 together till the sights " back. sight " on the shore compass. Then 
 is the card of the dumb compass oriented to the magnetic mer- 
 idian. The reading of it by the lower index is the magnetic head- 
 
THE COMPASS. 29 
 
 ing of the ship. And the difference between the magnetic head- 
 ing and '^compass" heading is the deviation, if any, of the ship's 
 compass, for that heading. This is called the method by Recip- 
 rocal BeariugS, and is used in harbors or other places where a 
 distant azimuth mark cannot be found. 
 
 Finding Compass Errors. — The illustration of the use of the 
 dumb compass in the preceding article, gives the method of finding 
 compass errors in port, but the office of that compass is the same 
 for all methods of finding compass errors, viz., to refer ship's 
 head to the reference line, — which may be any line of known 
 bearing, — the results only being different, as when the difference 
 line is known by its magnetic bearing or by its astronomical 
 bearing. 
 
 Direct Method, hj Reference to a Distant Object. — In this 
 case, the work of finding compass errors is somewhat simplified. 
 The azimuth mark may be a distant building, headland or shore 
 line, but it must be so distant that the change in the place of the 
 compass, in swinging ship, will not cause any appreciable parallax, 
 — say two or three miles, or not less than about 100 times the 
 diameter of the circle made by the compass in swinging ship^ 
 
 The bearing of this line is found as in the preceding article, or 
 it may be found from a chart, but when found, the subsequent 
 work is that of the former case, except there are no reciprocal 
 bearings to make with the shore compass, — whence, it is called 
 the direct method. 
 
 Method of Recording Obseryations of Compass Errors and 
 deriving therefrom the Deviation. — The method of recording 
 the observations for finding deviation of our compasses, is some- 
 what different from that at sea, for the reason that our compasses 
 are differently mounted and differently equipped from those at sea. 
 
 There, the bearing of an azimuth mark or reference line can be 
 directly compared with the bearing of the same line, as indicated 
 by the ship's compass, and on the same instrument. But our com- 
 passes, as before seen, are so arranged that the bearings of objects 
 cannot be taken with them, they can only show the apparent bear- 
 ing of ship's head, hence the necessity of a different form of record. 
 
 Example: The table on the following page, gives the observa- 
 tions of the Steamer Huron, swung for deviation, at South Haven, 
 by the author, June 26, 1876. (See form I). The first column 
 gives the bearings of ship's head, as indicated by her compass. 
 
30 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 The second column gives the bearings of ship's head, as indicated 
 by the ^ 'shore compass" (or rather by solar transit with index 
 correcting for variation). The third column gives the difference 
 of these two columns, which is the deviation for the respective 
 bearings or headings of ship. 
 
 It will be seen by referring to columns 1 and 2, form I, that the 
 first course in the second quadrant of the first column, corresponds 
 to the last course of the first quadrant in the second column. Now, 
 in finding their difference, we must reckon both courses from the 
 same zero point, — no matter which — say the north, in this case. 
 Then, instead of S. 6 p. E., we would have X. 10 p. E., i. e., v/e 
 must take the supplement of the course. Then we can find the 
 difference of the two bearings, 2h p. 
 
 
 I. 
 
 
 II. 
 
 
 
 
 Deviations of 
 
 D 
 
 EVIATIONS OF 
 
 
 
 Steamer Huron. 
 
 Steamer Huron. 
 
 
 
 Head by 
 Ship's 
 
 Head by 
 
 ^^Shore" 
 
 Devia- 
 tions. 
 
 Head by 
 Ship's 
 
 Head bv 
 ^SShore'' 
 
 Devia- 
 tion o 
 
 
 Compass. 
 
 Compass. 
 
 Compass. 
 
 Compass. 
 
 LiLi 
 
 ' 
 
 
 Xorth. 
 
 N. fp.E. 
 
 fp.E. 
 
 
 
 f 
 
 ~Tv 
 
 . E. 
 
 1 
 
 X.lip.E. 
 
 N.H E. 
 
 i W. 
 
 H 
 
 li 
 
 i 
 
 W. 
 
 X.2| E. 
 
 N.ll E. 
 
 i w. 
 
 2f 
 
 H 
 
 * 
 
 W. 
 
 
 N.4i E. 
 
 X.2f E. 
 
 n w. 
 
 4i 
 
 2| 
 
 u 
 
 W. 
 
 
 N.5 E. 
 
 X.3} E, 
 
 If w. 
 
 5 
 
 3} 
 
 If 
 
 W. 
 
 
 N.6 E. 
 
 X.3I- E. 
 
 2J w. 
 
 6 
 
 3J 
 
 2* 
 
 W. 
 
 
 N.7| E. 
 
 ^M E. 
 
 2f W. 
 
 7i 
 
 oj 
 
 2f 
 
 W. 
 
 
 S.6 E. 
 S. 5 E. 
 
 N.7J E. 
 
 21 \y. 
 
 2i W. 
 
 10 
 11 
 
 8f 
 
 2J 
 
 2i 
 
 W- 
 
 
 S.7J E. 
 
 W. 
 
 2 
 
 S. 3f E. 
 
 S.5J E. 
 
 2f W. 
 
 12i 
 
 lOJ 
 
 2i 
 
 w. 
 
 
 S.3 E. 
 
 S.4X E. 
 
 H w. 
 
 13 
 
 11f 
 
 li 
 
 w. 
 
 
 S.2 E. 
 
 S.3J E. 
 
 IJ w. 
 
 14 
 
 121 
 
 IJ 
 
 w. 
 
 
 S.l E. 
 
 S.2 E. 
 
 i w. 
 
 15 
 161 
 
 14 
 
 16 
 
 1 
 
 w. 
 
 
 S. i W. 
 
 South. 
 
 i w. 
 
 w. 
 
 
 S. i W. 
 S.lf w. 
 
 S. f w. 
 s.lf w. 
 
 
 J E. 
 
 16f 
 
 16f 
 
 
 
 
 
 I'f 
 
 l"f 
 
 i 
 
 E. 
 
 3 
 
 s.2 J W. 
 
 s.3 J W. 
 
 1 E. 
 
 18J 
 
 m 
 
 f 
 
 E. 
 
 
 S. 3 W. 
 
 S.3i W. 
 
 1 E. 
 
 19 
 
 191- 
 
 |- 
 
 E. 
 
 
 S.3t W. 
 
 S.4| W. 
 
 U E. 
 
 19| 
 
 20f 
 
 1f 
 
 E. 
 
 
 S.41 W. 
 
 S.5J W. 
 
 If E. 
 
 201 
 
 211- 
 
 If 
 
 E. 
 
 
 S.5^ W. 
 
 S.6J ^y. 
 
 S.6J W. 
 
 If E. 
 If E. 
 
 21J 
 22J 
 
 221 
 241 
 
 1^ 
 If 
 
 E. 
 
 
 N.7f W. 
 
 E. 
 
 
 S.7f w. 
 
 N.6| W. 
 N.4| W. 
 
 H E. 
 2J E. 
 
 23f 
 
 25J 
 
 25f 
 271 
 
 1* 
 
 28 
 
 E. 
 
 
 N,6- W. 
 
 E. 
 
 4 
 
 N.3i W. 
 
 N.IJ W. 
 
 If E. 
 
 28f 
 
 301 
 
 If 
 
 E. 
 
 
 N.lf W. 
 
 N. ^ W. 
 
 1* E. 
 
 30f 
 
 31J 
 
 u 
 
 E. 
 
THE COMPASS. 31 
 
 And so in the third and fourth quadrants. 
 
 Writing the Courses by Azimuth. — This method is given in 
 form II of the preceding table, which consists in writing them 
 continuously in points, from zero at the north, around by the east, 
 etc., to 32 points. 
 
 This method is very simple, when the card is graduated con- 
 tinuously around the circle, and greatly facilitates the finding of 
 the difference of the courses given by the two compasses. On that 
 account, it is desirable to reduce the courses, as ordinarily given, 
 to azimuth readings, as follows: 
 
 1. In the first quadrant, i. e., from the north to the east, 
 write the courses as they are given from the card, — the courses 
 being given by their numerals. 
 
 2. In the second quadrant, E. to S., write the supplement of 
 the compass reading. 
 
 3. In the third quadrant, S. to W., add 16 p. to the compass 
 reading. 
 
 4. Add 24 p. to the compliment of the compass reading, in 
 the fourth quadrant. 
 
 Thus, the azimuth reading for S. 2 p. E. is 14 p.; for S. 2 p. 
 W., it is 18 p.; for N. 7 p. W., it is 25 p. 
 
 We remind the student that supplement is what the course 
 lacks of 16 p. or 180°, and compliment is what it lacks of 8 p. or 
 90°. 
 
 Naming the Deviations, will require a little careful attention, 
 from the fact that each two adjacent quadrants are read in opposite 
 directions; which may cause the young student to stumble in this 
 matter, if he is not on his guard. 
 
 He should first go over the table and write the difference of the 
 courses given in the first and second columns, without regard to 
 name or sign in the third column. Then consider which way 
 (right or left) the card of the ''shore" compass must be turned to 
 make its readings agree with those of the ship's compass. That 
 direction is the name to be applied to the differences for the 
 deviations. 
 
 Thus, in the first quadrant, the readings of the ''shore" com- 
 pass for any given course, are less than those of the ship's compass 
 for the same course, and if we turn the card of the "shore" compass 
 to the leftjthe two readings may be made to agree, — i. e., the devi- 
 ation is west by the amount of the difference of the readings. 
 
32 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Again, in the second quadrant, the reading of the *^shore^' 
 compass for any particular course, is greater than that of the 
 ship's compass, and yet the card of the * 'shore" compass must be 
 turned to the left to make the readings of the two compasses 
 agree, — so that there, also, we have westerly deviation. 
 
 The same results will be found between the third and fourth 
 quadrants. This comes from reading the alternate quadrants in 
 opposite directions. 
 
 Deviation Curve. — We are told by the books, to take the devi- 
 ations on the consecutive points, when swinging ship. But this, 
 while possible, is at times totally impracticable, and we take such 
 as we can get. Thus, it will be seen that the headings given of 
 the Huron, in the preceding table, are very irregular as to their 
 intervals. Still, it is important to have the deviation for regular 
 intervals, as by points, so that the deviation for intermediate 
 courses, as for parts of a point, may be interpolated for steering 
 purposes. 
 
 Many schemes have been devised for the attainment of this 
 object, — and mostly of the graphical type. Among the best of 
 these methods, are those of J. E,. Napier, Esq., F. R. S., and 
 Archibald Smith, M. A., of the Liverpool Compass Committee. 
 (See Admiralty Manual, for 1874). 
 
 Some sixteen years ago (1891 now), the author of this Manual 
 devised a scheme for the representation of deviation and the con- 
 version of compass courses, which on being submitted to Prof. 
 J. H. C. Coffin and Commodore A. W. McCormick, U. S. X., is 
 endorsed and recommended by both of these eminent men, as 
 being a decided improvement on that of Mr. Napier, accordingly 
 I give the method of constructing it and of using it. 
 
 Pearson's Diagram, was invented for the representation of 
 deviation, and for the conversion of compass errors. 
 
 This diagram (see plate I) shows two scales, — one vertical and 
 one horizontal. 
 
 The vertical scale represents azimuth, and is to be read down- 
 wards, — such reading corresponding to the reading of the compass 
 card from the left to the right, as we read a watch dial. It is 
 simply the circumference of the card, developed by rolling it down 
 the page and marking the points. 
 
THE COMPASS. 33 
 
 This scale may be read in points and parts, or in degrees, as 
 desired, — the horozontal lines across the page being the gradua- 
 tion marks. 
 
 The horizontal scale, — or rather the two scales, one at the 
 head, the other at the foot of the page, — is for setting off varia- 
 tion and deviation. 
 
 The thick line drawn vertically through the middle of the page, 
 represents the magnetic meridian, to which deviation is referred. 
 
 The thin lines drawn obliquely across the page, from the left, 
 upward and to the right, are for the purpose of performing addi- 
 tion and subtraction graphically, of variation and deviation. 
 
 A thin line drawn vertically through the page, parallel with 
 the magnetic meridian, and cutting the variation scale at any 
 point, is called the local meridian, corresponding to that 
 variation. 
 
 Any two convenient scales may be taken, at pleasure, for the 
 variation and the azimuth scales, — the condition being that the 
 oblique lines for performing addition and subtraction, pass 
 through points in each scale, having the same numerical value. 
 Thus, if an oblique line pass through the azimuth line at, say N. 
 1 p. E., then it must also pass through the variation or horizontal 
 scale at 11^° from the zero of that scale, etc. 
 
 It is observed that easterly variation is placed to the left of the 
 magnetic meridian, and westerly variation to the right. This is 
 for the purpose of locating the magnetic meridian to the right of 
 the true meridian for easterly variation, and to the left of the true 
 meridian for westerly variation, as it should be. 
 
 But, deviation being referred to the magnetic meridian, must 
 be set off from that line, according to its name, — to the right for 
 easterly and to the left for westerly deviation. 
 
 Construction of the Deviation Curve. — The deviations having 
 been satisfactorily found,- as on page 29, set them off to the right 
 or left of the magnetic meridian, according to their name (see plate 
 II). Thus, when ship's compass needle indicates north, it is known 
 to be out of place to the right by f p. Take f p. in the dividers 
 from the variation scale at head of page, and set it off to the right 
 on the line of north, from the magnetic meridian, and make a dot 
 surrounded with a small circle, thus ©. 
 
 Again, when ship's head is IJ p. E. of N., the needle is out J p. 
 to the left. As before, take J p. in the dividers and set it off 
 
34 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 on azimuth, IST. IJ- p. E. from the magnetic meridian, making the 
 mark (©) as before. 
 
 In this manner, set off all the deviations, according to their 
 name, — to the right for easterly deviations and to the left for 
 westerly deviations, on the horizontal lines corresponding to their 
 azimuth, and from the magnetic meridian as zero point. 
 
 Through the points set off, draw a fair freehand curve, giving 
 and taking a little at times to make the curve fair, and you have 
 the desired deviation curve. 
 
 It will be observed that about half of the curve is on either side 
 of the magnetic meridian. On this account, this deviation is 
 called semi-circular. 
 
 It will be observed, too, that the two parts are about equal, as 
 the curve crosses the meridian when ship heads N. by E., also 
 when it heads S. by W., but this is not always the case. And when 
 they are not equal, it is not always that good reversals can be had 
 in compensating the compass for deviation, as will be seen here- 
 after. 
 
 Use of the Deviation Curve, Reduction of Courses. — The 
 
 deviation curve being constructed, we are prepared to solve many 
 questions that vex the ship-master who is compelled to work with 
 a deviated compass. A few examples will illustrate: 
 
 Example 1. At Grand Haven, Mich., mean var. 3° E. It is 
 desired to sail to Milwaukee, with Steamer Huron. Eequired 
 steering course for ship's compass, also the return steering course. 
 
 Solution: First, draw a line through var. 3° E., to represent 
 the true or chart meridian. On this line, take up the chart course 
 required to be made, viz., W. J S., as at a, aud project it onto the 
 curve at b, at W. by S. I S., which is the compass course desired^ 
 — Ans. S. 6J p. W. 
 
 To find the return course: Reverse the given chart course, W. 
 J S., to E. J N., and take it up on the true meridian as before, as 
 at c, and project the same onto the curve at d, when will be found 
 the required compass course for the return, viz., E. S. E. J S., or 
 S. 5 Jp. E. 
 
 Example 2. On the same diagram will be seen the deviation 
 curve of the U, vS. S. Monadnoc. Required the outward and the 
 return steering courses for the compass of this vessel for the same 
 voyage. 
 
THE COMPASS. 35 
 
 Solution: As with Example 1, take up the chart course, W. i 
 S., on the true meridian, as at a, and project the same onto the 
 curve of the Monadnoc, as seen at b^, in azimuth W. J N.^N. 7J 
 p. AV. 
 
 For the return course. Reverse the chart course, W. J S., to 
 E. J X., and as before, project the same onto the curve, as at c^, 
 where we find the required return course, viz., E. by X. f X., or 
 N. 6i p. E. 
 
 Discussion of the Preceding Problem. — It will be remem- 
 bered that w^hen the card is turned to the right of its normal 
 place, the readings for the bearing of a given object are too small, 
 when regarded as azimuth (as all courses are eventually regarded), 
 and consequently we must take up a course smaller than the 
 compass would indicate by the amount of disturbance, which 
 deduction is made by moving up the page of the diagram; other- 
 wise it would take us too far to right. 
 
 For the same reason, when the card is out of place to left, its 
 readings for the bearing of an object are tOO large, and without 
 reduction, the card would take us too far to left. So that in this 
 case we must increase the readings by the card by the amount of 
 disturbance, to attain any given course. This is attained by mov- 
 ing down the page. 
 
 Tariation, is seen to be the distance between the magnetic 
 meridian and the true meridian. 
 
 Deviation, is the distance between the magnetic meridian and 
 the curve, at any point measured on the horizontal line passing 
 through the point, and by the variation scale. 
 
 " Correction," or Total Error, is the distance between the 
 true meridian and the curve. 
 
 These quantities always come together algebraically, westerly 
 deviation and variation being called minus ( — ), and easterly are 
 called plus ( + ). 
 
 Correction for Leeway, can also be made by means of this 
 diagram. 
 
 Example 3. At Grand Haven, Mich., mean var. 3° E. Required 
 the outward and the return compass courses for each of the 
 steamers Huron and Monadnoc, taking into consideration a port- 
 hand leeway of | p. — the chart course to Chicago being S. W. J S. 
 
 Solution: Take up chart course, S. W. J S., on the true mer- 
 idian (3° to left of magnetic meridian) and move down the page 
 
36 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 I p. to compensate leeway. Then project the course thus found, 
 viz., S. W. i W., onto the curve at e, when we find S. W. by S. 
 for the Huron; and projecting onto the curve of the Monadnoc, 
 we find S. W. J W. for that vessel's compass. 
 
 Keturning, with the same wind, our leeway will be to starboard, 
 then moving up the page f p. to N. E. f N. (the course having 
 been reversed) and going to the curves, we find for the Huron, 
 N. E. f E., and for the Monadnoc, N. N. E. J E., for their res- 
 pective compass courses for the return voyage. 
 
 Example 4. At Duluth; var. 8° E. ; wish to make a chart course 
 N. E. by E. J E., and looking for a port-hand leeway of J p. 
 What is the compass course of the Huron and for the Monadnoc, 
 as deduced from their respective deviations? 
 
 Solution: On a true meridian drawn through var. 8° E., take 
 up the desired chart course and move down the page J p., to 
 compensate the port leeway. Project the point thus found, onto 
 the curve for each vessel, and we find for the Huron, E. J X. at 
 g; and for the Monadnoc, N. E. f E. at g^, as their respective 
 steering courses. 
 
 Deviations of Steamer Huron. 
 
 Head by 
 
 
 Head by 
 
 
 Ship's 
 
 Deviation. 
 
 Ship's 
 
 Deviation. 
 
 Compass. 
 
 
 Compass. 
 
 
 North. 
 
 7FE. 
 
 South. 
 
 5 ° W. 
 
 N. by E. 
 
 
 
 s. by ^y. 
 
 
 
 N. N. E. 
 
 6 W. 
 
 s. s. w. 
 
 5J E. 
 
 N. E. by N. 
 
 11} W. 
 
 S. W. by S. 
 
 11 E. 
 
 N. E. 
 
 17 W. 
 
 S. W. 
 
 13i E. 
 
 N. E. by E. 
 
 21 W. 
 
 S.W.bvW. 
 
 14 E. 
 
 E. N. E. 
 
 24 W. 
 
 W. S. W. 
 
 18} E. 
 
 E. by N. 
 
 26* W. 
 
 W. by S. 
 
 21 E. 
 
 East. 
 
 28^ AV. 
 
 West. 
 
 23 E. 
 
 E. by S. 
 
 28-1 ^y. 
 
 W. bv X. 
 
 24J E. 
 
 E. S. E. 
 
 28" W. 
 
 W. N. W. 
 
 24J E. 
 
 S. E. by E. 
 
 27 W. 
 
 X.W.byW. 
 
 24 E. 
 
 S. E. 
 
 25 W. 
 
 X. W. 
 
 22 E. 
 
 S. E. bv S. 
 
 21J W. 
 
 X.W.byN. 
 
 19^ E. 
 
 s. s. e: 
 
 18 W. 
 
 X. X. W. 
 
 15 E. 
 
 S. by E. 
 
 12 W. 
 
 X. bv W. 
 
 lU E. 
 
 South. 
 
 5 W. 
 
 Xorth. 
 
 6" E. 
 
 It is believed the above illustrations will fully explain the oper- 
 ation of reducing compass courses. 
 
THE COMPASS. 37 
 
 Another use of the curve, besides the reduction of courses, is 
 found in the facility with which it gives the Deviation on Con- 
 secntive Points of the card when observations have been made 
 at irregular intervals, thus making the deviations available for a 
 steering card. To do this, we have only to measure the deviations 
 on each point, by taking the distance between the curve and the 
 magnetic meridian, with a pair of dividers, refer them to the vari- 
 ation scale at the head of the page, and tabulate the results. 
 Thus, we find at north, the deviation curve to the right of the 
 magnetic meridian. Applying the distance to the scale at the 
 head of the diagram, we find it 7J°, which we write in the devia- 
 tion column, with its name, E. 
 
 Shaping the Course. — The four questions just discussed con- 
 stitute the problem of ''shaping the course," and assumes the 
 following form: With a compass that is deviated, what course 
 shall be taken to attain a given chart course ? 
 
 Rule, using the diagram: Project the given chart course from 
 the true meridian onto the curve. At the intersection, is found 
 the compass course desired, to attain the required chart course. 
 
 To Find the Course ''made good," is to find what chart 
 course has been actually attained, by sailing with a compass that 
 is deviated. 
 
 Knle. Project the compass course from the curve onto the 
 true meridian. At the intersection will be found the equivalent 
 chart course. 
 
 This problem, which is seen to be the reverse of the first ques- 
 tion, is wanted at sea, in making up the ''day's work" for finding 
 place of ship, but is seldom or never wanted on the lakes, for the 
 reason that the place of ship, when outside, is rarely, if ever, 
 looked for, as it is looked for at sea. 
 
 The runs being short, — only a few hours at most — points, as 
 lighthouses, are seen so frequently, and their co-ordinates of place 
 being known from ihe list of lights, there is no necessity for 
 "reducing a traverse," as at sea, to find place of ship, and no 
 necessity for finding the "course made good." 
 
 The Log-Line and Time-Glass. — These are also essential parts 
 of the equipment for keeping account of the place of ship at sea. 
 
 The log-line is adapted to the sea-mile of 6080 feet, with a time- 
 glass of 28 seconds. 
 
38 A MANUAI. OF NAVIGATION FOR THE LAKES. 
 
 Theoretically, the time-glass was supposed to run 30 seconds 
 and the sea-mile to be 6087 feet, but as the circle of the equator is 
 larger than any other great circle of the earth, vessels always 
 found themselves ahead of their reckoning, by using the equa- 
 torial mile as the unit of distance. 
 
 To correct this inconvenience, the time-glass has been reduced 
 to 28 seconds, and the sea-mile has been reduced to 6080 feet, 
 which corresponds now nearly with the minute of a circle on the 
 mean diameter of the earth. 
 
 These values are now adopted by nearly all maritime people. 
 Then taking the same part of 6080 feet, that 28 seconds are of one 
 hour, the length of the ^'knot" on the log-line would be 47 feet. 
 Some navigators make it more or less, accordingly as they think 
 their ship over-runs or under-runs her reckoning. 
 
 The U. S. navy make the length of the log-line 45 feet, for a 
 28 second glass. 
 
 On and about the great lakes, the statute mile of 5280 feet is 
 regarded as the unit of measure for distance. Accordingly, for a 
 28 second glass, the knot on the log-line should be 40 feet, or 43 
 feet for a 30 second glass. 
 
 If the patent log, adapted to the sea-mile be used, its readings 
 may be reduced to indicate statute miles, by multiplying the indi- 
 cated distance by 1.15, which is adding 15% to the indicated 
 distance. 
 
 As the modern patent log depends on the *'pitch^' of its screw 
 to give it the right number of turns for indicating the distance, 
 the labor of getting the right pitch is somewhat tedious, it being a 
 tentative operation. 
 
 But it is not necessary that it should show the number of turns 
 for the correct distance. It is quite enough to know the number 
 of turns made for a known distance. 
 
 Suppose, for a distance known to be 76 miles, the dial indicates 
 80 miles, — that is, it indicates 80-76ths of the distance. Invert 
 this fraction and make 76-80ths of 80 miles, for the correct dis- 
 tance, i. e., multiply the indicated distance by the reciprocal of 
 the screw's rate. 
 
 Thus, suppose the log shows only 13 miles when it should show 
 15 miles. Then, inverting the ratio 13-15ths to 15-13ths and 
 multiplying by 15, we get the correct distance. Whence, 15-13ths 
 
THE COMPASS. 39 
 
 or 1.154 is the factor or co-efficient by which to multiply all indi- 
 cations of this log. This is called finding the co -efficient of 
 the log. 
 
 Distance by Propeller Wheel. — Another method of measur- 
 ing the ship's rate of sailing, is that by means of the propeller 
 wheel. 
 
 The vessel, when in her usual trim, is run over a known distance, 
 noting the total time and the number of revolutions per minute. 
 
 The known distance being reduced to feet and divided by the 
 total number of revolutions in the given time, gives the ''net 
 pitch" of the wheel in feet, i. e., the distance made good by one 
 revolution. 
 
 Example: A steamer making with her wheel 106 revolutions 
 per minute, for 64 minutes, makes a known distance of 12.4 miles. 
 Bequired net pitch of wheel. 
 
 Solution. 12.4X5280 ^ ^^ , ^. . -, ^ -, , 
 
 —^j-^-rjr~^=z9 .QQ feet. Jset pitch of wheel. 
 
 The net pitch being known, and multiplied by the number of 
 turns per minute, and by 60, the number of minutes in an hour, 
 then divided by the number of feet in one mile, 5280, gives the 
 rate of ship in miles per hour, — or if he divide by 6080 feet, he 
 will have the rate in nautical miles per hour. 
 
 In this manner the ship master can readily log his ship for dif- 
 ferent speed of engine, and thus learn his rate of sailing to a good 
 degree of certainty, and better than by a log whose ''rate" is not 
 known. 
 
CHAPTEK III. 
 
 The Sailings. 
 
 Navigation implies the conducting of a ship from one port to 
 another, and in its broadest sense, implies the solution of many 
 problems involving a knowledge of geography, nautical astronomy 
 and mathematics, to a large extent. 
 
 The terms Plane Sailing, Traverse Sailing, Parallel Sailing, 
 Middle Latitude Sailing, Mercator's Sailing, Current Sailing, etc., 
 indicate different methods of finding place of ship, rather than 
 any peculiarity in the manner of sailing, as the name might seem 
 to imply. 
 
 The result of any of these methods is called the place of ship 
 by dead reckoning', or the method of finding place of ship from 
 the course, the distance, the rate, time sailed, and a knowledge of 
 the place sailed from. It is also called the place of ship by account. 
 
 Plane Sailing, is the method of finding place of ship with 
 regard to place sailed from, by means of the co-ordinates, course 
 and distance, from the properties of the plane triangle. 
 
 The method of plane sailing has been denounced as being inac- 
 urate, but when we recollect that the course or ^' rhumb line'' 
 makes a constant angle with the several meridians w^hich it crosses, 
 we see that distance sailed, the difference of latitude, and the 
 departure, are correctly represented by the hypothenuse and sides 
 of a right triangle, in which the course is the angle opposite the 
 departure, and the hypothenuse is the distance. We give a few 
 definitions: 
 
 Equator. — A plane through the center of the earth and at right 
 angles to the earth's axis, is called the plane of the equator; and 
 the intersection of this plane wdth the surface of the earth, is 
 called the equator. 
 
 Zero, or Prime Meridian. — Any plane through the earth, and 
 containing the earth's axis, is called a meridian. Any and every 
 
PLANE SAILING. 41 
 
 point on the surface of the earth has its meridian. The meridian 
 passing through the observatory of Greenwich, is now assumed as 
 the prime meridian, or meridian from which longitude is reckoned 
 by most maritime people. 
 
 Greo^raphic Latitude, is the distance in arc, measured on the 
 meridian of any point, from the equator, — or the angle between 
 the plane of the equator and a vertical or plumb line passing 
 through the point; but, 
 
 Geocentric Latitude, is the angle included between the plane 
 of the equator and a line joining any point on the surface of the 
 earth with the center of the earth. Geographic latitude is always 
 slightly greater than geocentric latitude, in consequence of the 
 unequal diameters of the earth, this excess amounting to about 
 llj minutes at lat. 45°. Geocentric latitude is wanted in finding 
 longitude from the place of the moon, but geographic latitude is 
 always used in giving the place of any point, with regard to the 
 equator. 
 
 Long'itude, is distance in arc on the equator, east or west. ..Its 
 use is to refer any point on the surface of the earth to the prime 
 or zero meridian. Latitude and longitude together, are called the 
 geographical co-ordinates of place. 
 
 Course, is the angle between the meridian and the track of ship, 
 and is usually represented by its initial, C. Distance, is the 
 length of line sailed; and course and distance together, are called 
 the polar co-ordinates of 2)lace, — the course referring to the mer- 
 idian, and the distance referring to the place sailed from. 
 
 But this is special. It concerns only the two points mentioned, 
 — the place of ship and the place sailed from, — whence, the method 
 by geographical co-ordinates, is general. It not only refers the 
 places to each other, but to the equator and the prime meridian, 
 and thence to any other point on the surface of the earth. 
 
 Whence, before we can compare the location of places as found 
 by ^'course and distance," with that as determined by latitude 
 and longitude, we must reduce our polar co-ordinates to geograph- 
 ical co-ordinates. This reduction is made by means of a Traverse 
 Table, which may be called a table of right triangles. 
 
 In sailing on any oblique course, we make two components, — 
 one with regard to distance on the meridian, and one with regard 
 to distance east and west, on the prime vertical. 
 
42 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 The component, north and south, is called the difference of 
 latitude; and that made on the east and west line, is called depart- 
 ure. Just what these components are for any course and distance, 
 is shown by table I. 
 
 This table is computed for courses varying by J point, and for 
 distances varying by unity up to 10. 
 
 By removing the decimal point one place to the right, in any 
 column, we multiply the sum by 10; moving two places, we mul- 
 tiply by 100, etc. 
 
 By this device, the table is made to give the distance likely to 
 be sailed at any one run, and to occupy but a small part of the 
 space it otherwise would. 
 
 The courses are given up to four points in the left hand column, 
 — increasing from at the head to four points at the foot. 
 
 On the right hand, they are given from four points at the foot, 
 to eight points at the head. 
 
 The components are abbreviated at the head of the columns, to 
 diff. lat. and dep., for the courses in the left hand column, and at 
 the foot of the column for the courses given in the right hand 
 column. 
 
 It will be observed that the column that is marked lat. at the 
 head, has dep, at the foot. 
 
 The Manner of Using the Table, will be seen from the fol- 
 lowing examples: 
 
 1. Sailed N. N. E. J E., or N. 2^ E., 57 mHes. Kequired the 
 components of the course, i. e., the northings and the eastings, — 
 or diff. lat. and dep. 
 
 Opposite 2 J p. in the left hand column and under 5 in the lat. 
 column, we find 4.52. Eemoving the decimal point one place to 
 the right, we have 45.2 miles as the northings for 50 miles. 
 
 Under 7 in the column of lat., we find 6.3, which added to the 
 
 45.2 miles, make 51.5 miles as the total northings for the run of 
 b1 miles. 
 
 In the same manner, we find under 5 in the dep. column, 2.14. 
 Tlemoving the decimal point one place to the right, and we have 
 21.4 miles as the dep. or eastings due to 50 miles. Under 7 in the 
 dep. column, we find 2.9 miles, which added to the 21.4, wc have 
 
 24.3 miles as the total dep. or eastings due to 57 miles on that 
 course. 
 
PLANE SAILING. 
 
 43 
 
 N. 
 
 S. 
 
 E. 
 
 W. 
 
 
 21.7 
 52.9 
 
 38.8 
 
 2.6 
 
 40.6 
 
 9.7 
 
 
 113.4 
 
 2.6 
 
 50.3 
 
 2.6 
 
 47.7 
 
 2 Kequired the diff. lat. and dep. for the following courses 
 and distances, viz.: 
 
 1. S. W. by W.=S. 5 p. W. 46 miles, 
 
 2. S. iE. =S. ip. E. 53 '' 
 
 3. S.byW.JW.=S.lJp.W.40 '' 
 Ans. Diff. lat. or southings=113.4 miles. 
 
 Dep. or westings = 47.7 '^ 
 
 Let the student prepare a table with a column for each of the 
 components and a line for each of the courses, as in the example. 
 
 Having taken the components for each of the several courses 
 from the traverse table and arranged them in their respective 
 columns, add the numbers in each of the several columns, 
 writing their sum at the foot of the column. 
 
 Then the difference latitude will be the algebraic sum of the 
 northings and southings, and 
 
 The departure will be the algebraic sum of the eastings and 
 westings, whence. 
 
 Take the difference of the northings and southings for the dif- 
 ference of latitude, and 
 
 Take the difference of the eastings and westings for departure. 
 
 Thus, in the example, the sum of the southings is 113.4 miles, 
 and as there are no northings, this is all difference of latitude. 
 
 And the difference between the eastings and the westings is 
 found to be 47.7 miles westings. 
 
 Note. — The table of Natural Sines and Cosines, elsewhere 
 explained, is a traverse table for distance unity for all courses 
 varying by *V from up to 90°. The column of cosines corres- 
 ponding to difference of latitude, and the column of sine, to 
 departure. This table should be used when precision in regard to 
 the course is wanted. 
 
 The tabular components of the course, being multiplied by the 
 distance, gives the actual components. 
 
 Rhumb Line. — In consequence of the convergence of the mer- 
 idian, as we approach the pole, or recede from it, in sailing on any 
 
 *The intervals of our table vary by 5'. 
 
44 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 oblique course, the ship makes a curved line or track, called a 
 Rhumb Line. The course of ship is frequently called a Rhumb. 
 This curve is always convex toward the equator. 
 
 The constancy of the angle, or course, in crossing the several 
 meridians, while they are all inclined to each other, except on the 
 equator, is what gives curvature to the course, and it is this prop- 
 erty of the rhumb that makes the results of plane sailing rigor- 
 ourly correct, notwithstanding a spherical surface cannot be 
 developed on a plane. 
 
 Difference of Latitude, is the distance between two parallels 
 of latitude. 
 
 As given from a course, by means of the traverse table, it is 
 miles or minutes and is called northings or southings, accordingly 
 as ship has made northings or southings. But when reduced to 
 degrees by dividing by 60 or 69 J, it is called Difference of Latitude. 
 
 Difference of Longitude of two places, is the arc of the 
 equator intercepted by the two meridians that pass through the 
 two points or places. 
 
 Longitude, is distance in arc (degrees) measured on the equator, 
 east or west, from the zero meridian (Greenwich). 
 
 Reduction of Departure to Longitude. — It will be observed 
 that eastings or westings called Departure, as found by the pre- 
 ceding article, is really difference of longitude, — it being the 
 distance between two meridians, — though on a small circle instead 
 of the equator. 
 
 And Avhen w^e recollect that there are just as many degrees in a 
 small circle as in a large one, we see that some reduction must be 
 made before we can measure longitude on a small circle w^ith the 
 same unit (the sea mile) that is used in measuring it on a large 
 circle. 
 
 Departure, though strictly difference of longitude, is not called 
 such till this reduction is made. 
 
 Two methods of converting departure to difference of longitude 
 are the following: 
 
 First. We may reduce the measuring unit in the same propor- 
 tion as that by which the small circle has been reduced from the 
 large one; or 
 
 Second. We may expand or enlarge the departure in the same 
 proportion as that by which the equator is greater than the parallel, 
 
PLANE SAILING. 45 
 
 SO as to make it embrace as many degrees on the equator as it 
 
 otherwise would on the parallel. 
 
 Bearing in mind that the circumferences of circles vary as their 
 
 diameters, and that the cosine of any latitude is the radius of that 
 
 parallel, we have the following proportion: 
 
 -r>. J .. J length of any arc \ .. f length of correspond- 
 
 .cos. .. ^ on the equator / " \ ing arc on the parallel. 
 
 Then, bearing in mind that R is unity, and multiplying extremes 
 and means, we have the 
 
 Arc of any parallels r'^^ri^Tr'^'"? equatorial arc 
 ^xv; ^x .* J pc* u V. (^ multiplied by cosine of parallel. 
 
 The use of this equation is seen in the following problem: 
 Required the length of a degree of longitude in any latitude, 
 L, say 43°, that on the equator being 60 miles. 
 Ans. Arc=60Xcos. 43° 
 
 =60X7314=43.88 miles. 
 Or suppose we wieh to find the number of feet in 1^ of longitude 
 on parallel 43°, we would have to multiply the equatorial minute, 
 0087 feet, by the cosine of 43°, thus, 
 
 5087X.7314=4452 feet. Ans. 
 Thus we see that any unit of measure for longitude on the 
 equator, may be used for measuring longitude on any parallel, by 
 first multiplying that unit by the cosine of the parallel. 
 
 Second method. By inverting the terms of the proportion in 
 the preceding article, we have, 
 
 p J Tf f ^^^ ^^^ ^^ ^^^ 1 ^ corresponding arc on 
 L.OS. i.: J^:: | parallel of L. / ' t the equator. 
 
 Whence, 
 
 -o f any arc on the ) j ( corresponding arc on 
 
 ^ parallel of L. ^ * / the equator. 
 
 Or, 
 
 Cos. L I parallel of L. f \ the equator. 
 
 R { any arc on the } ^^ ( corresponding arc on 
 
 But, 
 
 =Secant. 
 
 Cos. 
 
 "Whence, we have the following Rule for converting departure 
 
 into difference of longitude, viz.: 
 
 Multiply the departure into the secant of the latitude. 
 
46 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Example: How much longitude will 60 miles of departure 
 embrace on parallel L=40°, 60°, and 80°? 
 Solution: 
 
 Secant 40°=1.305, then 60X1.305= 78.3 miles. ) 
 
 '' 60°=2.000, " 60X2.000=120.0 '' V Answers. 
 <i 80°=5.759, '' 60X5.759=345.5 '' ) 
 
 The student will observe that the longitude or difference of 
 longitude found above, is minutes, which must be divided by 60 
 to reduce them to degrees. 
 
 We could divide by the cosine of the L and get the same results, 
 but it is easier to multiply by secant. 
 
 Middle Latitude Sailing. — The methods of plane sailing, 
 though correct in their results, are incomplete in that they do not 
 determine the particular parallel on which departure shall be 
 reduced to difference of longitude, but, 
 
 Middle Latitude Sailing has for its object to determine on 
 what paralled the departure corresponding to any rhumb -line, 
 shall be converted into diference of longitude. 
 
 The methods of finding the components of a course, is identical 
 with that for Plane Sailing, and indeed, for all the sailings. 
 
 It is the practice among seamen, to assume the middle paralled 
 of any rhumb-line, as the parallel on which to make this reduc- 
 tion. This is well for short runs, but only roughly approximate 
 for small courses, in high latitudes, with large distances. 
 
 The true parallel, on which to make this reduction is somewhat 
 above the middle of latitude, — just how much is shown by a table 
 of corrections given at page 76 of Bowditch's Navigation, called 
 Workman's Table. The [construction of this table is fully 
 explained in Prof. Coffin^s Navigation, Problems 5 to 10 of Plane 
 Sailing, but too abstruse for an elementary work like this. 
 
 It is table XI. of this work. 
 
 Preparatory to the discussion and solution of problems, it is 
 well to introduce and explain some notation, for the purpose of 
 shortening our work. 
 
 C=the course. L==latitude left. 
 
 d=the distance. L^ ^latitude arrived at. 
 
 I =the difference of latitude. p=departure. 
 
 D=the difference of longitude. L^^corrected middle latitude. 
 
 Thus, in the triangle ABC. 
 
 C B being the meridian, the side. 
 
MIDDLE LATITUDE SAILING. 
 
 47 
 
 A C=the distance d. 
 
 C B=the difference latitude I , 
 
 A B=the departure p. 
 
 And when the departure is expanded from A B into m n, it is 
 called difference of longitude and represented hy D. 
 
 Let the student make himself familiar with this notation, as it 
 will help him greatly in getting clear apprehension of his work. 
 
 Example: Sailing from L^65° N on a 
 course 0=^. 42° E, distance d=648 miles. 
 Required difference of latitude I and 
 difference of longitude D. 
 
 Let the student construct the problem 
 carefully to scale. — constructing C by its 
 chord. 
 
 Solution: As in Plane Sailing, find 
 difference of latitude and departure. 
 
 Fig. 20. 
 
 1. Dep. p=dXsin C (=42°)=648X.6691=433.6^ or miles. 
 
 2. Diff. lat. I =dXcos C (=42°)=648X.7431=481.5^ or miles. 
 
 3. Or in degrees =481.5^60=8°. IJ^ 
 
 4. Lat. attained, Li=65°-f-8°.li^=73°.lJ^ 
 
 5. Middle lat. =(65°+73Mi)--2=69°.f 
 
 6. Correction to Middle lat. (see table) = 16^ 
 
 7. Corrected middle lat. L2=69°.|+16^=Say 69°.17^ 
 
 8. Then, by the rule of the preceeding article 
 Diff. of long. D=Sec. L^ Xp, or 
 
 2.8263X433.6=1225.5^ or 20°.25J^=D. 
 Thus we have our answer in equation (2) and in equation (8.) 
 1=481.2 miles, or 8°.1J^ 
 D= 20°.25}^ 
 
 Let the student go carefully over this work, looking all the 
 numbers out of the tables and performing all the indicated work. 
 By this means he will very quickly gain command of the 
 problems. 
 
 The student will observe that in equation (4) we added the 
 difference of latitude to L to find L^ . If we had made southings 
 instead of northings, with our course, we should have used the — 
 instead of the + sign. 
 
48 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Instead of multiplying the departure p. by the see. of L, we 
 could have devided by cosine of L^ thus: 
 
 433.G--.3538=1225.5^ or 20°.25J^ as before. 
 
 A table for reducing departure to difference of longitude is 
 given (table X.) that will give results slightly less than the pre- 
 ceeding solution, yet more nearly correct, as it is adapted to 
 the periodical or actual form of the earth, rather than that of the 
 sphere, as is the table of cosines. 
 
 It is calculated for parallels 30^ apart, from to 80°. When 
 it is desired to be more precise, the divisor for internediate 
 minutes may be readily interporlated. The divisors will be found 
 to be slightly greater than the cosines of the corresponding 
 parallels. Thus in the preceeding case, the divisor for L2= 
 69°,17^=.355, instead of .3538 and gives D=20°,21J^ instead of 
 20°,25J^ 
 
 In using this table, we have merely to divide the departure by 
 the value of one minute for the parallel, as given by the table. 
 
 Parallel Sailing, is the finding of the place of ship where it 
 sails on a parallel of latitude. 
 
 In this case, the entire distance is departure, the converting of 
 which into difference of longitude is already explained, and there- 
 fore requires no further attention. 
 
 Mercator's Sailing*. — To devise a means of representing cor- 
 rectly, large areas of the earth's surface, and at the same time to 
 simplify the reduction of departure to difference of longitude, 
 Girard Mercator, in 1566, invented a chart called 3Iercator's 
 Chart. This chart, from its many merits, has come into general 
 use, the world over, by maritime people. 
 
 To construct this chart, he first expanded all the parallels as we 
 have done, thereby making the meridianal distance on all parallels 
 equal between any two meridians, and thereby making the mer- 
 idians all parallel instead of convergent toward the poles, as on 
 the sphere. Then, to preserve the relative positions and the rela- 
 tive magnitude of objects, he expanded the meridians in the same 
 proportion. 
 
 Each minute of the meridian, from the equator up to the limit 
 of the chart, — usually 80° of latitude, — was multiplied by the 
 secant of its middle latitude, precisely as we expanded our depart- 
 ure to difference of longitude (see page 47) and the sum of these 
 augmented minutes, up to any parallel, was called the meridianal 
 
MERCATOR S SAILING. 
 
 49 
 
 parts for that parallel,—!, e., the expanded meridian. Thus, 
 the actual distance from the equator to the parallel 42° would be, 
 
 42X^0=2520 miles, geographic, 
 but when each mile is multiplied by the secant of its middle lati- 
 tude and added into one sum, we have for parallel 42°, 2782 mer- 
 idianal parts. 
 
 To distinguish between the two measurements, the expanded 
 distance, on the meridian lines is called Augmented Latitude, 
 while that on the sphere, as usually measured, is called Proper 
 Latitude. 
 
 The difference of two augmented latitudes is called Meridianal 
 Difference of Latitude. Thus, the meridianal parts for 42° 
 (abbreviated to M. P.) is 2781.7. The M. P. for 44° is 2945 8, and 
 their difference is 2945.8 — 2781 .7=164.1 , which is the meridianal 
 difference of latitude between 42° and 44°, whereas their proper 
 difference of latitude would be but 120. But, if the two parallels 
 be on opposite sides of the equator, then their sum is the merid- 
 ianal difference of latitude. 
 
 The course on a Mercator's chart is represented by a straight 
 line. This is a consequence of parallel meridians. 
 
 The relation of the parts is seen in the following figure: 
 
 Difi Long 
 
 In the triangle ABC, 
 CB=Proper Diff. Latitude =1 . 
 CA=Distance =d, 
 
 BA=Departure =p, 
 
 C ] =Augmented Diff. Latitude=m. 
 
 a 1 =Augmented Departure ) -p. 
 
 =^Dift'erence of Longitude ) * 
 
 Course=angle ACB =C. 
 
 From the properties of similar 
 triangles, we have, 
 
 CI '.1 a::R:tan. C, whence, 
 ] aX-R-=Cl Xtan. C, 
 but \ a is diff. long. D; R is unity 
 and c 1 is the augmented difference 
 of latitude, whence, 
 
 D=c] Xtan. C=m tan. C. 
 That is to say. 
 
 The difference of longitude is found by multiplying the aug- 
 mented difference of latitude by the tangent of the course. 
 
50 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Thus we find the difference of longitude without the necessity 
 of first finding the departure. This is one of the advantages of 
 Mercator's sailing. But we cannot measure distances by scale on 
 this chart. 
 
 Problem: Given the course C=X 26° E, the distance d=142 
 miles, L=28° N. Required L^ and D. 
 
 Solution: Preparatory to finding L^ , we must find 1, as in plane 
 sailing. Then, augmenting 1, we have m. 
 
 lr=-dXcos, C=142X.8988=rl27.6^=2°, 7.6^ 
 Li=-L+1 =28+2°, 7.6^=30°, 7.6^ Ans. 
 
 Meridianal parts (M. P.) of Li=1897.1 
 
 M. P. of L =1751.2 
 
 Meridianal difference of lat.=m=145.9'' Then, 
 Diff. long. D=mXtan. C (=26°) 
 
 =145.9X.488=71.2^=1°. 11.2^ Ans. 
 Second Solution. Working this example by middle latitude 
 sailing, will show the agreement of the two methods. 
 As in plane sailing, 
 
 1=2°, 7.6^ and L^ =30°, 7.6^ 
 
 Middle lat.=(L+Li)--2=(28°+30°, 7.6^^2=29°, 3.8^ 
 Correction to middle latitude (see table) == V 
 
 Whence, L-^ =29°, 5^ say, 
 
 p==dXsin. C=142X.4384 = 62.24 miles. 
 
 D=pXsecant L^ (=29°, 50=62.24^X1.144= 71.2^ 
 
 = 1°, 11.2^ Ans. 
 Or the same result as before. 
 
 The student must keep fresh in mind the notation of pages 46 
 and 47. 
 
 Note. — As the tables of sines, tangents, etc., given in this work, 
 vary by 5^ of arc, the results obtainable with them will not cor- 
 respond strictly with the results given here. 
 
 Problem: Given two places by their geographical co-ordinates 
 of place. 
 
 L =46°, 10^ S. Long. 46°, 30^ E. 
 
 Li=52°, 15^ S. Long. 51°, 10^ E. 
 Required C and d from L to Li. 
 
 Solution: From the co-ordinates of place Ave have the base and 
 perpendicular of a right triangle from which to find the hypoth- 
 
mercator's sailing, 51 
 
 enuse, which is the d, and the angle opposite the departure, 
 
 which is C. 
 
 M. P. of L 1 (=52°, 150 =3689.6 
 
 M. P. of L (=46°, 100 =3130.0 
 
 Meridian difference of latitude=ni = 559.6 miles. 
 
 Diff. long, D=(51°, 10^—46°, 300=4°, 40^=280 miles. 
 By equation 2, page 9, 
 
 Tan. C=D^-meridian difference of latitude 
 '' **=280^-^559.6=.5003=tan. 26°, 35^ Ans. 
 Proper difference of lat.=(52°, 15^—46°, 100=6°, 5^=365 miles. 
 Distance d=] Xsecant C (=26°. 350 
 
 =365X1.1182=408.1 miles. Ans. 
 
 . / C=S. 26°, 35^ E. 
 ^^s- \ d=408.1 miles. 
 It will be observed that d is found from the proper difference 
 of latitude 1. This is necessary because oblique distances cannot 
 be measured on Mercator's chart. 
 
 Solution by middle latitude sailing: 
 Proper difference of latitude=52°, 15^—46°, 10^=6°, 5^=365=1. 
 Middle latitude=(52°, 15^+46°, 100-^2=49°, 12 J^ 
 Correction for middle latitude (see table)=5^. 
 Corrected middle latitude L=2=49°, 17J^. 
 
 Differenceof longitude D=(51°, 10^—46°, 30^)=4°, 40^=280 miles. 
 Departure p=cos. L^XD. 
 
 ==.6522X280=182.6 miles. 
 Tan. C=p^l=182.6^365=.5003=tan. 26°, 35^ Answer, 
 as before. Also, 
 Distance d=lXsec. C (=26°, 350 
 
 =365X1.1182=408.1 miles. Ans., as before. 
 Examples for exercise: 
 
 1. Sailed from L=42°, 30^ N., and long. 58°, 51^ W. S. W. by 
 S., 591 miles. Required the latitude and longitude in. 
 
 Ans. Latitude 34°, 19^ N. 
 
 Longitude 65°, 51^ W. 
 
 2. A ship sailed from L=49°, 57^ N., and long. 30°, 00^ W., 
 on course C=S. 39° W., till she arrives at Li=45°, 31^ N. Re- 
 quired the distance sailed and the longitude in. 
 
 Ans, Distance=342.3 miles. 
 
 Longitude in=35'^, 21^ W. 
 
52 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Let the student construct these problems to scale, carefully, 
 before attempting numerical solution. 
 
 Traverse Sailing, — When a vessel sails on a number of courses 
 in making a run, she is said to make a traverse or irregular track. 
 And the finding of the equivalent of the traverse in one course 
 and distance, is called traverse sailing. While the operation or 
 the solution of the question is called the reduction of the tra- 
 verse, or finding the course and distance "made good," or reduc- 
 ing a day's work. 
 
 For moderate distances, the method of plane sailing is satis- 
 factory. But on longer voyages, some one of the other sailings 
 must be combined with it for the purpose of keeping account of 
 the longitude. It is one of the most useful of the several 
 msthods. An example will illustrate: 
 
 Sailed N. N. E.=N. 2 p. E., 140 miles, 
 
 Thence, N. E. by E.=X. 5 p. E., 48 '' 
 Thence, east, - - 26 ^* 
 
 Thence, S. by W.=S. 1 p. W., 36 '' 
 Thence, S.S. W. J W.=S. 3J p. W\, 76 '' 
 Required the equivalent sing-Ie course and distan^^e, or course 
 and distance "made good." 
 
 First, prepare a table in which to arrange the several courses, 
 with their distances and their several components, — providing a 
 column also for changing the courses into degrees, as follows: 
 
 
 Course. 
 
 Degrees. 
 
 Dist. 
 
 North' gs 
 
 South'gs 
 
 Eastings 
 
 W^est'gs. 
 
 1 
 
 N. 2p. E. 
 
 22°, 30^ 
 
 140 
 
 .924 
 
 129.4 
 
 
 .383 
 
 53.6 
 
 
 2 
 
 N. 5 p. E. 
 
 56°, Id' 
 
 48 
 
 .555 
 
 26.6 
 
 
 .831 
 
 39.9 
 
 
 3 
 
 East. 
 
 90°, 00^ 
 
 26 
 
 
 
 1.000 
 
 26.0 
 
 
 4 
 
 S. 1 p. W. 
 
 11°, 15^ 
 
 36 
 
 
 .981 
 
 35.3 
 
 
 .195 
 
 7.0 
 
 5 
 
 S.2Jp.W. 
 
 28°, 07^ 
 
 76 
 
 
 .882 
 
 67.0 
 
 
 .471 
 35.8 
 
 
 
 156.0 
 
 102.3 
 
 119.5 
 
 42.8 
 
 
 
 102.3 
 
 
 42.8 
 
 
 
 53.7 
 
 76.7 
 
 
 Having arranged the courses in order, with their degrees and 
 distances in their proper columns, take the cosines of the courses 
 
 from a table of natural sines and cosines, and write them in the 
 
TRAVERSE SAILING. 53 
 
 upper left hand of the space for the northings or southings for 
 that course, as the case may be. 
 
 And write the sines of the courses in the column for departure 
 — eastings or Avestings, as the case may be; thus, 
 
 In the upper left hand of the space for northings, is written the 
 cosine of 22°, 30^; and in the column for eastings, is written the 
 sine of 22°, 30^ 
 
 These are the factors with which to multiply the distance, 140 
 miles, for the components of course; and so for all the courses. 
 
 Having multiplied each distance into the cosine and sine of its 
 course, and arranged the products in their respective columns, 
 write the sum of the components of each column at the foot of the 
 same, and 
 
 Find the difference of the northings and southings and the dif- 
 ference of the eastings and westings, thus, 
 
 In this case, we find the northings in excess of the southings 
 53.7 miles; and the eastings exceed the westings 76.7 miles. 
 
 Thus far, the problem is merely another form of plane sailing. 
 We have now our difference of latitude 1=53.7 miles, and depart- 
 ure 76.7 miles, from which to find the course C and the distance 
 d, whence, 
 
 p-Hl=76.7--54.7=1.4021=tan. 54°, 30^=course, 
 and as the course takes its name from its components, we have 
 the course, 
 
 X. 54°, 30^ E„ or X. E. J E., 
 and multiplying northings by secant of C, we have, 
 Distance=54.7Xl.T22=94.2 miles. 
 
 Instead of taking our components from a traverse table, we 
 have taken the cosines and sines to distance unity, and multiplied 
 by their respective distances. 
 
 This is for the purpose of showing a more precise way of com- 
 puting the components of a course, as the ordinary traverse tables 
 are not computed for anything lower than degrees, — though this 
 is not designed to supercede the use of the traverse table for 
 ordinary work. 
 
 If, now, the question of longitude is involved, we must introduce 
 middle latitude sailing, as in the following 
 
54 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Example: From latitude 41°, 12^ N., longitude 20° W., make 
 the following traverse: Required the latitude and longitude 
 attained, and the course and distance ^'made good," 
 
 1. S. W. by W. 21 miles. 2. S. W. J S, 31 miles. 
 3. W.S.W.JS. 16 '^ 4. S. IE. 18 '' 
 
 5. S. W. i W. 14 '^ 6. W. J N. 30 '' 
 
 As in the former case, rule seven columns for courses, etc. 
 
 Angle 
 
 Dist. 
 
 North 
 
 gs 
 
 South 'gs 
 
 Eastings 
 
 AVest'gs. 
 
 1. 
 
 S.W.byW. 
 
 =5 p. 
 
 21 
 
 
 
 11.7 
 
 
 17.5 
 
 2. 
 
 s. w. ^ s. 
 
 =3^ p. 
 
 31 
 
 
 
 24.0 
 
 
 19.7 
 
 3. 
 
 W.S.W.JS. 
 
 =5Jp. 
 
 16 
 
 
 
 7.i> 
 
 
 14.1 
 
 4. 
 
 S. f E. 
 
 = fp. 
 
 18 
 
 
 
 17.8 
 
 2.6 
 
 
 o. 
 
 s. w. 1 ^y. 
 
 =4ip. 
 
 14 
 
 
 
 9.4 
 
 
 10.4 
 
 6. 
 
 W. J N. 
 
 ==7J p. 
 
 30 
 
 2.9 
 
 
 
 
 29.8 
 
 
 
 
 
 2.y 
 
 
 70.4 
 2.9 
 
 2.6 
 
 91.5 
 
 2.6 
 
 
 Southi 
 
 ngs^67.5 
 
 Westings 
 
 =88.9 
 
 We find we have made difference of latitude southings 67.5 
 miles, and departure westings 88.9 miles. Dividing the southings 
 by 60, we have, 
 
 67.5-r-60=l°, 7Y southings in latilude, or, 
 latitude attained is 
 
 41°, 12^— 1.7J=40°, 4y ^.=LK 
 Then, dividing departure p by difference of latitude 1, we have, 
 
 p^l^88.9--67.5=rl.k7=tan. 52°, 48^=course, 
 and because the components of our course are southings and 
 westings, the name of our course is S. 52°, 48'' W. 
 
 Multiplying difference of latitude by secant of C, we have dis- 
 tance d=67.5Xl-655=111.7 miles. 
 By middle latitude sailing, we have, 
 
 D=pXsec. L 2=88.9X1.318=117.2 miles, 
 
 =1°, 57^ diff. long, westings. 
 Then the longitude in 
 
 =20°+l°, 57^=21°, 57^ W. 
 Collecting our results, we have, 
 
 Li=40°, 4J^N. 
 C=S. 52°, 48^ W. 
 d=111.7 miles. 
 Longitude in=21°, 57^ W. 
 It will be observed that w^e have in this problem taken half the 
 sum of the extreme latitudes for the middle latitude L^. This is 
 
CURRENT SAILING. 55 
 
 the usual practice for small distances. But for large distances, or 
 with small C, it is necessary to apply the correction of table XI. 
 
 Or the following rule may be used for finding the parallel on 
 which the departure p must be augmented for difference of 
 longitude. 
 
 Take the parallel whose cosine is half the sum of the cosines of 
 the extreme parallels, for the middle parallel. (This rule is 
 original). 
 
 This, although not rigorously correct, is practically so for all 
 ordinary cases. It gives the L^ very slightly too large for L less 
 than 45°, and slightly too small for L larger than 45°. 
 
 The student should solve all his questions by at least two meth- 
 ods, as a means of checking against mistakes, — one of which 
 should be by construction, for fixing the problem clearly in 
 the mind; another is by inspection for some of the parts. 
 
 Thus, after we have found the 1 and p for a traverse, we can 
 search the latitude and departure columns of a traverse table till 
 we find these two components in the same line. The corres- 
 ponding distance will be found in the ^'distance" column on the 
 left, and the course will be found at head or foot of page or 
 column. 
 
 But this plan, though rigorously correct in principle, will not 
 generally be found satisfactory. 
 
 The traverse tables are not computed generally to arcs varying 
 by less than one degree, so that the two components can rarely be 
 found precisely, — hence some interpolation will be required. 
 
 A better method is to divide both the components by the dif- 
 ference of latitude 1, Then, by a principle of trigonometry, we 
 have the tangent of the course C, which may be found from the 
 table of Natural Sines, Tangents, etc. And from the same table, 
 the secant of C is found, which gives us the distance d for unity. 
 Then, multiplying sec. C by 1, we have the d desired, as in the 
 preceding examples. 
 
 Current Sailing. — The effect of a current on a vessel is the 
 same as that of another course and distance, the course being the 
 direction of the current, and the distance being the rate per unit 
 of time, — as an hour, — multiplied by the time sailed in the same. 
 
 If a vessel sails with a current, she will be ahead of her reck- 
 oning, by the amount of the motion of the current, during the 
 
oG 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 time of sailing in it. And if she sail againfet the current, she will 
 be behind her reckoning by the same amount. 
 
 If she sail across the current, she will be carried with it through 
 the distance the current moves while the ship is in it. 
 
 The direction of the current, with regard to the meridian, is 
 called the set ; and the rate at which it runs per hour is called 
 the drift, 
 
 From the above conditions, it is seen that in all cases, when 
 sailing with a current, the set and drift must be regarded as an 
 independent course and distance. Also that time is an element 
 to be considered. 
 
 Example: A ship made the following traverse in a current 
 setting X. by W., at the rate of two miles per hour. 
 
 1. S. W. J W., 2 hours, 8 miles per hour=16 miles, 
 
 2. W. JS. 3^-7^^ '' —21 " 
 
 3. W. byN. 3^-6*^ '^ =18 '' 
 Required the course and distance ^'made good.'^ 
 
 
 
 Dist. 
 
 X. 
 
 S. 
 
 E. 
 
 W. 
 
 S. \Y. J W 
 
 50°, 37^ 
 
 84°, 22^ 
 78°, 45^ 
 11°, 15^ 
 
 16 
 21 
 
 18 
 16 
 
 3.5 
 15.7 
 
 10.1 
 
 1.1 
 
 
 12.4 
 
 W. JS 
 
 "^O.d 
 
 W. bvN 
 
 17.6 
 
 N.byW 
 
 3.1 
 
 
 
 Current 
 
 
 
 19.2 
 11.2 
 
 11.2 
 
 
 n^A) 
 
 
 miles. 
 
 -= 8.0 miles. 
 Course=]S'. 81°, 34^ W. Distance, 54.7 miles. 
 
 Example 2: A ship sails S. 17° E. for two hours, at two miles 
 per hour, as indicated by ship's log; thence S. 18° W. four hours, 
 at the rate of seven miles per hour; and during the whole time, 
 the current sets X. 76°, at the rate of two miles per hour. Required 
 the course and distance ''made good." 
 
 Ans. C=S. 21°, 49^ W. Distance==42 miles. 
 
 Note. — It will be observed that this is merely the application 
 of traverse sailing, with an extra course and distance introduced 
 into the traverse. 
 
 Oblique Sailing. — It will be observed that, up to the present, 
 all our determinations for place of ship have been made by means 
 of the properties of the right plane triangle. Some questions 
 require the use of the properties of oblique triangles. The 
 
OBLIQUE SAILING. 
 
 57 
 
 finding of place of ship by this means, or the determining of ques- 
 tions in navigation by this means, is called oblique sailillg*. 
 
 It is used chiefly in the survey of harbors, — in the location of 
 shoals, with regard to their bearing from other objects, — or in 
 finding compass errors. A few examples will illustrate. 
 
 But these questions will involve the use of some means on board 
 ship, and as but very few of our lake vessels are provided with 
 such appliances, I give the following method for the 
 
 Construction ^'f the Dumb Card, which should be on every 
 ship on the lakes: 
 
 Let the ship's carpenter describe a circle, say six or eight feet 
 in diameter, with 
 center C in center 
 line of ship. 
 
 Through C, and 
 at right angles to 
 center line, draw the 
 line 0. 
 
 Divide each J of 
 the circle into eight 
 equal parts, and by 
 means of chalk-line 
 or straight-edge, 
 transfer these points 
 onto the rail, and 
 make a clean, deep 
 mark at the inter- 
 section. Number these marks from to 4, from forward and from 
 abeam, on each side of the head of ship; and number from to 
 1, 2, 3, etc., points abaft the beam. 
 
 Let the marks on the rail be numbered with the number of the 
 point, by driving brass or tin headed nails to indicate the number. 
 
 Let the center of the circle be marked, as by driving some nails 
 at the intersection of the fore-and-aft and 'thwart-ship lines. 
 
 Then, to use this card, the eye being over the center and look- 
 ing over the rail, the bearing of any object from ship is readily 
 seen; and the bearing is referred to the nearest zero line, — 
 thus we would say an object appears IJ points abaft the port 
 
 Fig. 22. 
 
58 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 beam, 2 points forward of the starboard beam, or 3 points off the 
 port bow, as the case may be. 
 
 Problem: A ship-master being about to sail, wishes to examine 
 his compass, as to its accuracy. He observes from his chart that 
 when 30 miles out on his proposed voyage, he will be 4 miles to 
 the left of a certain lighthouse that stands on a headland. When 
 Hearing the light, he observed it to be If points forward of the 
 starboard beam. After sailing 6 miles, as indicated by the log, 
 and on the same course, the light appeared IJ points abaft the 
 beam. Having shaped his course, on the supposition that his 
 compass was correct, he wishes to know from the above observa- 
 tions if it is so. If not, which way is it out, and how much. 
 
 Solution: Construct the problem as per figure. Draw any right 
 line AB to represent course of ship, and on it take AB=6 miles, 
 to any convenient scale, At A draw 
 the line AC, forward of the beam If p. 
 And at B,*draw BC, abaft the beam IJ 
 p. They w^ill intersect at C. 
 
 It will be observed that each of the 
 
 angles B and A, in the triangle ABC, 
 
 is the compliment of the observed angle, 
 
 i, e., the angle at A=8 p. — If p.=6J p., 
 
 ^•* etc. 
 
 Then, bearing in mind that the sum 
 of the three angles of any plane triangle 
 ^^' is 16 p., we have only to take the com- 
 
 pliment of the sums of the angles A and B, to know C=3J p. 
 
 Then, in the triangle ABC, knowing one side and the three 
 angles, we have a case for the **sine proportion,' ' by which 
 we have, 
 
 Sin. C:AB::sin. B : AC 
 
 : : sin. A : BC, or 
 
 Sin. 3J p. (=.5958) :6 miles :: sin. 6 J p. (=.9569): AC (=9.63 miles) 
 
 :: sin. 6J p. (=.9416):BC (=9.48 miles). 
 
 See table V, for the sines, etc., of points and parts, and let the 
 student verify by careful construction, and let him perform the 
 numerical work here indicated. 
 
UBLIQUJ 
 
 59 
 
 Our question requires us to know the height CD of the triangle. 
 This can be measured by scale, or computed numerically. 
 
 By equation 3, page 9, 
 
 CD==ACXsin. A {=6^ p.) 
 3=9.63X.9-A16=9.06 miles. 
 
 By the conditions of our question, we should be 4 miles to the 
 left of C, but the above work shows us to be 9.06 miles to left, — 
 that is to say, our compass has taken us to left of our true course 
 O.06 miles in 30. Then, by equation 2, page 9, we have, 
 
 Tan. course=5--30=.1666=tan. 9°, 29^ 
 So that our compass is out to the left 9J°=f p. 
 
 In the preceding solution, it would have been sufficient to make 
 one proportion for one side of the triangle ABC; but tinding both 
 sides, gave means of checking our work. Thus, either side, AC 
 or BC, multiplied into the sine of its adjacent angle, should pro- 
 duce the perpendicular CD=9.06 miles. 
 
 Problem. The following problem is of the same character as 
 the preceding, except the bearings are referred to the meridian 
 instead of to center line of \ 
 
 ship: 
 
 A port, n, bears X. E. J 
 ]N". from a port, m. At 26 
 miles out from m, toward 
 B, directly abreast the port 
 beam, is a light, distant 6 
 miles, when the ship is on 
 the right course. The mas- 
 ter having shaped his 
 course on the supposition 
 that his compass was cor- 
 rect, when nearing the 
 light, took its bearing, N. 
 by W. i W.--N. IJ p. W. 
 After sailing on same 
 course 8 miles further, as ^^' 
 
 indicated by ship's log, he took a second bearing, W. by X. | N. 
 ^^N. 6J p. W. Eequired to know the compass error, if any, — 
 liow much and which way. 
 
 Solution by Construction: Draw any line, as AE, through the 
 page, vertically, for the meridian. From any point on this line, 
 
60 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 as at A, set off the course of ship, 3J p. to right of the meridian 
 and 8 miles long, to any convenient scale. From the same point 
 A, set off the bearing of the light C, IJ p. to left of meridian. 
 From B set off BC 6J p. to the left of the meridian, or, Avhat is 
 the same thing, make the angle ABC=to the supplement of the 
 bearings AB and AC, viz., 6J p, Then by scale, we find, 
 CD=7.8 miles. Answer. 
 Solution by Oblique Trigonometry: In the triangle ABC, the 
 Angle at A=e5 p. or 3 J -^1 J, 
 Angle at B=6} p. or 16 p— (3J+6}), 
 Angle at C=4f p. or 16 p. — (5-f 6}). 
 Then, by the sine proportion: 
 
 Sin. C: AB :: sin. A : BC, 
 
 :: sin. B : AC, or, 
 Sin. 4J p. (=.8032) : 8 :: sin. ^ p. (=.8315) : BC (=8.28 miles), 
 :: sin. 6} p. (—.9416): AC (=9.38 miles). 
 Then, either side, AC or BC, multiplied by the sine of its adjacent 
 angle, gives the perpendicular, thus, 
 
 AC (=9.38 miles)Xsin. 5 p. (=.8315)=CD (=7.79 miles), 
 and BC (=8.28 miles)Xsin. 6i p. (=.9416)=CD (=7.79 miles). 
 
 But, by the conditions of our problem, CD should be 6 miles^ 
 that is to say, our compass has taken us to the right, say 1.8 miles 
 in 26, which, by right trigonometry, corresponds to an angle 
 of 4°. 
 
 Answer. Compass out to the right 4°. 
 
 Many other questions could be proposed for solution by oblique 
 trigonometry, but they would be of a class that seldom or never 
 occur in practice, — are more curious than useful, — so we spend no 
 time with them. Following are a few miscellaneous questions: 
 
 Distance of an Obiect by two Bearings. — The two preceding 
 problems are at the foundation of table XII. They are deemed 
 of so much importance to ship masters who have occasion to 
 round headlands in the night, that a table has been prepared from 
 which the distance of a light, or other object from a ship, — as also 
 the line of the ship's course, at right angles from the light or 
 object, may be told in advance, or before reaching the vicinity of 
 the '^danger line.'' 
 
 But the table here presented is quite different from that used at 
 sea, and for the following reason: There, the bearings of the 
 
OBLIQUE SAILING. 61 
 
 object or light are referred to the meridian, by means of the stand- 
 ard compass, which is furnished with a movable ring and sights 
 for the purpose. 
 
 But, on the lakes, our compasses being boxed up in the pilot 
 house, are not available for such work, even if they had the ring 
 and sights. We can only take the bearing of the object from the 
 ship's center line, by means of the ship's dumb compass, — reading 
 the bearing directly from the card, when she is fortunate enough 
 to have one, — which, indeed, is the better way. 
 
 The table is constructed generally by solving a number of tri- 
 angles, varying in their angles by J or J point, through such 
 limits as would embrace all the cases likely to occur, both for the 
 side opposite the first bearing and for the perpendicular distance 
 of the base or line of the ship's bearing, produced, from the 
 light or object, for a distance of unity. 
 
 The results are tabulated in column under the first bearings 
 taken, and in line of the second bearings as factors by which to 
 multiply the distance run between the times of observation for the 
 distances sought, — the larger product giving the distance from 
 ship to light at the time of the second observation; the smaller 
 product giving the distance to light or object when it comes 
 abeam, — or the height of the triangle, as it is technically termed. 
 
 In the table which I give (table XII) I introduce the factor, as 
 above, for the distance of the light from ship at the time of second 
 observation. But, instead of the second factor, I introduce the 
 sine of the second bearing, to be multiplied by distance of ship 
 from light, for the perpendicular distance of object, or height of 
 the triangle. 
 
 An example will illustrate the use of the table: 
 
 Being about to pass a headland in the night, the track by which, 
 as given by the chart, lies two miles to the right, and not knowing 
 whether my compass is correct or whether I was in the proper 
 track when shaping course, I wish to know if my course will take 
 me the proper distance to the right of the light. Soon after 
 making the light, I found it to bear 2 points to the left of the 
 ship's heading. After sailing 3J miles, as indicated by ship's log, 
 it bore 4J points to left. Eequired the distance of ship from the 
 light at time of second observation, also the perpendicular dis- 
 tance of the track of ship from the light. (See Fig. 25). 
 
62 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Solution: Draw the indefinite line A m to represent the ship's 
 course. From any point A in this line, draw the line AC two 
 points to the left of A m, and from A set off to B the distance 
 sailed, 3 J miles. From B draw BC 4 J p. to the left of ship's 
 course, meeting AC in C. Then is C the place of the light. 
 
 From table XII, in column of 2 p. and in line of 4J p., will be 
 found the decimal .81. This, multiplied by the distance, 3J miles, 
 gives us BC. the distance of light from ship=2.83 miles. Thia 
 
 again multiplied by the sine of 4J p. 
 (=:.773)=2.19 miles=DC, the dis- 
 tance of ship's track from the light. 
 This problem is also available in 
 finding compass errors. 
 
 Course aud Distance from the 
 Co-ordinates of Place. — The won- 
 derful clearness and fullness of 
 detail in our lake charts are pur- 
 chased at the expense of one con- 
 venience, and that is the finding of 
 course and distance, in some cases. 
 
 The scale of the chart is so large 
 
 as to make it impracticable to give 
 
 us a full or entire lake on one sheet; 
 
 as a consequence, they are given in 
 
 sections, which must be combined 
 
 before courses can be obtained from. 
 
 Fig. 25. them in all cases. 
 
 This combination is inconvenient at times, nevertheless the course 
 
 between ports represented on different sections can be readily 
 
 obtained from their *^ co-ordinates of place," as given in the list 
 
 of lights. An example w^ill illustrate: 
 
 At Sturgeon Bay Canal. Required the course and distance to 
 Michigan City. 
 
 Solution: From the U. S. List of Lights for the Lakes, we find 
 Sturgeon Bay in latitude 44°, 47^, longitude 87°, 18^. Michigan 
 City in latitude 41°, 43^ longitude 86°, 54^ 
 
 The difference of latitude is 3°, 04^=212 miles, 
 The difference of longitude is 24^=20 miles, 
 for the mean latitude 43°, 15^. 
 Whence, in sailing froni Sturgeon Bay to Michigan City, we must 
 
OBLIQUE SAILING. 
 
 63 
 
 ^•0 
 
 '/ZF 
 
 make southings 212 miles, and eastings 20 miles. The southings 
 
 are found by multiplying the difference of latitude in degrees by 
 
 the value of 1 degree=69.15 miles, giving 212 
 
 miles. The eastings are found by multiplying 
 
 the difference of longitude, 24^ by the value 
 
 of 1 minute of longitude, for the mean 
 
 latitude, as given by table X, giving us 20 
 
 miles. 
 
 Constructing a triangle, as in Fig. 26, 
 
 with base 212 and with perpendicular 20, and 
 
 measuring the angle at A with protractor 
 
 or scale of chart, we find the angle say \ p. 
 
 That is to sav, our course is S. J E. 
 
 
 Then, by plane trigonometry, multiplying 
 
 the base 212 by the secant of course (^=1.0048, 
 
 table III), we have 213 miles for distance. 
 
 By plane trigonometry, also, the course 
 may be found, for 
 
 20--212=.0943=tan. 5°, 25^=say \ p. 
 This, it must be remembered, is the chart 
 course, which must be modified by variation, 
 — in this case \ p. to the right, — making the 
 magnetic course S. f E.; and this must 
 again be modified by deviation, if any, also by 
 leeway and for current, if any. 
 
 Fig. 26. 
 
CHAPTEK ly. 
 
 Construction of Charts. 
 
 Construction and Use of Mercator's Chart. — The principles 
 underlying the construction of this chart have been examined 
 under the head of Mercator's Sailing. It only remains to make 
 the practical application. 
 
 Suppose we were to construct a Mercator's chart of the territory 
 embracing the great lakes, — say from latitude 40° to 50°, and 
 from longitude 75° to 95° W. We must prepare the skeleton or 
 blank, as follows: (See table on following page). 
 
 1. Write down in a column the degrees and parts of a degree 
 that are required to be represented on the sides of the map, — as 
 the whole degrees, J° or the ^°. (See table IX). 
 
 2. From a table of meridianal parts (abbreviated to M. P.) 
 write the M. P's corresponding to each degree and part of a 
 degree, in an adjoining column. 
 
 3. From the M. P's of the highest latitude, subtract those of 
 the lowest latitude. Thus, the M. P's of 50° (--3474.5), the M. P's 
 of 40° (=2622.7), their difference is 891.8, which is the depth of 
 our map in latitude, in the scale units that rejDresent 1°- 
 
 4. From the M. P's of 50°, take the M. P's of the whole 
 degrees, 49°, 48°, 47°, etc., setting down the differences opposite 
 their respective latitudes. Thus each difference will represent the 
 distance between the consecutive parallels that are one degree 
 apart, and the place of each parallel can be marked on the side of 
 the map at one placing of the scale, after we 
 
 5. Add the several differences consecutively, for the height 
 up to the consecutive parallels. Thus, 
 
 The M. P's for 50°=3474.5 
 '' '' '' '' 49°=3382.1 
 
 92.4, seen at foot of 3d column, 
 M. P. for 49°— M. P. for 48°=90.6, as seen in 3d column, etc. 
 
MEKCATOR S CHART. 
 
 65 
 
 In this manner will the widths of single degrees be found and 
 recorded in the 3d column. Then, adding consecutively, as per 
 (5), we have the height of the first parallel from base of map= 
 78.9 scale units. Then 78.9-i-80.1=159=height of 42° parallel 
 from base of map; 159+81.4=240.4 M. P's for the height of the 
 43d parallel, etc. In this manner was the fourth column pro- 
 duced, that shows the height of each parallel from base of map. 
 
 6. In the widths for the J degrees, subtract the M. P's of 49J° 
 from those of 50°, and we get 46.5 M. P's, found at the foot of 
 the 5th column. The M. P's for 49J°— those for 49°=45.9 M. P's, 
 seen in the 5th column. And in this manner was the width of 
 €ach consecutive half degree found. 
 
 Table of Elements for a Mercator's Chart. 
 Latitude 40° to 50°. 
 
 Lat. 
 
 Meridianal 
 
 Widths of 
 
 Heights in 
 
 Widths of 
 
 Parts. 
 
 Degrees. 
 
 Latitude. 
 
 Half Deg. 
 
 40 
 
 2622.7 
 
 
 
 39.3 
 
 i 
 
 2662.0 
 
 
 
 39.6 
 
 41 
 
 2701.6 
 
 78.9 
 
 78.9 
 
 39.9 
 
 i 
 
 2741.5 
 
 
 
 40.2 
 
 42 
 
 2781.7 
 
 80.1 
 
 159.0 
 
 40,6 
 
 1 
 
 2822.3 
 
 
 
 40.8 
 
 43 
 
 2863.1 
 
 81.4 
 
 240.4 
 
 41.2 
 
 i 
 
 2904.3 
 
 
 
 41.5 
 
 44 
 
 2945.8 
 
 82.7 
 
 323.1 
 
 41.9 
 
 J 
 
 2987.7 
 
 
 
 42.3 
 
 45 
 
 3030.0 
 
 84.2 
 
 407.3 
 
 42.6 
 
 i 
 
 3072.6 
 
 
 
 43.0 
 
 46 
 
 3115.6 
 
 85.6 
 
 492.8 
 
 43.4 
 
 i 
 
 3159.0 
 
 
 
 43.7 
 
 47 
 
 3202.7 
 
 87.1 
 
 580.0 
 
 44.2 
 
 i 
 
 3246.9 
 
 
 
 44.6 
 
 48 
 
 3291.5 
 
 88.8 
 
 668.8 
 
 45.1 
 
 1 
 
 3336.6 
 
 
 
 45.5 
 
 49 
 
 3382.1 
 
 90.6 
 
 759.4 
 
 45.9 
 
 i 
 
 3428,0 
 
 
 
 46,5 
 
 50 
 
 3474.5 
 
 92.4 
 
 851.8 
 
 
 The M. P's for half degrees were found by taking the difference 
 of the consecutive half degrees in columns 1 and 2. 
 
 The table for the values of the degrees of latitude, being pre- 
 pared, we are ready to construct the framework or skeleto-n of our 
 chart. 
 
66 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 1. Assume any convenient scale. — preferably one that is 
 decimally divided, and set off on a horizontal line, at the foot of 
 the chart (the south side,) the amount of longitude required, — 
 in this case, 20° or 120 miles (1 mile being the scale unit), mark- 
 ing the points for the meridians, — each degree or each alternate 
 degree, as wanted. 
 
 2. At such extremity of the line representing the width of the 
 map in longitude, erect a perpendicular for the extreme meridian 
 or sides of the chart. 
 
 3. On each of these meridians, set up the distances given in 
 the preceding table, for the places of the several latitudes, com- 
 mencing at the bottom of the sheet, thus. 
 
 Set up 78.9 M. P's for the place of the 41° parallel, 
 
 u 1590 u u u a 490 
 
 a 240.4 '' '' " '' 43° '' etc. 
 
 Thus we have all the parallels located at their proper places, to 
 represent the augmented latitude. 
 
 4. We can now locate the places for the half degrees from 
 their scale values, in the column for the widths of half degrees. 
 
 The framework or skeleton is now ready to take the location of 
 places, — as towns, coast-lines, rivers, islands, boundaries, etc.,. 
 which are located from their known places of latitude and long- 
 itude, by means of two T squares, — one locating the latitude, the 
 other the longitude, — and their intersection being marked by a 
 needle-point or sharp pencil. Thus, a number of points in the 
 boundary of a lake or bay being located and a fair line traced 
 from one to another, the shore line is located, etc. 
 
 It is to be remarked that the "meridiaual parts" given in our 
 table IX, are those that have been in use a long time, as first con- 
 structed, on the supposition that the earth is a sphere. But more 
 modern w^orks compute them for the earth regarded as a spheroid^ 
 — making them slightly smaller than for the sphere, for any given 
 latitude. 
 
 Bearings and Distances. — The bearing* between any two 
 points on a Mercator's chart, is very readily found. 
 
 It is only necessary to draw a straight line between the two 
 points and apply a protractor to any meridian crossed by this 
 line, to read the bearing. This, and the straight ''rhumb line,'' 
 are two of its conveniencies. But not so with distance. This 
 cannot be measured by scale, on a Mercator's chart, except in 
 
PLANE CHART. 67 
 
 the direction of longitude; and this measurement must be multi- 
 plied by the cosine of the latitude before it is available for use, 
 for it will be remembered that the whole map, away from the 
 equator, has been expanded. 
 
 Scale measurements caiiiiot be made ill any oblique direction, 
 
 because the scale varies from the equator toward the pole. Thus, 
 at 40° of latitude, a half degree is only 39.3 M. P., while at 50°, 
 it is 46.5 M. P., — and this is an objection. 
 
 Construction of the Plane Chart. — In the plane or rectang- 
 ular chart, the meridians are parallel lines at a uniform distance 
 apart. 
 
 This distance, for one degree, is found by multiplying the 
 equatorial distance, 60 miles geographic or 69.15 statute, by the 
 cosine of the latitude for which the map is made. 
 
 The following considerations will show the amount of error for 
 such a chart: 
 
 First. It must be remembered that the meridians at the equa- 
 tor are parallel, and at the poles they have their maximum inclin- 
 ation, which is the whole difference of longitude, and that between 
 these limits, the inclination varies as the sine of the latitude. 
 
 Example: Suppose we wish to make a rectangular or plane 
 chart for the latitude of 42°, for an area of one or two counties, 
 or say 30 miles square. Multiplying 60, the number of geograph- 
 ical miles in one degree, by the sine of 42° (=.6691) we have 40.14 
 geographic miles for one degree of longitude, or 46.28 statute 
 miles, and the error resulting from their being parallel, would be 
 60^Xsin. 42° (=.6691)=40^ for 1° of longitude, i. e., in a block 
 30 miles square, the south side would be about 55 rods too Small, 
 while on the north it would be that amount too large. 
 
 The Conical Projection. — In the conical projection, the mer- 
 idians are all right lines, but they are inclined by the amount due 
 to the central latitude of the map. There are two methods for 
 this projection, — the tangent and the secant. The latter being 
 the more accurate, is the one we will illustrate. 
 
 In Fig. 27, let ABC be the arc of latitude to be embraced by 
 the map. Set off from each end of the arc of latitude, one fourth 
 of its length, to a and a^, and through these points draw the 
 secant intersecting the earth's axis produced in D. Then, the 
 cone, of which aa^ D is an element, will, when rolled out or 
 
A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 developed, be a portion of a circle and the radials through it, and 
 of which the secant aa^ D is one, will represent the meridians of 
 the sphere, and circles described through it from D as a center, 
 will be the parallels of latitude. The points whose latitude is a or 
 a^, would be correctly represented in magnitude, while those near 
 A and C would be too large and those near B would be represented 
 too small. 
 
 The following example will illustrate, viz., to construct the lines 
 for a map embracing latitude 40° to 50°, and 75° to 95° of long- 
 itude, i. e., the vicinity of the great lakes. 
 Dividing the difference of latitude, 10°, 
 into four parts ,we find the two parallels, 
 a and ai, to be five degrees apart, and two 
 and one-half degrees from the extremity 
 of the map; and a=42J°, and ai=47J°. 
 
 The length of one degree of longitude 
 on the parallel a 
 
 =cos. 42J (=.7373)X60=44.24 miles, 
 and on a^, 
 
 =cos. 47J {=.6756)X60==40.54 miles, 
 and their distance apart on the meridians, 
 
 ==^X60-=300 miles. 
 And the radius of the developed parallel a, 
 is the cotangent of 42 J° multiplied by the 
 radius of the earth; also that of a^ is co- 
 tangent of 47J° multiplied by the earth's 
 radius. These being too large to be used 
 as sweeps with which to describe the parallels, we must devise 
 some other means of sweeping arcs of circles. 
 
 Fig. 27. 
 
 Draw a line vertically through the middle of the map, to 
 represent the middle meridian. From head of the map, set 
 down 150 miles for the place of the parallel 47J°. Below this, set 
 down 300 miles more for the place of the parallel 42J°, and below 
 that, set down 150 for the parallel 40°, limiting the south side of 
 the map. 
 
 From a, (see Fig. 28) with one or two times the width of one 
 degree of longitude, 44.24 miles, as computed above, accordingly 
 as a meridian is wanted for every alternate degree, describe the 
 arc of a circle, both to right and left. 
 
ORIENTING SHIP. 
 
 69 
 
 In the same manner, from a^, with the same multiple of the 
 width, 40.54 miles, describe an arc on each side of the meridian, 
 and through these arcs draw the meridians, as for 83° and 87° of 
 longitude. 
 
 With twice theii distances, sweep again from same centers, for 
 the meridians 81° and 89°, and three times their distances, sweep 
 arcs for 79° and 91°, etc. 
 
 Produce the two outside meridians of the map till they meet 
 the middle meridian, from which point sweep the parallels of 
 latitude 40°, 42°, 44°, etc. 
 
 The above is the most practical and readily available of the 
 many methods in use for representing areas of several hundred 
 miles square. A more accurate method, called the PolyCOllic 
 Projection, is in use by the U. S. Coast and Geodetic Survey, 
 but as it is somewhat abstruse in its construction, we do not illus- 
 trate it. 
 
 Orieiitiiig" Ship. — Two methods, the Direct and Eeciprocal, of 
 finding compass errors, have been treated, together with the appli- 
 ances for the work. But the subject is of such importance, that 
 we give a number more, in- 
 cluding the Orienting of Ship, 
 for the purpose of compen- 
 sating ship's compass. 
 
 There are many opportun- 
 ities of finding lines of ' 'known 
 bearing," as between inter- 
 visible lights, or between two 
 headlands, or between a light 
 and a headland. Such lines 
 are said to have a port-hand 
 or a starboard-hand bearing, 
 accordingly as the line is to the left hand or to the right of the 
 meridian, as the observer looks to one of the objects. 
 
 Thus, the line Skilligille-Waugoshanee, bears X. 2J p. E. The 
 observer standing in this line and looking either to the north or 
 the south, will see one of the lights to the right of the meridian, 
 whence, the light is said to have a starboard-hand bearing. 
 This notation will be found very convenient for defining the bear- 
 ing of lines. 
 
 IS- </d ^/ 
 
 ?^ ?s ^0 & 79 71 
 Fi^. 28. 
 
70 
 
 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Example: The ship crosses a line known from the chart, to have 
 a port-hand bearing of 2 J p. At the moment of crossing, the line 
 is seen J p. abaft the port beam, as indicated by the dumb card, 
 described on page 57; and the ship's compass indicated ship to be 
 headed E. by N. J N., and the chart gave the variation E. J p. 
 Required whether the compass is in error, which way and how 
 much. 
 
 Solution: Draw two lines, X. S. and E. W., to represent the 
 meridian and the prime vertical, at right angles to each other. 
 Through the intersection of these lines, draw the reference line, 
 to the left of the meridian 2 J p., and draw the magnetic meridian 
 
 J p. to the right of the true mer- 
 idian and the E. W. line. Then, 
 by observation, the ship's head 
 (indicated by the arrow-point) 
 was 8^^ p. to the right of the ref- 
 erence line. From this, subtract 
 2| p., the distance of the mag- 
 netic meridian to the right of the 
 true meridian, and find ship's 
 head to be 5J p. to the right of 
 magnetic meridian, leaving 2 J p. 
 between ship's head and magnetic 
 E.; but the compass shows only 
 IJ p., whence compass is out to 
 left J p. Thus the dumb card 
 gives great facility and clearness 
 to this class of questions. 
 
 Example 2: The line St. Hel- 
 
 ena-McGulpin's Pt,, has a port- 
 
 A steamer, in crossing this line, 
 
 The compass indicated 
 
 Fig. 29. 
 
 liand course of 44J°, or say 4 p. 
 
 observed it 6 J p, off the starboard bow. 
 
 ship's head at S. W. by W., variation J p. to the right. Required 
 
 -compass error, if any. 
 
 Solution: As in the former example, draw the cardinal lines 
 and the line representing the magnetic meridian. Then, by obser- 
 vation, the ship's head was 6J p. to the left of the reference line, 
 or lOJ p. to the left of the N., or 5 J p. to the right of the S. But 
 hy ship's compass, the ship's head was only 5 p. to the right of 
 
ORIENTINCi SHIP. 
 
 71 
 
 S., whence, bv deviation and variation, it is J p. out to right 
 Deducting variation J p., the needle is out to right f p. 
 
 These two problems will show the method of treating this class 
 of questions, and the great utility of the dumb card. 
 
 With the dumb compass, described elsewhere, there would be 
 no computation whatever. The sights being set to the bearing of 
 the reference line and then turned into that line, the true heading 
 of ship is read from the card. 
 
 Orienting Sliip by Amplitudes. — The bearing of the sun at 
 sunrise or sunset, when referred to the prime vertical, i. e., to the 
 E. or W., is called the amplitude of the sun. 
 
 This angle or bearing depends on the latitude of the observer 
 and on the declination of the sun. It is a problem of spher- 
 ical trigonometry, and is solved 
 by the following equation: 
 
 Sin. amplitude^ 
 
 sec. lat.Xsin. dec, or 
 =sin. dec.^-cos, lat. 
 Example: In latitude 43° 
 N., with sun's declination 21"^ 
 ^., what is the bearing of the 
 sun at sunset or sunrise. 
 Solution: 
 
 Sec. L (=43°)=. 1.367 
 
 Sin. D (=21°)= .3584 
 
 5448 
 10936 
 , 6835 
 4101 
 
 Amplitude 
 
 =sin. 29°,- 20^=. 4899308 
 
 Fig. 30. 
 
 As the bearing takes its name from the decimation, the sun will 
 bear E. 29°, 20^ N. at sunrise, and W. 29°, 20^ N. at sunset,— or 
 better, :N. 60°, 40^ E. and N. 60°, 40^ W. 
 
 The dumb compass makes this reference line available for 
 orienting ship, with great facility. Thus, 
 
 1. Set the index to the reading of the amplitude and turn the 
 card under the index, to the right or to the left, by the amount 
 
72 A MANUAL OF NAViaATION FOR THE LAKES. 
 
 of variation for the locality. This reduces the true amplitude to 
 a magnetic amplitude, which is what is wanted when we are look- 
 ing for compass deviation or when we are compensating for devi- 
 ation. . 
 
 An example will best illustrate, though at the risk of some rep- 
 etition: 
 
 It IS required to orient ship at Buffalo. Lat. 43° X., sun's dec. 
 
 Solution: From the table of amplitudes (table IV), select the 
 amplitude 25°. corresponding to the given latitude and declina- 
 ation. The dumb compass being in position, with the zero line 
 of the lower index, or lubber line, set toward the head of the ves- 
 sel, and the north side of the card approximately to the north, set 
 the index of the right bar to read the amplitude, viz., E. 25° N. 
 for sunrise, or W. 25° X. for sunset. 
 
 Turn the card under the index 5° to the left. Clamp the index 
 and card together and sight to the sun. Then is the card oriented 
 to the magnetic meridian. 
 
 Hold the card and sights pointing to the sun and give ship the 
 ^ 'wheel'' till ship's head is at the desired cardinal point, as indi- 
 cated by the reading of the lower index. Then will ship be 
 oriented to the magnetic meridian. 
 
 The two preceding methods of finding true bearing of ship's 
 course, are eminently practical and of easy attainment, readily 
 available and requiring but little or no mathematical work; but 
 each has its inconvenience. 
 
 In the first, the ship must recover her place in the reference 
 line after each observation, in order to avoid parallax. 
 
 In the second, by amplitudes, according to the usual practice, 
 the time for work is too limited. The sun soon leaves the place 
 defined by the amplitude, when observations for compass error 
 . must cease. But there is a property in the rate of the change of 
 the sun's place, by which this inconvenience may be avoided, at 
 least for some time, and that is, 
 
 A Constancy in the Rate of Change of the Sun's Azimuth. 
 
 It is found that, with latitude and declination of the same name. 
 
ORIENTING SHIP. 73 
 
 the rate of change of the sun's azimuth, between sunrise and the 
 time of his crossing the prime vertical, is practically constant for 
 any latitude above 40". Thus, Avith latitude 37°, declination 0°, 
 the rate of change is one degree in 6 minutes, 40 seconds. With 
 declination 12°, the rate is 1 degree in 6 minutes, 20 seconds. 
 Again, in latitude 45°, with declination 0°, the change of the sun's 
 azimuth is 1 degree in 5 minutes, 50 seconds. With declination 
 12°, it is 1 degree in 5 minutes, 30 seconds, and with declination 
 20°, it is 1 degree in 5 minutes, 30 seconds. And this rate will 
 hold for a few minutes after passing the prime vertical, after which 
 it increases rapidly. 
 
 This property in the rate of the change of the sun's azimuth, 
 between sunrise and the prime vertical, or between the prime 
 vertical and sunset, makes it easy to construct a table that is in 
 all respects a veritable table of the sun's time azimuth. 
 
 To facilitate this work, I have computed the rate of change for 
 each latitude, and written it immediately under the latitude of 
 each of the columns of table lY. 
 
 An example wdll best illustrate. With latitude 43° and declin- 
 ation 15°, alike, the amplitude 20°, 44^, — say 21°. Observe the 
 w^atch time of the sun's rising and change this amplitude to azi- 
 muth by taking the compliment. Then, say at 5 A. M., the sun 
 rises IS". 69° E.; at 5:06 A. m., lie is at N. 70° E.; at 5:12 a. m., he 
 is x^. 71° E., etc., till we get 90° of azimuth, when the sun is ia 
 the prime vertical and the rate of change begins to increase. The 
 evening w^ould do as well, but then the observer would have to 
 know the time of sunset and the error of his watch on apparent 
 time. — things not always known. 
 
 By Time Azimuths of the Sun, — The relations of latitude and 
 declination, and local apparent time, are such that these elements 
 being known, the bearing of the sun can be readily deduced. 
 These bearings are called Time Azimuths. 
 
 They are deemed of such importance at sea that tables of them 
 for all declinations up to 80°, and for time up to the length of the 
 longest day, have been computed. Among the best of these azi- 
 muth tables, are those of Major General E. Shortrede, F.K.A.S., 
 a part of which we give in this work, with explanations for their 
 use. (See table VIj. 
 
74 A MANUAL. OF NAVIGATION FOR THE LAKES. 
 
 These tables require a precise knowledge of the local apparent 
 time, and, as a consequence, at sea, require daily observations for 
 that element. 
 
 But with us on the lakes, or along the coast, now that we have 
 Standard time established, and the latitude and longitude of all 
 the lights along the coast and lakes being known, the case is very 
 different. The astronomical work required at sea for local ap- 
 parent time, is dispensed with on the lakes, — for the reason that 
 it is done for all time, and to a degree of precision that we could 
 not hope to attain, — in the IT. S. List of Lights. 
 
 This with the equation of time as furnished in the Nautical 
 Almanac, and our Standard or Mean Time, which is now 
 available in nearly every R. R. Station, gives us the means of 
 knowing the apparent time at any locality, with great precision, 
 whence the Azimuth tables are available to us without the 
 astronomical instruments, and without the nautical service, 
 requisite for their use at sea. 
 
 Local Apparent Time. — As our watches are supposed to be 
 regulated to Standard or Mean time for some particular meridian, 
 and as we want the local apparent time, it is necessary to find the 
 error of our watch on such time. 
 
 And as we are to find this error from the difference of longitude 
 between the meridian of the observer, and that to which our 
 watch is regulated, the reduction of arc or longitude to time, and 
 the reverse, will be involved in the question. (See table VIII.) 
 
 This reduction of Arc to Time, and the reverse is made by 
 means of margined tables to the tables of natural Sines Tangents, 
 etc. which see. The outer column on the left contains the time 
 for the degrees at the top of the page, and the minutes of arc 
 on the left. (See large works on Navigation.) 
 
 The outer column on the right, gives the time for the degrees 
 at the foot of the page, and the minutee of arc on the right. 
 
 In the time columns the hours under 3, will be found at the 
 head of the column. The minutes in black faced figures, and 
 the seconds in common figures in the column. 
 
 The hours from 3 to 6 will be found at the foot of the column. 
 
ORIENTING SHIP. /O 
 
 Thus, for 34°, 28^ of arc, the corresponding time=2 h. 17 m. 52 s. 
 Thus, for 76°, 16^ of arc, the corresponding time=5 h. 5 m. 4 s. 
 
 The first step in this problem, is to find that part of the error 
 of our watch due to difference of longitude. 
 
 This is found by taking the difference of longitude of different 
 places, as given in the List of Lights, on the lakes, and reducing 
 the same to time at the rate of one hour for 15° of longitude, one 
 minute of time for 15^ of longitude, etc. or 4 m. to 1°, 
 
 Thus the longitude of Chicago is given as 87°, 37^, while the 
 longitude of the standard time Avatch is 90°, — the difference of 
 longitude is 2°, 23^. This, reduced to time by table XIII, is 9 m, 
 32 s. That is to say the difference between standard time, and 
 the local mean time of Chicago, is 9 m. 32 s. 
 
 And because the meridian of the standard time 90°, is West oi' 
 that of Chicago, the watch is slow, and requires the {-\-,) plus 
 correction to reduce it to the local mean time of Chicago. 
 
 Again, — the longitude of Buffalo is given as 78°, 54^, while the 
 longitude of the standard time watch, — to which Buffalo mean 
 time is referred, is 75°, — and their difference is 3° 54^, which 
 reduced to time, is 15 m. 36 s. But in this case, the standard 
 meridian is East of the local meridian, — whence the time correc- 
 tion must be subtractive to reduce the standard to the local mean 
 time, — and the minus sign is prefixed to the time correction. In 
 this manner was table VII prepared for most places on the lakes. 
 
 Having found the error of our standard time watch on the 
 mean time of any locality, it remains to find the error on ''Appar- 
 ent Time," i. e., the error on the time indicated by the sun. 
 
 Apparent time does not progress uniformly as does mean time. 
 As a consequence, it is sometimes fast and sometimes slow of the 
 mean time clock. Just how much fast or slow is shown by the 
 Nautical Almanac, under the heading "Equation of Time," tech- 
 nically called the Eq. of time. 
 
 The sign prefixed to this correction of time, — or equation of 
 time, — or eq. of time, — is that for reducing apparent time to mean 
 time. Thus, when the sun is slow, the eq. of time has the + sign 
 prefixed to reduce apparent time to mean time, and when he is 
 fast, the — sign is applied. 
 
76 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 But in our case, we wish to reduce our mean time to apparent, 
 i. e., we wish to find the error of our mean local time on apparent 
 local time, whence the above signs must be reversed. Accord- 
 ingly, in table V, we have prefixed to the time correction, or eq. 
 of time, the sign requisite to reduce mean time to apparent time, 
 whence, 
 
 To find the error of the standard time watch on local apparent 
 time, we have only to take the algebraic sum of the two errors, or 
 corrections of tables V and VTI, observing the signs. 
 
 A few examples will illustrate this problem: 
 
 1. At Chicago, June loth (no matter what year), required the 
 error of the standard time watch on local apparent time, for the 
 purpose of finding azimuth from the sun. 
 
 Solution : 
 
 Time correction for diff. long, (table VII), + 9 m., 32 s. 
 Eq. of time (table Y), - - — 10 s. 
 
 Error of watch on apparent time, - + 9 m., 22 s. 
 
 That is to say, we must count the time forward 9 m., 22 s., on the 
 watch, to get apparent local time. 
 
 2. At Chicago, November 1, required the error of standard 
 time on local apparent time. 
 Solution: 
 
 Time correction for diff. of long, (table YII), + 9m., 32s. 
 Eq. of time (table Y), - - - +16m., 20s. 
 
 Error of watch on local apparent time, -^25m., 52s. 
 
 As before, we must read our watch forward, say 26 m., to get local 
 apparent time. 
 
 3. At Cleveland, May 24, required the error of the standard 
 time watch on local apparent time. 
 Solution: 
 
 Time correction for diff. long, (table YII), — 26 m.,48 s. 
 
 Eq. of time (table Y), - - + 3 m., 26 s. 
 
 Error of watch on local apparent time, — 23 m., 22 s. 
 Here we must count backwards on our watch to get local appar- 
 ent time, 23 m., 22 s. 
 
 It will be observed we have not given the year of the date in 
 our examples. This is not necessary for purposes of azimuth, for 
 
ORIENTING SHIP. 77 
 
 the corrections being the mean corrections, — equations, as they 
 are called, — for four years, the error cannot be more than the 
 change of declination for J day, and cannot, therefore, appreci- 
 ably vitiate the resulting azimuth. 
 
 Explanation of the Tables of Time Azimuths. — The tables 
 are arranged with the declinations in a column on the left of the 
 page, and azimuths in a line at the head of the column, the hour 
 anole corresponding to any declination and azimuth, being found 
 in the line of the one and column of the other. 
 
 The declination 0, is written in the middle of the column, and 
 it increases upwards and downwards to 24°, varying by intervals 
 of 2°. The upper part of the column is marked + to correspond 
 Avith latitude of the same name; the lower part is marked — to 
 correspond with latitude of different name. 
 
 Definite azimuths are assumed, and the corresponding H. A. 
 (hour angles) are computed for each degree of latitude and for 
 each second degree of declination, and written in columns under 
 their respective azimuths (Z), and in lines opposite their respect- 
 ive declinations, the degrees of the latitude being written at the 
 head of the page. 
 
 The table commenees with the sun at apparent noon, giving on 
 the first page, asknuths (Z) for intervals of 5°, and on the second 
 page, at intervals of 3°. 
 
 Azimuths, as well as declinations, might have been given at 
 smaller intervals, but these intervals were assumed to keep the 
 tables from being too voluminous, while they are yet small enough 
 to m.ake interpolation easy. The folloA\dng definitions will help 
 to a more ready apprehension of the manner of using the tables : 
 
 Upper and Lower Pole. — The upper pole is that one having 
 the same name as the latitude of the observer. Thus, with us, the 
 north pole is the upper pole, while the south is the lower pole. 
 
 North and South Meridian. — That part of the observer's 
 meridian between himself and the north pole, is called the north 
 meridian, while that part between himself and the south pole, is 
 called the south meridian. The meridian opposite in longitude, 
 is called the nether meridian. 
 
 Supplement of H. A., is what the hour angle lacks of 12 hours, 
 or 180°. 
 
78 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 P. M. and A. M, — In popular language, these initials iraplv the 
 after part or the fore part of the day. But in this question, A.M. 
 implies that part of the forenoon within 90° of the meridian; 
 while P. M. implies that part of the afternoon within 90° of the 
 meridian. 
 
 Morning', is that part of the forenoon between sunrise and the 
 prime vertical, or 90° from no on; and Evening is that part of the 
 day between the prime vertical and sunset. These terms do not 
 occur, except with latitude and declination of the same name. 
 
 Azimuths are reckoned from that pole that is on the same side 
 of the prime vertical as the sun. Thus, with the stin in N. declin- 
 ation, we reckon azimuth from the north, morning* and evening^ 
 i. e., before the sun passes the prime vertical in the morning, and 
 after he passes it in the P. M. 
 
 Zenith and ]?fadir. — Zenith is the pole of the observer's hori- 
 zon, while Xadir is the pole of the nether horizon. So that when 
 the sun is more than six hours from the meridian of the observer, 
 it is less than six hours from the nether meridian, and conse- 
 quently above the nether horizon; and the supplement for the H. 
 A. for the evening is the H. A. for the morning. 
 
 As the terms latitude, azimuth, declination and H. A , do not, in 
 themselves, show immediately when the sun is above or below the 
 horizon, this information is given by the l)Iack line in the several 
 columns. The terms below the black line belong to the sun when 
 below the horizon, and as a consequence, to the evening and the 
 morning. 
 
 To Use the Tables. — The tables answer such questions as the 
 following : 
 
 At what time in the day will the sun be at a given azimuth Z 
 from the observer ? Or, at a given time in the day, what is the 
 bearing of the sun? In reply to the latter question, we will say : 
 
 If the latitude and declination are of the same name, i. e., both 
 I^. or both S.. find the declination in the upper part of the page, 
 on the left, with the latitude at the head of the page. Then fol- 
 low the line of the declination till the H. A., or time nearest to it, 
 is found. Then, over the column containing the H. A. thus found, 
 will be the azimuth Z. 
 
 1. Thus, in latitude 43, with sun in declination 20°, both N., 
 what will be the bearing Z of the sun at 34 m. P. M.? 
 
ORIENTING SHIP. /^ 
 
 The latitude and declination being of the same name, we l,ook 
 for (he declination on the upper left of the page, with the given 
 lat tude at the head. Then, following the line till we find the 
 given H. A., 34 m., and looking to the head of the column, we 
 find Z=20°. That is to say, the sun will then bear S. 20° W. Or 
 at 11 h., 2G m. A. M., the sun would bear S. 20° E. 
 
 2. In latitude 46°, with declination 18°, both N., what will be 
 the bearing of the sun at 7 o'clock in the morning? 
 
 Taking up the declination in the upper left of the page and fol- 
 lowing the line of H. A. to the right, we find that the sun does 
 not cross the prime vertical till it is 4 h., 56 m., from noon, so the 
 sun is still on the morning side of the prime vertical. Looking ia 
 the line of 16° declination, in the lower part of the page, we find 
 at 7 h,, 04 m., the sun is in the prime vertical, i. e., due east. 
 
 3. What will be the bearing of the sun at 8 o'clock A. M., at 
 10 and 11, apparent time? 
 
 As the H. A. in these are the times from noon, we must look 
 for the supplement of A. M. time. Thus, at H, A. 4 h., 08 m., 
 
 Z=81°; for H. A. 2 h., 01 m., Z=50°; for H. A. 1 h., 06 m., Z=rrr30°. 
 
 With the preceding general information concerning the table, 
 
 the following more concise instructions, adapted to our particular 
 
 latitude, viz,, the coast and lakes of the U, S., will be permissable: 
 
 To And the H. A. for Morning', with the sun in X. declina- 
 tion. Look in line of the declination for the given latitude, and 
 under the black line, where will be found the H. A. from sunrise 
 to the prime meridian, with the corresponding Z at the head of 
 the column. 
 
 The H. A. will be found increasing, as also the Z, from left to 
 right, till the sun crosses the prime vertical. At the crossing, the 
 morning changes to A. M., and the supplement of the clock time 
 becomes the H. A. which is found in the line of the declination 
 in the upper part of the page, then 
 
 The A. M. H. A's will be found in the upper part of the page, 
 with the corresponding Z at the head of the column, both the Z 
 and TLA. diminishing from right to left, from prime vertical to 
 noon, where they vanish. 
 
 KoTE.— It must not be forgotten that while the sun is north of 
 the prime vertical, the Z must be measured from the north mer- 
 
80 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 idian, or, if read from the south meridian, the Z will be found in 
 the corresponding column at foot of page. 
 
 Thus, for latitude 43° N.. declination 20° N., in the morning at 
 4 h., 46 m., apparent time, the sun is 63° from the north meridian, 
 i. e., its Z is N. 63° E. Ail h., 32 m., it is due east, i. e., it is in 
 the prime vertical, and its Z is 90°, as seen both at the foot and 
 the head of the right hand column. The supplement of this time, 
 or 4 h., 28 m., shows the same thing, in the same column, in the 
 upper part of the page, in line of the given declination, 20°. 
 
 With the Sun in South Declination, we shall never have use 
 for the H. A's under the black line. 
 
 The supplement of the clock time will be the H. A. from sun- 
 rise to noon, where it is 0. 
 
 These commence in the line of the given declination, imme- 
 diately to the left of the black line, and diminish to the left, — as 
 do also the Z's. 
 
 In the p. 31., the clock time is the H. A., which will increase 
 from left to right, — from noon to sunset, — which will be found at 
 the black line or immediately above it. 
 
 Example: Thus, with latitude 43° N. and declination 8° S. the 
 last P. M. H. A. given is 5 h., 24 m., with corresponding Z=78°. 
 The H. A. and Z would be the same for A. M., but the clock time 
 would be the supplement of the H. A., or 6 h., 36 m. 
 
 Interpolation, — We may want Z for intervals, or at times 
 intermediate between those given in the table. These are easily 
 obtained. 
 
 The difference between two consecutive H. A. in the same line, 
 is the difference or interval between the two corresponding Z's. 
 This, divided by the difference of the Z's, gives the time due to a 
 change of 1° in the Z. 
 
 Thus, for latitude 36°, declination 20°, both + the H. A. for 
 Z=60, is 1 h., 38 m. For Z=63°, the H. A. 1 h.^47 m., i. e., in 
 9 m. Z has changed 3°, or 1° in 3 m. 
 
 "Whence, having found the H. A. for Z, we may set the index 
 of our dumb compass, or azimuth ring forward 1 degree for each 
 3 minutes, or for each 4 minutes, or 5 minutes, as the case may be. 
 By forward, is meant the motion from left to right, as we read the 
 dial of a watch, — supposing the eye of observer at the center of 
 card. 
 
 Interpolation for Declination. — This is done with the same 
 facility. Thus in latitude 43°, declination 19°, with names alike, 
 
ORIENTING SHIP. 81 
 
 or +. The H. A. for 60° azimuth is required. Here, the H. A. 
 for declination 20° is 2 h., 10 m. For declination 22°, the H. A. 
 is 2 h., 02 m. The difference is 8 m., whence the. H. A. corres- 
 ponding to declination i9° is 2 h., 6 m., for Z:==60°. This can all 
 be done mentally. • 
 
 The rate of change of azimuth is not alike in all parts of the 
 day. It is slowest at the prime vertical and most rapid at the 
 meridian, as before shown. 
 
 Preparation in Advance. — The ship-master or compass- 
 adjuster who contemplates to swing ship for compass errors, par- 
 ticularly in the A. M. part of the day, should find the error of his 
 watch on local apparent time in advance. Take the supplement 
 of the H. A. in advance and reduce them to his Avatch time, — 
 thereby saving time by being ready for any favorable moment, 
 and avoiding liability to mistake by not having too many things 
 on the mind at once. 
 
 A convenient way of interpolating is to find the rate at the azi- 
 muth, Z changes, between two consecutive H. A., as given in the 
 tables, precisely as given in the methods by amplitudes (page 71). 
 Thus, with latitude 43° N., declination 18° N., the 
 H. A. for Z=66° is 2 h., 40 m. 
 '' '' ^'=69° is 2 h., 52 m. 
 
 3° 12 m., or Z changes 1° in 4 m. 
 
 H.A. for Z=15° is 27 m. 
 '' '' ''=20^ is 37 m. 
 
 b^ 10 m., or Z changes 1° in 2 m., etc. 
 
 The error of the watch on local apparent time being found, and 
 
 the H. A. for any particular Z, or azimuth, being looked out of 
 
 the tables, the work of orienting ship is precisely that explained 
 
 in former articles. 
 
 The method of orienting ship by means of time 
 azimuths will cost the student more study, more diligent and 
 patient application than any other of the several methods enumer- 
 ated for orienting ship, but it is also the most useful, the most 
 satisfactory and the most generally available; and now that we 
 have standard time established, it is peculiarly adapted to use on 
 the lakes and coast, where ii is available without the astronomical 
 work requisite for its use at sea. 
 
CHAPTEK V. 
 
 Terrestrial Magnetism and the Magnetism of Iron in 
 
 Vessels, as Affecting the Compass Needle; and 
 
 THE Correction of Co3ipass Errors. 
 
 Of magnetism we know little more than that it is one of the 
 many imponderable agents by which the material universe is act- 
 uated. Some of the laws governing its mode of action have been 
 discovered by observation. 
 
 Prof. Bartlett, formerly of the Military Academy of West Point, 
 gives us to understand that what is called Terrestrial 3Iagnetisiii 
 is generated by a thermal wave constantly flowing from east to 
 west, caused by the constant change in temperature of that portion 
 of the earth's surface exposed to the sun's rays, together with the 
 axial motion of the earth. 
 
 And electricians tell us that one of the properties of a magnet- 
 ized needle is to take position directly across or at right angles to 
 a current of electricity, when brought in proximity to such current. 
 
 As a consequence, if this current were strictly parallel w^ith the 
 geographical equator, the magnetic needle w^ould coincide with 
 the geographical meridian and there would be no variation, but 
 unfortunately, this is not the case. 
 
 For some reason, supposed to be the nonsymmetry of the topo- 
 graphical conditions contiguous along the track or to inequality 
 of conducting power of different parts of the earth's surface, this 
 current is diverted from a parallelism with the earth's equator, 
 
MAGNETIC EQUATOR. b6 
 
 — resulting in what is called the 'S^ariation" of the needle, — the 
 current being sometimes to the right, and again to the left of the 
 equator, in its motion from east to west, 
 
 A little to the west of Africa, in longitude about 0, this current, 
 or rather the center of this current, crosses the equator, coming 
 from the north and taking a southwesterly direction, touches 
 South America near latitude 18° S. From this point in its west- 
 erly course, it tends gradually northward, reaching the equator 
 near longitude 70° W. from Greenwich, and giving westerly var- 
 iation in the Indian and Atlantic Oceans, Africa, most of Europe, 
 and that part of North America east of the great lakes, wtth a 
 small portion of eastern South America. And easterly variation 
 to the entire Pacilic Ocean, most of the two Americas, and Asia. 
 
 But this terrestrial magnetism is not the only force that is active 
 in giving position to the needle. The magnetism in the iron of 
 the ship is the principal disturbing element that makes trouble tor 
 navigators, with regard to their compasses. 
 
 Now that the practice is becoming general, on the lakes, of 
 compensating ships' compasses for deviation, by the use of oppos- 
 ing magnets, it is important that those who are to do this work 
 should have a little idea of the manner this local magnetism is 
 deduced from terrestrial magnetisms, and of some of the laws of 
 its action. Accordingly I offer the following few elementary facts 
 and definitions : 
 
 Magnetic Equator. — There is a point in the current spoken 
 of, in which the magnetic needle assumes a horizontal position 
 when suspended in the magnetic meridian. To the north of this 
 point the north end of the needle dips, i. e., inclmes downward, 
 while to the south, the south end dips. The line on which the 
 needle is horizontal is called the magnetic equator, inconsequence 
 of proximity to, and analogy to the geographical equator. The 
 Dip is the angle of inclination to the plane of the horizon and 
 increases from the magnetic equator to the magnetic pole, each 
 way, — but at an irregular rate, — when it is vertical, or 90°. 
 
 Magnetic Latitude, is distance either north or south, defined 
 by the dip of the locality. Thus, along the south shore of Lake 
 Erie, or the south part of Lake Michigan, the north end of the 
 unbalanced magnetized needle would dip about 73°. These local- 
 ities would then be said to be in 73° of magnetic latitude, though 
 their geographic latitude is only about 42° north. 
 
84 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 Line of Force. — The opposite poles of a magnet being at its 
 extremities, the direction which the needle takes shows the direc- 
 tion of the action of the forces residing in the two ends, or oppo- 
 site poles. — such direction being called the line of force. 
 
 It must not be supposed that, as between the forces of the two 
 opposite poles, there is anj force of translation, but merely that 
 of direction. The maximum intensity of magnetism in any locality 
 being in this line of force. 
 
 Neutral Plane. — If a plane be conceived to intersect the line 
 of force at right angles, it will show the position of minimum 
 force for that locality, or rather the position of no magnetic force. 
 It is in this plane that magnetic forces change sign in passing from 
 one ])ole to the other, — whence it has come to be called the neu- 
 tral ])lane. 
 
 If any object susceptible to the earth's influence, as a bar of 
 soft iron, make any angle with the plane, the intensity of the 
 magnetic action will be proportioned to the sine of this angle. 
 Thus, at the points before mentioned, in latitude 43°, if a bar 
 of soft iron be laid in a horizontal position and in the plane of 
 the magnet!,, meridian, it will be 17° from the neutral plane. 
 Then as the sine of 17° is .2923, it will be acted on by less than .3 
 of the earth's magnetism at that place. 
 
 If the rod is up-ended to the vertical, it will be 73° from the 
 plane and will therefore be acted upon by .956 of the earth's 
 magnet force, — this being the sine of 73°. While if the rod be 
 placed horizontally in an east and west line, it will be in the neu- 
 tral plane, and therefore will not be acted on at all. This variable 
 intensity of the earth's magnetic action makes great trouble for 
 navigators. 
 
 Components of Magnetic Force. — The needles in use in all 
 compasses are horizontal, while, as we have seen, the earth's 
 magnetic force acts in an inclined direction. It is therefore 
 necessary to regard the earth's magnetic force as the resultant of 
 two other forces, — one horizontal and one perpendicular, or 
 rather, vertical. 
 
 The Horizontal Component, is that part of the earth's mag- 
 netic force that orients the needle, while the vertical component 
 acts constantly in a vertical direction, in the magnetic meridian, 
 — varying the dip with the change of magnetic latitude, and, as 
 we shall see hereafter, indirectly disturbing the needle. 
 
INDUCED MAGNETISM. 85 
 
 The Relatire Magnitude of these two forces is shown on 
 Plate II, Figs, 2 and 3, bv a plan which I have devised from a 
 chart by Mr. R. J. Evans. 
 
 Fig. 2 shows the varying intensity of the total force between the 
 magnetic equator and the magnetic pole, for the meridian of about 
 170° W. and passing near Behring's Strait, — that at the equator 
 being unity. 
 
 It is known to be about 2.3 times as great at the magnetic pole 
 as at the equator, — and between these limits it has been found to 
 vary nearly as the square root of 1 plus three times the square of 
 the sine of the magnetic latitude. 
 
 The curve bounding the section of Fig. 2 is formed by the pre- 
 ceding law, and shows that the intensity of the earth's magnetism 
 increases faster on first leaving the equator than when nearing the 
 pole. 
 
 Fig. 3 represents the corresponding horizontal component. 
 This is a maximum at the equator, where the whole force is hor- 
 izontal. At the magnetic pole, where the line of force is vertical, 
 this horizontal component vanishes. Between these limits it varies 
 as the total force at the locality multiplied by the cosine of the 
 dip. 
 
 The curve bounding the section of Fig. 3, is made by the above 
 proportion, — the result showing that for some distance from the 
 equator, toward the pole, the horizontal or directive component 
 holds its own with but little diminution, after which it falls off 
 rapidly on nearing the pole, where it vanishes. 
 
 Those who have read Dr. Kane's report of his expedition in the 
 Arctic Ocean, will remember the trouble he had with his compass 
 needle, for want of directive force. 
 
 Fig. 4 represents the Vertical Component. This is nothing 
 at the magnetic equator, where the needle is horizontal, and has 
 its maximum, — which is the total force, — at the magnetic pole, 
 where the unbalanced magnetic neeedle stands vertical. Between 
 these limits it varies as the total force of any locality multiplied 
 by the sine of the dip for that locality. 
 
 Induced Magnetism. — In consequence of the constant stream 
 of magnetism flowing around the earth, all fuliginous matter 
 within its influence becomes charged with magnetism by induction. 
 Whence, in our (north) latitude, the lower portions of all masses 
 of iron, such as columns, vertical shafts, smoke funnels, vertical 
 
S6 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 rods of any kind, — being below the neutral plane, — are on the 
 polar side of that plane and therefore have the same magnetism 
 that is in the north end of the needle. While the upper parts 
 being on the equatorial side of this plane, have that kind of mag- 
 netism that is in the south end of the needle. 
 
 Hard and Soft Iron. — The influence of this induced magnetism 
 on different kinds of iron, is seen in different effects on the com- 
 pass needle. 
 
 Hard Iron is slow to receive magnetism by induction, and as 
 slow to part with it. If a bar of hard iron, having its longer 
 dimension approximately parallel with the ^Mine of force," be 
 violently treated, as by bending, twisting, chipping, filing, or in 
 putting lathe-work upon it, its magnetism will not only be 
 increased, but comparatively fixed. 
 
 Soft Iron on the contrary, becomes magnetic almost imme- 
 diately when its longer dimension is placed in a position approx- 
 imately parallel with the *^ line of force." It also has its intensity 
 increased and comparatively fixed by violent treatment. 
 
 Fixed or Permanent Magnetism, is that which remains con- 
 stant, or practically so, in the body to which it pertains, as in a 
 hardened steel magnet, or in the so-called loadstone. Magnetism 
 in vessels is not permanent. Hard iron retains its magnetism some 
 time, but not so permanently as steel. 
 
 The plates of vessels, boilers, smoke funnels, tubes, etc., being 
 worked cold, then hammered in riveting and caulking, have their 
 magnetism partially fixed, producing a magnetism called sub- 
 permanent, i. e., composed partly of permanent and partly of 
 transient magnetism. This transient part uf mechanism passes 
 away rapidly from a new vessel, by use, making much trouble for 
 the navigator. 
 
 Different Effects of the Different Magnetisms.— The differ- 
 ence m the effects produced by a fixed magnet, as that of a bar of 
 hard steel magnetized, and that produced by a bar of soft iron 
 that is made magnetic by induction, is seen in the following 
 experiment: 
 
 Apply the end of a soft iron rod to a compass needle, the rod 
 being in a level position, at the same height of the compass and 
 in an east and west direction. The rod, in such position, being 
 in the neutral plane, is not acted upon by the earth's magnetism, 
 and as a consequence, cannot affect the needle. !Now, up-end the 
 
INDUCED MAGNETISM. 87 
 
 rod to a vertical position, keeping one end,— say the lower end, — 
 on a level with the compass. The lower end of the rod, in this 
 position, is below the neutral plane, and therefore is charged by 
 induction with the magnetism that is in the north end of the 
 needle, as will be shown by its repelling the north end of the 
 needle. 
 
 If now, we lower the rod away, canting it till the upper end 
 comes below the level of the compass, — still keeping it vertical, — 
 we shall bring the south magnetism of the rod to act on the 
 needle, which will be shown by its attraction of the needle, thus 
 giving the opposite result, and showing that it is not any pecu- 
 liarity of the rod that produces this change, but the almost 
 instantaneous effects of the earth's magnetism on the rod. 
 
 Effect of Direction of Ship's Head, while Building". — It has 
 been seen from observation that iron ships receive a permanent 
 magnetic character from the direction in which the head is located 
 during the building. In a vessel built with head to the magnetic 
 north, the magnetic meridian would coincide with the ship's plane 
 symmetry, — or with her center line, as it is called. 
 
 The forward lower part of the vessel, being below the neutral 
 plane (in north latitude), would have that kind of magnetism that 
 is in the north end of the needle, while the after end of the ves- 
 sel would have south magnetism, — and so for other directions of 
 the ship's head. 
 
 Semi-circular Deviation. — In swinging ship, the fixed mag- 
 netism of the head iron retains its relative position with the other 
 forces and is therefore on one side of the compass needle in one- 
 half of the revolution, and on the other side during the other 
 half, thereby producing deviation through 180° of azimuth on 
 each side of the magnetic meridian, — whence the name semi- 
 circular deviation. 
 
 The deviation due to the effects of hard iron, is far from being 
 constant. It falls off rapidly with the use of the vessel during the 
 first few months of service. As a consequence, a new iron ship 
 cannot be depended on lo carry her adjustments long, without 
 some change. 
 
 The magnetism resulting from vertical component, being of 
 constant sign and constant intensity in the same latitude, also 
 produces a semi-circular deviation. And this is always combined 
 with that produced by the hard iron, and is made less in amount 
 than the latter. 
 
88 A MANUAI. OF NAVIGATIOX FOR THE LAKES. 
 
 These two parts, which always come together by addition, have 
 different properties. 
 
 That depending on hard iron is not constant in its intensity, 
 because of the inability of the iron to hold its magnetism perman- 
 ently. Hence the term, ^'sub-permanent magnetism." While 
 that depending on the vertical component of induced magnetism, 
 is constant in the same latitude. 
 
 Quadrant al Deyiatioii, is the result of magnetic action 
 induced by the horizontal component in the soft iron in the ship 
 and it is constant in amount, or nearly so, — being proportional to 
 horizontal component of the locality. And, as will be presently 
 seen, it changes sign in each quadrant. Commencing in the first 
 quadrant, N. to E., the signs are -j-, — , -f-j — j whence the name 
 **quadrantal deviation." 
 
 Graphical Illustration. — The several parts of which devia- 
 tion is composed, are illustrated by Fig. 1, of Plate II, 
 
 The two segmental areas, 4-B, — B, represent that part of the 
 semi -circular deviation due to hard iron, and which is generally 
 the larger part. 
 
 The curve bounding this arc, is a curye of sines, i. e., if the 
 deviation at east or west be multiplied by the sine of the azimuth 
 of any point in the curve, measured from the Kode of the curve, 
 or the place where the curve crosses the magnetic meridian, the 
 product w^ill be the deviation at that point. 
 
 Thus, if a deviation at E. be 20°, that for X. N. E. would be 
 (see table III) 
 
 sin. 2 p. (=r.384)X20°= 7.7^; and for X. E. 
 would be sin. 4 p. (==.707)X20°=14.1°; and for E. N. E. 
 would be sin. 6 p. (-=.924)X20°=18.5°. 
 
 These distances set off from the magnetic mei^idian, will give 
 points in the curve for the semi-circular deviation. 
 
 The Lnnular Seg-ments, +C, — C, show the part of the semi- 
 circular deviation due to the effects of the vertical component. 
 The boundry of this area is also a curve of sines. 
 
 The four lunular segments, -j-D, — D, +D, — D, show how 
 quadrantal deviation is combined with semi-circular deviation to 
 produce the total error of the compass. These, also, are curves 
 of sines. 
 
 The Law of Change of Signs will be seen from the following 
 considerations: 
 
QUADKANTAL DEVIATION. 89 
 
 When the head of the ship is to the magnetic north, the mag- 
 netism of the ship and that of the earth conspire in one direction 
 on the needle, so that, though the directive force be increased, 
 there is no deviation, — whence the magnetic north is a nodal 
 point of this curve, or nearly so. 
 
 When the ship's head is east magnetic, the horizontal com- 
 ponent is in the neutral plane and thus does not act, and as a con- 
 sequence, there is no deviation, — whence, the east is also a nodal 
 point for this curve. 
 
 In the same manner the south and west may be shown to be 
 nodal points. 
 
 The several parts of deviation above enumerated, are called 
 *'Co-efflcieiits." They are A, B, C, D, E^, of which we have 
 shown B, C and D. 
 
 *'A" is that part of compass error due to what may be called 
 Index error, as when the zero line of compass is not parallel with 
 center line of ship or when the card is not cemented onto the 
 needle in the proper place, — and is usually too small to notice. 
 
 **E" is a small error remaining in the octants after the quad- 
 ranted deviation D, has been corrected. It is seldom seen except 
 with very long compass needles, and is generally so small as not 
 to be worth noticing. 
 
 The preceding illustrations are on the supposition that the sev- 
 eral causes of disturbance are symmetrically arranged to each 
 other on the two sides of the ship's plane of symmetry, called also 
 her ''midship plane;" and that the vertical magnetic plane that 
 coincided with the plane of the magnetic meridian during the 
 building of ship, coincides also with this 'midship plane, — and 
 that the ship's compass is in this plane. But all these conditions 
 seldom or never prevail at the same time. 
 
 And this trouble is aggravated by the habit of masters and 
 builders of placing their compasses to one side of the center line 
 of ship, 
 
 The result is, these several parts of deviation, which always 
 come together by addition, algebraically, are shifted in azimuth 
 with regard to each other, so that the nodes of the curve are not 
 at the magnetic cardinal points, as they otherwise would be, 
 nearly. 
 
 This is illustrated in the deviation curves of the ''Trident" and 
 "Warrior," Plate I. 
 
90 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 In the case of the ''Trident,'^ it will be seen that the curve 
 crosses the magnetic meridian at N. J E. and at S. ^ W. — thus 
 making one semi-circle f ])oint longer than the other. 
 
 The curve of the ^' Warrior," crosses the meridian at N. 3 p. W. 
 and at S. f p. E., thus making the distance between nodes, 2f p. 
 greater on one side than on the other; — which makes it impossible 
 to bring the needle to place at both North and South or East and 
 West as the case might be, by means of opposing magnets. And 
 this often brings the compass adjuster into disrepute by people 
 who are ignorant of the above facts, and who think that because 
 he cannot get good ** reversals" in all cases, that therefore he 
 does not know his business. 
 
 A Yivid llliistratioii of the effect of combining quadrantal^ 
 with semi-circular deviation, is seen also in the deviation curves 
 of the "Trident" and '^Warrior" above referred to. 
 
 It will be remembered that the signs of B and C, when west, 
 are — , and that the sign of D is +, — , +, — , commencing in 
 the first quadrant, and going around by the East, South and 
 West, to the North. 
 
 Constructing the curve bounding the two parts B and C, (as on 
 Page 88), — taking the deviation at West, 23°, as the height of 
 the curve, we find for intervals of 2 p. of azimuth, the height of 
 €urve as follows, viz: (See table III.) 
 
 Sin. 2 p. (=.383) = 8°.8. 
 
 Sin. 4 p. (=.707) X23°=16°.3. 
 Sin. 6 p. (=.924) =21°.3. 
 
 Setting off these distances, by means of the scale at head or 
 foot of the plate, we have the place of the curve shown by the 
 dotted line, thus showing the area embraced by the dotted or 
 broken line, to be reduced in the first quadrant, and making the 
 deviation curve slightly concave at N. E. 
 
 At S. E. the deviation curve is sharply convex, because the 
 two signs being alike ( — ), their parts come together by addition. 
 
 At S. W. the two signs being alike (+), show the deviation 
 curve to be sharply convex. 
 
 At the N. W. the two signs being unlike, (Xj — )> the two parts 
 come together by subtraction, and we have a lean concave curve, 
 as in the first quadrant. Thus: 
 
 The total deviation in the two northern quadrants, is greater 
 than the deviation in their diagonally opposite quadrants by 
 twice the mean quadrantal deviation of those quadrants. 
 
HEELING DEVIATION. 91 
 
 The above property affords a ready means of separating quad- 
 rantal deviation from the total deviation. Eule: Take half the 
 arithmetical difference of the diagonally opposite quadrantal 
 deviations, for the mean quadrantal deviation. 
 
 And in constructing the curve, observe that the heights, or 
 ordinates of the curve vary as the sine of twice the azimuth of 
 ship's heading, measured from the point midway between the 
 magnetic meridian and the compass m.eridian. 
 
 The Yariation of the Intensity of Magnetism, for distance 
 is known to be Inversely as the Square of the distance. But 
 when two magnetic forces act on each other, the intensity of their 
 mutual or joint force varies Inversely as the third course of 
 their distance. 
 
 Thus — if two magnetic bodies acting on each other at a given 
 distance produce a given deviation, then at twice the distance, 
 the deviation will be only J as much; at three times the distance, 
 it will be only ~ as much; at ten times the distance, ^^ as 
 much, etc. 
 
 This law shows us how a small element of disturbance near by, 
 can make more trouble than a shipload of iron a little ways off. 
 
 Heeling Deviation. — So far, our consideration of compass 
 errors have been considered with regards vessels on an ^^even 
 beam. But there is an error resulting from the lifting or heeling 
 of the vessel, called Heeling Deviation, 
 
 When the vessel is heeled to starboard or port, the vertical 
 longitudinal plane, containing the compass, is shifted to leeward 
 of that containing the general center of magnetic effort of the 
 ship. As a consequence, the relations existing with ship on an 
 even beam are disturbed. 
 
 But, fortunately, our lake vessels have so much more beam for 
 their tonnage and draft of water than sea-going vessels, that this 
 disturbance will now give us much trouble. 
 
 This error is at its maximum when the ship's center line is 
 parallel with the compass needle, as with ship's head IST. or S. 
 and nothing with head E. or W. magnetic. 
 
 Mechanical Correction of Deviation, by Means of Mag- 
 nets. — Because the two parts B and C, of deviation are semi- 
 circular, or nearly so, they may be compensated, or neutralized 
 mechanically by introducing magnets acting in contrary direction. 
 
92 A MANUAI. OF NAVIGATION FOR THE LAKES. 
 
 Having oriented the vessel by any of the methods heretofore 
 explained, with ship's head to any one of the cardinal points, — 
 say to the North, — mark on the deck under the compass, and on 
 the wall of the pilot house, if near, the intersection of the mag- 
 netic meridian plane with them, that passes through the compass. 
 This line will be parallel to ship's center line. 
 
 Also, at right angles to this line, draw the intersection of the 
 transverse vertical plane passing through the centre of compass, — 
 marking the trace of same on the side of pilot house, when 
 near, — showing a possible place for a magnet. 
 
 Place a magnet with its center on the fore-and-aft line of the 
 deck, moving from or towards compass, — changing ends if neces- 
 sary, — till the compass point correctly, — and fasten tempor- 
 arily, keeping the magnet perpendicular to the meridian plane. 
 
 If the deck is too far off from the compass, for the strength of 
 the magnet, apply a larger magnet, or use two magnets, — or 
 fasten one to the side or front of the pilot house. 
 
 Then, ship's head being swung to magnetic East ci AVest, 
 again bring the needle to its normal place, as befor?, — being 
 careful to keep the magnet truly fore-and-aft, with its center on 
 the transverse line on the deck, — and fasten temporarily. 
 
 If, now the fixed magnetism of the ship represented by X^ 
 and — B in Fig. 1, of Plate II, be in its proper place, with its 
 zero points at the North and South, coinciding with the nodal 
 points of the other two parts C and D, then compensation for 
 the semi-circular deviation will be complete, and ship will 
 *' reverse bearings" on the cardinal points, and in the quadrants, 
 except for the error due to quadrantal deviation. 
 
 But this is too often not the case. This part B may be shifted 
 in azimuth with regard to the other parts C and D, so as to give 
 a material error of a J or possibly a whole point on reversal of 
 ship. In this case no amount of "fixing" or "fudging" will 
 avail us. If Ave bring needle to place on one cardinal point, it 
 will be out on the other, — and if we bring it to place on the 
 other, it will be out on the one. 
 
 In this case we can do nothing more in the way of correction, 
 but resort to a Deviation Card. Swing ship again to verify the 
 work, and secure the magnets permanently. 
 
 If the quadrantal deviation (for there is some in all ships), is 
 to be corrected by deviation card, which is the better way, swing 
 
DEVIATION CARD. 
 
 93 
 
 ship again, putting ship's head carefully to all the alternate 
 points of the card, magnetic by means of the desired compass, 
 and note the reading carefully of the ship's compass, and record 
 them. 
 
 If there was no error at the cardinal points, the errors found 
 in the quarters, will rarely exceed a J point. But if these courses 
 at the cardinal letters, are reversed, these errors will affect the 
 •quadrantal errors. 
 
 The following is a deviation card from actual practice, the 
 original error being over 2^ p. 
 
 The first and third columns giving the magnetic course desired; 
 the second column giving the correction, which must be read 
 with the course in the magnetic column, to get the compass 
 course desired. 
 
 Thus, to get S. E. magnetic, we take S. E. J S. per compass; 
 and for IS". W. magnetic, we take N. W. J N. per compass, etc. 
 
 Steering Card. 
 
 Magnetic 
 
 Correc- 
 
 Magnetic 
 
 Correc- 
 
 Course. 
 
 tion. 
 
 Course. 
 
 tion. 
 
 Compass 
 
 Course. 
 
 Compass Course. 
 
 North. 
 
 
 South. 
 
 
 K. N.E. 
 
 
 s. s. w. 
 
 
 N. E. 
 
 
 s. w. 
 
 
 E. N. E. 
 
 
 w. s. w. 
 
 
 East. 
 
 
 West. 
 
 
 E. 8. E. 
 
 J S. 
 
 W.N.W. 
 
 JN. 
 
 S. E. 
 
 i s. 
 
 N. W. 
 
 }N. 
 
 S. S. E. 
 
 i s. 
 
 N. N. W. 
 
 JN. 
 
 South. 
 
 
 North. 
 
 
 Mechanical Correction of Quadrantal Deviation, is made 
 hy means of soft iron, or cast iron. Experience has shown that 
 cylinders of cast iron, 3 to 3i inches in diameter, and 
 9 to 12 inches lon^, with hemispherical ends, and placed on a 
 level with the compass card, and with their ends pointing radially 
 to the center of the card, give the best results. 
 
 Nails and chain in boxes have been used; cast iron balls, also, 
 have been used with satisfactory results. 
 
94 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 The Correction. — The semi-circular deviation having been 
 corrected, set ship's head to one of the inter-cardinal points, — 
 say N. E. — ^^magnetic. 
 
 Place one of the cylinders to the north of the card, on a 'evel 
 with the card, and with end pointing to the center of the 
 compass. This corrector should be directly in line of the needle 
 when brought to its normal place. 
 
 Place another cylinder to the east or Avest side of the compass, 
 in the same manner, as may be necessary to make the compass 
 point correctly. 
 
 Now, keeping the ends of the correctors at the same distance 
 from the card, — move them both outward or inward till the com- 
 pass points correctly, — and the work is done, — secure correctors. 
 
 Theoretically, this correction should be nearly perfect, but 
 practically it may be very imperfect, as the result of a non- 
 symmetrical arrangement of soft iron in the ship. 
 
 Correction of the Heeling Error. — This is made by means 
 of a Vertical Magnet, under the center of the compass. 
 
 Small vessels may be readily heeled 8° to 10°, when a magnet 
 may be applied in a line that would be vertical when ship is on 
 an even beam, — with that end up that brings the needle to 
 place, — and varying the distance, by moving it up or down, till 
 the needle shall point correctly, when it may be secured. 
 
 Large vessels cannot be readily heeled. In this case resort 
 must be had to a Magnetic Survey of the ship. But the dis- 
 cussion of such a survey is beyond our purpose. For this 
 information the student is referred to the Admiralty Manual 
 for 1874. 
 
 Practical Conclusions. — The following are some of conclu- 
 sions drawn from the scientific investigations and long practical 
 experience of Messrs. Smith & Evans, of the Liverpool Compass 
 Committee. 
 
 (I.) All mechanically corrected compasses, should have com- 
 pound needles, — or two parallel needles, whose extremities are 
 60° apart. 
 
 (II.) If single needle compasses are to be corrected, the 
 needles should not be over six or seven inches in length, 
 
 (III.) A correcting magnet on the same level of the compass, 
 should not be nearer to the center of the needle, than six times 
 the leno^th of the needle. 
 
PRACTICAL CONCLUSIONS. 95 
 
 (TV.) In making corrections for quadrantal deviation, the soft 
 iron correctors should not be brought nearer to the center of the 
 compass, than two times the length of the needle. 
 
 (Y.) No compass that is to be corrected by magnets, should be 
 placed where the original deviation is more than two points. 
 
 (A"I.) The compass should not be near either end of an iron 
 ship. And if the decks are of iron, they should be provided 
 with a hatch immediately below the compass, of a width IJ times 
 the height of compass above deck. 
 
 (VII.) If the compass of an iron ship is to be carried amid- 
 ships, the direction of the head of ship, during the building, is 
 not important. 
 
 Remark, — The preceding is but a meager outline of the sub- 
 ject, ''Terrestrial Magnetism," wdth ships, and their compasses, 
 which alone would require a large volume, but it is believed to be 
 enough to give the shipmaster some idea of the forces that 
 disturb his compass, and to put him on his guard in the care of 
 that valuable instrument. 
 
 For further information on this topic, he is referred to 
 *' Magnetism of Ships" and '' Deviation of Compasses," published 
 by the Bureau of Navigation, — Navy Department, 1867. 
 
CHAPTER VI. 
 
 The Propeller Wheel. 
 
 In view of the great importance of the propeller wheel in pro- 
 moting the vast and growing commerce of the lakes, it is desirable 
 that a more general, and in some respects, a more correct know- 
 ledge of that indispensable agent be had by our maritime people. 
 Accordingly I offer the following concerning it. 
 
 Deffinitions. — The Length of the wheel is the distance from 
 the forward edge to the after edge of the blade, measured in a 
 direction parallel with the shaft, and usually is about ~ of the 
 diameter. In the early history of the wheel, it was much longer. 
 
 The Pitch is the distance made, by one revolution of a point, 
 with regard to the shaft, and is usually given in terms of the 
 wheel, — as a pitch of IJ or 1 J diameters. It is sometimes given in 
 feet, but that method gives no idea of the pitch — angle. 
 
 The Net Pitch is the distance made by the vessel, by one 
 revolution of the wheel. This is usually given in feet, for the 
 purpose of deducing the speed of ship from the number of revo- 
 lutions of the wheel, in a given time, though when used for 
 comparing the value of different wheels, it should be given in 
 terms of the diameter. It is the pitch as usually defined, dimin- 
 ished by slip. 
 
 Slip is the difference between the pitch of the wheel, and the 
 distance made by the vessel at one turn of the wheel, and is 
 expressed in feet, or as a per cent of the pitch, as wanted. 
 
 Example. If a wheel of 9 feet diameter have a pitch of IJ 
 diameters, propel a vessel 10 feet with one revolution, it is said to 
 have a pitch of 12 feet, — a net pitch of 10 feet, and slip of 2 feet, 
 or of 2^12=.166, or say ^. 
 
CENTER OF EFFORT. 97 
 
 Negative Slip. — Sometimes the vessel shows a speed greater 
 than that due to the pitch of her wheel. This apparent 
 paradox, — for it is only apparent, — comes from measuring the 
 pitch at the end of blade instead of measuring it at a point called 
 'the Center of Effort, situated about — to — of the length of 
 blade inboard from the end, as will be hereafter explained. 
 
 It is pitiful, yet laughable, to see the absurd theories that have 
 been advanced, even by learned professors, to account for this 
 apparent anomaly. 
 
 True Screw. — In this screw, the working face is a warped 
 surface, generated by a line having two motions, — one angular at 
 a uniform rate, and the other, a motion of translation at a uniform 
 rate, along a right line, and always making the same angle with 
 that line. 
 
 Illustration. — In Fig. 31, let m. n. represent the center line of 
 shaft, or hub of wheel, and let the line n. a. A. represent the 
 generating line. 
 
 Then if this line be moved uniformly along the line n. m., and 
 with a uniform angular motion along A. C. as indicated by the 
 arrow points, it will generate the helicoidal surface, A. c. m. n. 
 forming one surface of the blade of a true screw, — A. C. being 
 the Helix. 
 
 Pitch Angle.— In Fig. 31, let A. B. be the arc of a circle 
 whose plane is at right angles to the line m. n. or shaft, — and the 
 arc A. C. the helix generated by the point A. in its revolution 
 about the line m. n. — then is B. A. C. the pitch angle, and 
 B. C, parallel with and equal to n. m., is the pitch corresponding 
 to Avidth of blade. 
 
 Center of Effort is that point in the length of the blade, 
 having the same amount of work on either side of it, and, in the 
 true screw, is at — of length of blade from center of wheel, — and 
 not at — as told us in engineering books. 
 
 If the wheel were quiessent, supporting a weight uniformly 
 distributed over the disk area of the same, — then the center of 
 effort would be at — of radius from center, as said. But the cir- 
 cular motion throws this center outboard. 
 
 The disc area of the wheel varies as the square of the diameter. 
 Also the intensity of percussion varies as the square of velocity 
 of the several points, in the radius of the wheel. And the 
 amount of work done varies with the product of these two 
 
98 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 factors, — whence the total work up to any point in the radius of 
 the wheel, varies as the fourth power of the distance of such 
 point from the center of wheel. 
 
 As a consequence, the center of effort is at a point equal to the 
 fourth root of J := — of R., tliat is, in the helix a. b. — of rad- 
 ius from n. m. 
 
 The angle which this helix a. b. makes with the plane of the 
 arc A. B. and which is at right angles to the shaft, is the angle 
 that should always be used in any calculations concerning the 
 efficiency of wheels, or the amount of power wanted for 
 propulsion. 
 
 AVhen the pitch is measured at that point, we shall never hear 
 of negative slip. 
 
 Expansion of Pitch, Radial and Axial, is a change of the 
 pitch from that of the true screw. 
 
 If the blade (Fig. 31) be made narrower at the inboard end, by 
 lifting the point n. towards m. the pitch of the blade at that end 
 will be reduced. 
 
 This change is called Radial Expansion, and is introduced by 
 many wheel makers, for the purpose of throwing the work of this 
 end of blade out to where the pitch-angle is smaller, to avoid 
 part of the loss from oblique action. 
 
 But there is a limit to this reduction, it must not exceed the 
 prospective slip. The object is to relieve the inboard end entirely 
 of work, which is attained by making the pitch the same as the 
 net pitch. If this is made less than the net pitch, the blade takes 
 water on the forward face, — doing negative work. 
 
 Axial Expansion, is an increment of pitch from the forward 
 edge to after edge of blade, making the working face concave. 
 
 The object is to take hold of the water gradually, and to in- 
 crease the pressure by increasing the pitch gradually, in conse- 
 quence of which the helix AC. (Fig. 31) is concave towards AB. 
 
 This expansion is all right in the hands of those who know 
 how to use it, for it has its limit, which is the prospective slip. 
 If the expansion exceed this limit the blade takes water on the 
 forward face, doing negative work. Besides, the propelling of 
 ship is done with the after part of blade, where the pitch angle is 
 greater, — and where, consequently loss from oblique action is 
 greater. 
 
KAPIAIi EXPANSION. 
 
 99 
 
 This is probably the most disasterous leak the coal-bunkers of 
 the lakes have ever been called on to pay. This mistake has 
 been general. 
 
 Fortunately, our best wheel makers know better now than to 
 exceed this limit, — while others, perhaps afraid of it, do not give 
 their blades any axial expansion at all, — which is perhaps as 
 well, for the smaller the slip, the more nearly must the blade be 
 a true screw. 
 
 ■yvV 
 
 Fig. 31. 
 
 By means of radial expansion, the center of effort is moved 
 outward from its normal place in the true screw, but just how 
 much, owing to the different ways of doing it, and the different 
 degrees to which it is done, it would be difficult to say. The prob- 
 ability is we may locate the center of effort between — and ^^ of 
 the distance from center to end of blade. 
 
100 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 To Measure the Pitch, is frequently required for the pur- 
 pose of comparing different wheels. It is of importance for 
 every engineer, master and owner of a steamer, to know the 
 pitch of his wheel, so as to know, in the event of a break, how to 
 order a new one. 
 
 This involves a knowledge of the angle BAG. (Fig. 31.) or of 
 the sides of that triangle. 
 
 The angle may be found by applying a carpenter's bevel, so as 
 to have one arm of the bevel on the helix AC. and the other on 
 the plane AB. and both at right angles to the radial at the point 
 of application, as at A. 
 
 The angle being found, construct on paper, as BAG. From 
 any point B. set up the perpendicular BC. Then is BC. the 
 pitch due to the developed helix BG. 
 
 Divide the base AB. by 3.14, then we have the diameter of a 
 circle whose circumference is AB. Divide BG. by the quotient 
 thus found, when we have the pitch in terms of the diameter of 
 the wheel. Example. 
 
 Suppose we find AB. in (Fig. 32.) by scale measure to be 25J 
 inches, and BC. to be 12 inches, then 
 
 25J^3.14=8 and 
 
 12 -^ 8 =1J. That is, our wheel has a 
 
 pitch of IJ diameters. It is the pitch at the end of blade, and is 
 the pitch by which wheels are usually defined. But wheels must 
 be compared by means of the angle at the center of effort, called 
 
 The Working' Pitch Angle, which is always greater than 
 that found at end of blade, by — to —^ depending on the amount 
 of radial expansion. 
 
 This angle is readily found in degrees, — thus 
 
 Divide the ratio of pitch to diameter, as 1, IJ, IJ, etc., by 
 3.1416, and we have the tangent of pitch angle at end of blade. 
 
 Increase this tangent by — or — , as may be required and we 
 have tangent of the Working Pitch Angle. 
 
 Example for a wheel whose pitch is IJ diameters: 
 1.5--3.14163=.4774=Tan. 25.° 31^ 
 which is angle at end of blade. 
 
 Increasing this tangent by its — , we have 
 
 .4774X'-^^^--=.o374=:=Tan. 28.° 08^ 
 for angle at c Alter of effort. 
 
 Or measure by protractor, or scale of chords. 
 
HOLDING POWER. 101 
 
 The "HoldJug* Power," or the ability of the wheel to furnish 
 inertia for the engine to work upon, depends on the disc area of 
 the wheel, and the depth of water draAvn by it. vSo that when 
 the wheel runs ''awash," the Holding Power varies as the 
 cube of the diameter, or 
 
 If the draft of the wheel is greater than the diameter, then the 
 Holding Power will vary as the square of the diameter, mul- 
 tiplied by the draft. 
 
 Illustration. — If a wheel of given diameter give a certain 
 amount of inertia, running *' awash/' then a wheel twice as large 
 under the same condition, will give eig'llt times as much; and a 
 unit of area in the large wheel, as a circular foot of the disc area, 
 wdll have twice as much holding power as the same unit in the 
 small wheel, for it has twice the weight of water on it. 
 
 The Holding Power per I. H. P., furnished by different wheel 
 makers, is widely different in the amount, — varying from 1 to 5 
 cylindric feet. 
 
 The minimum furnished by the best designers at present, is the 
 inertia to be derived from about 1 cylindric foot of water per 
 I. H. P., though the average of wheels on the lakes, is consider- 
 able above that. 
 
 The wheels of the steamers ''Alaska,'" "Spree," and the U. S. 
 Steamship "Maine," are so proportioned as to give about 1 
 cylindiic foot per I. H. P., — whence, to determine the diam- 
 eter of the wheel, in feet for that amount of inertia, or holding 
 power, we have only to take the cube root of the !• H. P. 
 
 Or, if the wheel be deeply submerged, we may divide the 
 I. H. P. by the draft, and take the square root of the quotient. 
 Example: 
 
 The wheels of the U. S. Steamship "Maine," are calculated to 
 work off 9,000 Horse Power, or 4,500 each. And the vessel is to 
 draw 21 J feet of water. Dividing the 4,500 by the draft, which 
 is say 20 feet, we have 225, as the square of the diameter, — the 
 square root of which is 15 feet, as provided by her designers. 
 
 Difficulty in ProYiding" Holding- Power for small vessels. 
 AVe know from Geometry, that similar solids or volumes vary as 
 the third power of any of their similar dimensions. Also, that 
 surface of similar figures are as the second power of their similar 
 sides. 
 
102 A MANUAL, OF NAVIGATION FOR THE LAKES. 
 
 And we know that when two vessels are similar, and loaded to 
 the same per cent, of their depth of hold, that their displacement 
 volumes are similar, and therefore that their wet surfaces are 
 similar, and have the relation of the squares of any of their sim- 
 ilar dimensions. Whence, the power required for them will be 
 as the squares of those dimensions, while as we have seen, the 
 ^^ holding power'' varies as the cube of those dimensions. Hence 
 we see that the holding power changes twice as fast as the 
 requirements for propelling power. Illustration. 
 
 Say a vessel with 8 feet draft, is provided with a satisfactory 
 power, and wheel, 
 
 A vessel of 16 feet draft, would have four times as much 
 power, if provided in proportion to increase of surface, while 
 the inertia available, is eig'llt times as much, i. e. the inertia 
 available, varies twice as fast as the power required. 
 
 This law is not generally known, and what is worse, it cannot 
 be repealed, nor circumvented. 
 
 If it w^ere better understood, people would know why a ton of 
 displacement in a small vessel, requires so much more power for 
 a given speed, than it does in a large one, — which is simply be- 
 cause the inertia required, is not obtainable. 
 
 With this rate of change, between the power required, and the 
 inertia available, there is some point, of course, where the inertia 
 to be had, Avill meet the requirements, when the wheel is running 
 ^' awash." 
 
 This condition will be found with vessels drawing 7 to 8 feet of 
 water, i. e. with such vessels provided say wdth 7 feet wheels, all 
 the inertia required, is available, but not more. 
 
 With vessels drawing more water, the supply increases faster 
 than the demand, — i. e. as the cube of the draft, while the 
 amount wanted is only as the square of the draft. 
 
 AVith vessels of smaller draft, the supply falls short of the 
 requirements, by the same ratio. Example: 
 
 With a vessel of only 4 feet draft, the power required, to be in 
 the proportion of their surfaces if the vessels were similar, would 
 be as the square of J, which is J, while the inertia available 
 would the cube of J=J, that is, the supply falls short of the 
 demand, in this case, in the ratio of J to J, or 1 to 2, i. e. we can 
 only get half of what we want. 
 
USEFUL AND LOST WORK. 
 
 103 
 
 Many plans have been devised to surmount this difficulty, — 
 chief among which is the increase of the pitch. This introduces 
 two other troubles, loss by oblique action of blade, and a serious 
 lateral pull, by the great amount of lost work in the lower blade, 
 resulting in loss by great slip. 
 
 Fig. 32. 
 
 Two wheels have been introduced, but this is very expensive, 
 besides it increases the exposure to injury. The expense is seen 
 when we multiply the diameter of the one wheel, say 8 feet, by 
 the cube root of J=.794, and find that we must use two wheels, 
 6 J feet diameter, to get the holding power afforded by one 8 feet 
 diameter, when the wheels run ^^ awash." 
 
 Useful and Lost Work. — To represent these parts, let the 
 line DC. in Fig. 32, be the helix at center of effort. 
 
104 A MANUAL. OF NAVIGATION FOR THE LAKES. 
 
 From any point in this line, and at right angles to it, draw the 
 line c. d. to represent the total work of blade. Divide into 100 
 equal parts. Through d. draw d. b. parallel with BC, and 
 through c. draw b. c. parallel with D. B. 
 
 Then will b. d. represent the amount and direction of the use- 
 ful work, and b. c. those of the lost work. 
 
 The total, or 100 parts of work must be divided on these lines, 
 in proportion to their squares. This is done by demitting a per- 
 pendicular from b. onto d. c, showing in this case, the useful 
 work to be about — of the total work. The lost work, repre- 
 sented by b. c. is expended entirely in lateral work, like a paddle 
 wheel turned fore-and-aft and which is very much increased by 
 increasing the pitch angle, is seen to be — , in this example. 
 
 The triangles BCD. and b. c. d. being alike, the side b. c. cor- 
 responding to pitch of blade, is the Sine of the Pilch Angle, 
 while d. c. is cosine, whence from the preceding, we have 
 
 Cosine^ of the working pitch angle, measures the useful work, 
 and siue^ measures the lost work, in per cent of the whole. 
 In this manner was the following table made: 
 
 Pitch in 
 Diameter. 
 
 Angle. 
 
 Per Cent. Loss. 
 
 1 
 
 19° 
 
 .11 
 
 li 
 
 24° 
 
 .16 
 
 H 
 
 28° 
 
 .22 
 
 1} 
 
 32° 
 
 .28 
 
 2 
 
 36° 
 
 .35 
 
 Thus, it is seen from the table that we more than treble the 
 loss from oblique action of blade, by doubling the pitch. 
 
 Advantage of a Large TVlieeL — It is not infrequently desir- 
 able to make the wheel larger in diameter than the preceding 
 considerations would indicate, not necessarily for the purpose of 
 getting more inertia for the engine to work upon, — but to keep 
 the pitch-angle within economic limits. 
 
 When it is desired to obtain great speed, we must have great 
 pitch in some form. We may either increase the piston -speed, 
 by which more revolutions of the wheel are attained in a given 
 time, or we increase the diameter of the wheel, — keeping the 
 
ADVANTAGE OF A LARGE WHEEL. 105 
 
 pitch-angle constant, — till the desired pitch is attained. Another 
 method, practiced by some, — who have yet their first lesson tO' 
 learn concerning the propeller wheel, — is to increase the pitch- 
 angle. But as this increases the loss from oblique action of 
 blade very fast, after a pitch of about IJ diameters has been 
 attained, it is not to be tolerated, above that limit. 
 
 It is true that increase of diameter is objectionable on some 
 accounts, — as increased first cost, — increased exposure to injury, — 
 increase of friction; — but reduction of loss by oblique action of 
 blade, i. e. a saving of power, gained thereby; is greater than all. 
 
Explanation of the Tables. 
 
 I^^ Is a traverse table, giving the differences of latitude and 
 departure for distances up to 10, and bv courses varying by 4 
 point up to 8 points. For illustration of their use, see pages 
 41 to 43. 
 
 II. Is a table of the natural functions, — Sine and Cosine,— 
 Tangent and Cotangent, — Secant and Cosecant, for every 5 of the 
 quadrant, and to four places of decimals, — more decimals, or 
 shorter intervals of arc, for purposes of azimuth, with our lake 
 compasses being entirely unnecessary. 
 
 For further explanations of their use, see pages 11 to 13. 
 
 III. Is a table of rhumbs, varying by J point, for converting 
 points to degrees, and the reverse; — with their Sines and Cosines, 
 Tangents and Secants, — useful in making computations for angles 
 in points, etc., instead of in degrees, — as in Table II. 
 
 IV. Is a table of amplitudes of the sun, for the latitude of the 
 lakes, with the time of his rising and setting. Also the rate of 
 the change of his azimuth from sunrise, to the time of his cross- 
 ing the prime vertical, when the latitude and declination are 
 alike. 
 
 This rate being practically uniform for all declinations in any 
 latitude above about 40°, affords the means for deducing a table 
 of time azimuth, good for one to three hours, from a single 
 amplitude; — thereby vastly enlarging the utihty of amplitudes. 
 
 For further explanations see pages 71 to 73. 
 
EXPLANATION OF THE TABLES. 107 
 
 V. Is a table of the sun's declination to the nearest minute, 
 for the current year (1891), with the corresponding equation of 
 time, — the sign prefixed, being that for reducing mean time to 
 apparent time; — wanted with the table of time azimuth, when 
 orienting ship. 
 
 This table is never precisely correct, except for the particular 
 date for which it was computed; but as it is the mean between 
 two leap-years, it can never be far wrong, — the change of declin- 
 ation for 12 hours, being the maximum error that can occur 
 during any four years, — after which interval, the table is very 
 nearly correct again. 
 
 Ignoring this small error, — as we may for all purposes of azi- 
 muth, this table will be available for a great many years. 
 
 VI. Is a table of time azimuths of the sun, for the latitude of 
 the lakes. Useful in orienting ship for compass errors. — See 
 pages 74, 77, 78. 
 
 YII. Is a table of time corrections for reducing standard time 
 to local mean time. It is computed by reducing the difference of 
 longitude between any locality, and the longitude of the standard 
 time, to time, and applying the sign that reduces the standard 
 time to local mean time. Thus: 
 
 The difference of longitude between the light-house at Chicago, 
 and the longitude of central standard time, is 90^ — 87°, 37^=2°, 
 2o^ , or 9^^, 32^ of time. Then because the standard meridian is 
 ^vest of Chicago, this difference must be applied with the plus 
 sign, i. e. we must add 9^^, 32^ to central standard time, to obtain 
 the mean local time for Chicago. In this manner was the table 
 constructed. It is wanted with the equation of time in using the 
 tables of time azimuths. See page 74. 
 
 VIII. Is a table of chords for setting off, or measuring angles. 
 For particulars see pages 12-13. 
 
 IX. Is a table of meridianal parts, for the construction of 
 Mercator's chart. It is given for every 2 minutes of latitude 
 from to 75°. The intermediate minute is readily found by 
 interpolation, Avhen wanted. See pages 48 to 52. 
 
 X. This table gives the value of 1 minute of longitude for 
 €ach degree of latitude up to 30°, and for each half degree 
 thence up to 75°. It is wanted in many questions of dead reckon- 
 ing for reducing departure to difference of longitude. See 
 pages 44 to 46. 
 
108 A MANUAL OF NAVIGATION FOR THE LAKES. 
 
 XI. Is a table for the correction of middle latitude, in middle 
 latitude sailing. See pages 46-47. 
 
 XII. Is a table for finding distance of objects from ship, from 
 two bearings, and distance sailed between them. Pages 59 to 61. 
 
 XIII. Is a table for reducing difference of longitude to time, 
 wanted in deducing local apparent time of a locality from the 
 longitude of the locality and standard time. 
 
 Tariatioil Cliartlet. — Wishing to amplify the subject of vari- 
 ation somewhat, I give the following as an explanation of the 
 chartlet of ''Magnetic Tariations." 
 
 First, to explain the terms right and left as applied to the 
 magnetic card. When the north end of the needle is to the right 
 of the true north, it is in east variation, and in west variation 
 when the north end is to the left of the true north. This is well 
 when the ship has northings in her course, but it is a more gen- 
 eral expression to say that the card is out to the right or left, as 
 the case may be, by variation. 
 
 It Avill be observed that in sailing either wp or down the lakes 
 we are continually crossing the isogonic curves. In going west 
 the card swings to the right 18J° between the foot of Lake 
 Ontario and the head of Lake Superior, — thereby leading the 
 vessel to the right or north. 
 
 In the same manner, in going from Duluth to Buffalo, the card 
 swings to the left, taking the vessel to the left, or north, as 
 before. 
 
 That is to say, — in going any way across these lines, we are 
 taken to the north by change of the variation. 
 
 This *^ change" of variation, is a very different thing from 
 variation itself. When we go west, with a westerly variation, we 
 are led to the Sonth, and to the north with an easterly varia- 
 tion, — unless correction is made, — and conversely. 
 
 But wdth the *' change," we are led to the north in any case. 
 And the track made by the ship as the result of this *^ change," 
 is a curve, like a railroad curve, precisely. And the straight 
 line joining the end of this curve, changes its direction, with a 
 change in the length of the curve, just half as fast, as the curve 
 changes its direction. Whence, 
 
 Correction is made for this *' change," by taking half the sum 
 of the variation at the two extremities of a run for the mean 
 variation to be applied at either end of the run. 
 
VARIATION CHARTLET. 109 
 
 Thus:— Toledo to Buffalo,— Variation at Toledo,=rO°. Varia- 
 tion at Buffalo, :=4J° W. The J sum=2J° W. is the mean 
 variation to be applied in going either way between these ports. 
 
 In ''Shaping Course," the following rule w411 always apply 
 for mean variation. 
 
 With variation to the right, apply correction to the left. 
 With variation to the left, apply correction to the right. 
 
 There is scarcely another place in the world where the 
 '^change" of variation must be taken into account as an indepen- 
 dent factor. This results from the fact that there is no other 
 place in the world where the isogonic curves are so close together, 
 or are crossed so directly by vessels, as on our lakes. 
 
1 
 
 H 
 
 hi 
 
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 EH 
 
 
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 d 
 
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 ^ 
 
 Q 
 
 1-1 CO O t- Ci 
 
 1-1 tH 1-. rH rn C^ C<1 !M (M C^ (M 
 
 
 ■^ 
 
 3 
 
 3.99 
 3.98 
 3.95 
 3.92 
 3.88 
 3.83 
 3.77 
 3.G9 
 3.G2 
 3.53 
 3.43 
 3 . 33 
 3.21 
 3.08 
 2.9G 
 2.83 
 
 d 
 Q 
 
 CO 
 
 d 
 
 t~riOuor:'-^^OXl-l'*t-c;C:^c^ 
 Tt<T*<-*xc<it-Oi-iC<i'*kr;oi:-c:Oi-< 
 
 rt 
 
 CO 
 
 rH-q^U.I-.X^^^^^^^^^^ 
 
 
 g^^^sss^^s^g^^jf^s^:^ 
 
 d 
 
 C<IC<IC<IC^1C^(M(MC^C<IC^C^S^;M'MC^C<> 
 
 N 
 
 
 098 
 19G 
 294 
 390 
 48() 
 580 
 G74 
 7()5 
 855 
 943 
 .03 
 .11 
 .19 
 .27 
 .34 
 .41 
 
 3 
 
 ca 
 
 
 
 c:crixt2-<*<i-ixuo-^o-H;C'-LOXT-i 
 c^cncic^cncsxxxt-t-cc^OTti-^ 
 
 d 
 
 P 
 
 
 - 
 
 d 
 Q 
 
 -<*ir;-^c::'!*<cricoX(Mt^'-'i-':c;cot-o 
 oo^^c5c^r:co-*'*out»jooot~ 
 
 3 
 
 - 
 
 
 3 
 
 ciiJocr;^crt-c<i-^^'NXT--cocOi-<t- 
 
 ClCiXXt-iC'*CMOXi0C0Ot-'*O 
 
 CiCiC5C5CiC;CiCiC^XXXXt-t-t- 
 
 d 
 
 
 
 5i ^ 
 
 tHC^COOiHC^COCtHCOMCi-IC^COC 
 rH C^ CO -si 
 
 
 ^1 
 

 T.A.BIjE II J^, 
 
 
 Natural Sines anc 
 
 Cosines. 
 
 ^ 
 
 0° 
 
 1° 
 
 2° 
 
 3° 
 
 4° 
 
 ' 
 
 Sine, 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosiae 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Oj .0000 
 
 Unit. 
 
 .0174 
 
 .9998 
 
 .0349 
 
 .9994 
 
 .0523 
 
 .9986 
 
 .0698 
 
 .9976 
 
 60 
 
 5^ 
 
 .0014 
 
 Unit. 
 
 .0189 
 
 .9998 
 
 .0363 
 
 .9993 
 
 .0538 
 
 .9985 
 
 .0712 
 
 .9975 
 
 55 
 
 10 
 
 .0029 
 
 Unit. 
 
 .0120 
 
 .9098 
 
 .0378 
 
 .9993 
 
 .0552 
 
 .9985 
 
 0727 
 
 .9974 
 
 50 
 
 15 
 
 .0044 
 
 .9999 
 
 .0218 
 
 .9998 
 
 .0393 
 
 .9992 
 
 .0567 
 
 .6984 
 
 .0741 
 
 .9973 
 
 45 
 
 20 
 
 .0058 
 
 .9999 
 
 0233 
 
 .9997 
 
 .0407 
 
 .9992 
 
 .0581 
 
 .9983 
 
 .0756 
 
 .9971 
 
 40 
 
 25 
 
 .0073 
 
 .9999 
 
 .0247 
 
 .9997 
 
 .0422 
 
 .9991 
 
 0596 
 
 9982 
 
 .0770 
 
 .9970 
 
 35 
 
 30 
 
 .0087 
 
 .9999 
 
 .0262 
 
 .9996 
 
 .0436 
 
 .9990 
 
 .0610 
 
 .9985 
 
 .0785 
 
 .9969 
 
 30 
 
 35 
 
 .0102 
 
 .9999 
 
 .0276 
 
 .9996 
 
 .0451 
 
 .9989 
 
 .0625 
 
 .9980 
 
 .0799 
 
 .9968 
 
 25 
 
 40 
 
 .0116 
 
 .9999 
 
 .0291 
 
 .9996 
 
 .0465 
 
 .9989 
 
 .0639 
 
 .9979 
 
 .0814 
 
 9968 
 
 20 
 
 45 
 
 .0131 
 
 .9999 
 
 .0305 
 
 .9995 
 
 .0480 
 
 .9988 
 
 .0654 
 
 .9978 
 
 .0828 
 
 .9966 
 
 15 
 
 50 
 
 .0145 
 
 .9998 
 
 .0399 
 
 .9995 
 
 .0494 
 
 .9988 
 
 .0668 
 
 .9977 
 
 .0843 
 
 .9964 
 
 10 
 
 55 
 
 .0160 
 
 .9998 
 
 .0334 
 
 .9994 
 
 .0509 
 
 .9987 
 
 .0683 
 
 .9976 
 
 .0857 
 
 .9963 
 
 5 
 
 60 
 
 .0174 
 
 .9998 
 
 .0349 
 
 .9994 
 
 .0523 
 
 .9986 
 
 .0697 
 
 .9975 
 
 .0872 
 
 .9962 
 
 
 
 > 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 bine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 ' 
 
 89° 
 
 88° 
 
 87° i 
 
 86° 
 
 85° 
 
 ~b 
 
 5" 
 
 6° 
 
 7° 
 
 8° 
 
 1 9° 
 
 ' 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 .0872 
 
 .9962 
 
 .1045 
 
 .9945 
 
 .1219 
 
 .9925 
 
 . 1392 
 
 .9903 
 
 .1564 
 
 .9877 
 
 60 
 
 5 
 
 .08.^6 
 
 .9961 
 
 .1060 
 
 .9944 
 
 .1233 
 
 .9924 
 
 .1406 
 
 .9901 
 
 ..1579 
 
 .9875 
 
 55 
 
 iq 
 
 .0900 
 
 .9959 
 
 .1074 
 
 .9942 
 
 .1248 
 
 .9922 
 
 .1420 
 
 .9899 
 
 .1.593 
 
 .9872 
 
 50 
 
 15 
 
 .0915 
 
 .99.=^8 
 
 .1089 
 
 .9941 
 
 .1262 
 
 .99_>0 
 
 .1435 
 
 .9896 
 
 .1607 
 
 .9870 
 
 45 
 
 20 
 
 .0929 
 
 .9957 
 
 .1103 
 
 .9939 
 
 .1276 
 
 .9918 
 
 .1449 
 
 .9894 
 
 .1622 
 
 .9868 
 
 40 
 
 25 
 
 .0944 
 
 .9055 
 
 .1118 
 
 .9937 
 
 .1291 
 
 .9916 
 
 .1464 
 
 .9892 
 
 .1636 
 
 .9865 
 
 35 
 
 30 
 
 .0958 
 
 .99.54 
 
 .1132 
 
 .9936 
 
 .1305 
 
 .9914 
 
 .1478 
 
 .9890 
 
 .16.50 
 
 .9863 
 
 30 
 
 35 
 
 .0973 
 
 .99.53 
 
 .1146 
 
 .9934 
 
 .1320 
 
 .9912 
 
 .1492 
 
 .9888 
 
 .1665 
 
 .9860 
 
 25 
 
 40 
 
 .0987 
 
 .9951 
 
 .1161 
 
 .9932 
 
 .1334 
 
 .9911 
 
 .1.507 
 
 .9886 
 
 .1679 
 
 .9858 
 
 20 
 
 45 
 
 .1002 
 
 .99.50 
 
 .1175 
 
 .9931 
 
 .1348 
 
 .9909 
 
 .1521 
 
 .9884 
 
 .1693 
 
 .9856 
 
 15 
 
 50 
 
 .1016 
 
 .9948 
 
 .1190 
 
 .9929 
 
 .1363 
 
 .9907 
 
 .1535 
 
 .9881 
 
 .1708 
 
 .9853 
 
 10 
 
 55 
 
 .1031 
 
 .9947 
 
 .1204 
 
 .9927 
 
 .1377 
 
 .9005 
 
 . 1550 
 
 .9879 
 
 .1722 
 
 .98.51 
 
 5 
 
 69 
 
 .1045 
 
 . 9945 
 
 .1219 
 
 .9925 
 
 .1392 
 
 .99 3 
 
 .1564 
 
 9877 
 
 .1765 
 
 9848 
 
 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 / 
 
 84° 
 
 83° 
 
 82° 
 
 8P 
 
 80° 
 
 ^ 
 
 10- 
 
 ii° 
 
 l^o 
 
 13- 
 
 14° 
 
 ' 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 .1736 
 
 .9848 
 
 .1908 
 
 .9816 
 
 ..2079 
 
 .97H1 
 
 .2249 
 
 .9744 
 
 .2419 
 
 .9703 
 
 60 
 
 5 
 
 .1751 
 
 .9845 
 
 .1922 
 
 .9813 
 
 .2093 
 
 .9778 
 
 .2264 
 
 .9740 
 
 .2433 
 
 .9699 
 
 55 
 
 10 
 
 .1765 
 
 .9843 
 
 ..1937 
 
 .9811 
 
 .2108 
 
 .9775 
 
 ..2278 
 
 .9737 
 
 .2447 
 
 .9696 
 
 50 
 
 15 
 
 .1779 
 
 .9S40 
 
 .1951 
 
 .9808 
 
 .2122 
 
 .9772 
 
 .2292 
 
 .9734 
 
 .2461 
 
 .9692 
 
 45 
 
 20 
 
 .1794 
 
 .8438 
 
 .1965 
 
 .9805 
 
 .2136 
 
 .9769 
 
 .2306 
 
 .9730 
 
 .2476 
 
 .9689 
 
 40 
 
 25 
 
 .1808 
 
 .9835 
 
 .1979 
 
 .9802 
 
 .2150 
 
 .9766 
 
 .2320 
 
 .9727 
 
 .2490 
 
 .9685 
 
 85 
 
 3^ 
 
 .1822 
 
 .9832 
 
 .1994 
 
 .9799 
 
 .2164 
 
 .9763 
 
 .2334 
 
 .9724 
 
 .2504 
 
 .9681 
 
 30 
 
 35 
 
 .1837 
 
 .9830 
 
 .2008 
 
 .9696 
 
 .2179 
 
 .9760 
 
 .2349 
 
 .9720 
 
 .2518 
 
 .9678 
 
 25 
 
 40 
 
 .1851 
 
 .9827 
 
 .2022 
 
 .9793 
 
 .2193 
 
 .9757 
 
 .2363 
 
 .9717 
 
 .253 J 
 
 .9674 
 
 20 
 
 45 
 
 .1865 
 
 .9824 
 
 .2036 
 
 .9790 
 
 .2207 
 
 .9753 
 
 .2377 
 
 .9713 
 
 .2546 
 
 .9670 
 
 15 
 
 5^ 
 
 .1879 
 
 .9822 
 
 .2051 
 
 .9787 
 
 .2221 
 
 .9750 
 
 .2.391 
 
 .9710 
 
 .2560 
 
 .9667 
 
 10 
 
 55 
 
 .1894 
 
 .9819 
 
 .2065 
 
 .9784 
 
 .2235 
 
 .9747 
 
 .2405 
 
 .9706 
 
 .2.574 
 
 .9663 
 
 5 
 
 60 
 
 1908 
 
 .9816 
 
 .2079 
 
 .9781 
 
 .2249 
 
 .9744 
 
 .2419 
 
 .9707 
 
 2588 
 
 .9659 
 
 
 
 ^ 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 ' 
 
 79° 
 
 78° 
 
 T7^ 
 
 76" 
 
 75° 
 
Natural Sines and Cosines — Continued. 
 
 ^ 
 
 10" 
 
 16° 
 
 17- 
 
 18° 
 
 19° 
 
 60 
 55 
 50 
 45 
 40 
 35 
 30 
 25 
 20 
 15 
 10 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 CoMne 
 
 Sine 
 
 Cosine 
 
 bine 
 
 Cosi' e 
 
 
 5 
 10 
 15 
 20 
 25 
 30 
 35 
 40 
 45 
 50 
 55 
 60 
 
 .2588 
 .2602 
 .2616 
 .2630 
 .2644 
 .2658 
 .2672 
 .2686 
 .2700 
 .2714 
 .2728 
 .2742 
 .2756 
 
 .9659 
 .9655 
 .9652 
 .9648 
 .9644 
 .9640 
 .9636 
 .9632 
 .9628 
 .9625 
 .9621 
 .9617 
 .9613 
 
 .2756 
 
 .2770 
 .2784 
 .2798 
 .2812 
 .2826 
 .2840 
 .2854 
 .2868 
 .2882 
 .2896 
 .2910 
 .2924 
 
 .9613 
 .9608 
 .9605 
 
 smo 
 
 .9.=)96 
 .9592 
 .9.588 
 .9.584 
 .9580 
 .9576 
 .9571 
 .9567 
 .9563 
 
 .2924 
 .2938 
 .29.51 
 ..2965 
 .2979 
 .2993 
 .3007 
 .3021 
 .3035 
 .2049 
 .3062 
 .3076 
 .3090 
 
 .9.563 
 .9559 
 .9554 
 .9550 
 .9546 
 .9541 
 .9537 
 .9533 
 .9528 
 .9.524 
 .9519 
 .9.515 
 .9511 
 
 ..3090 
 .3104 
 .3118 
 .3132 
 .3145 
 .3159 
 .3137 
 .3187 
 .3201 
 .3214 
 .3228 
 .3242 
 .3255 
 
 .9511 
 .9.506 
 .9.501 
 .9497 
 .9492 
 .9488 
 .9483 
 .9479 
 .9474 
 .9469 
 .9465 
 .9460 
 .9455 
 
 .3256 
 .3269 
 .3283 
 .3297 
 .3311 
 .3324 
 .3338 
 .3352 
 .3365 
 .3379 
 .3393 
 .3406 
 .3420 
 
 .9455 
 .94.50 
 .9446 
 .9441 
 .9436 
 .9431 
 .9426 
 .9421 
 .9417 
 .9412 
 .9407 
 .9402 
 .9397 
 
 \ 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 * Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosinej bine 
 
 / 
 
 74° 
 
 73° 
 
 72° 
 
 71° 
 
 70° 
 
 1 
 
 10 
 15 
 20 
 25 
 30 
 35 
 
 i 
 
 55 
 
 60 
 
 2U° 
 
 21° 
 
 22° 
 
 23° ) 24° 
 
 / 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 blue 
 
 Cosint 
 
 .3420 
 .3433 
 .3447 
 .3461 
 .3475 
 .3488 
 .3502 
 .3516 
 .3529 
 .3543 
 .3556 
 .3570 
 .3584 
 
 .9397 
 .9392 
 .9387 
 .9382 
 .9377 
 .9372 
 .9367 
 .9362 
 .9356 
 .9351 
 .9346 
 .9341 
 .9336 
 
 .3584 
 .3597 
 .3611 
 .3624 
 .3638 
 .3652 
 .3665 
 .3678 
 .3692 
 .3706 
 .3719 
 .3733 
 .3746 
 
 .9336 
 .9331 
 .9325 
 .9320 
 .9315 
 .9309 
 .9304 
 .9299 
 .9293 
 .9288 
 .9283 
 .9277 
 .9272 
 
 .3746 
 .3759 
 .3773 
 .3786 
 .3800 
 .3813 
 .3827 
 .3840 
 .3854 
 .3867 
 .3881 
 .3891 
 .3907 
 
 .9272 
 .9266 
 .9261 
 .9255 
 .9250 
 .9244 
 .9239 
 .9233 
 .9228 
 .9222 
 .9216 
 .9211 
 .9205 
 
 .3907 
 .3921 
 .3934 
 .3947 
 .3961 
 .3974 
 .3987 
 .4001 
 .4014 
 .4027 
 .4041 
 .4054 
 .4067 
 
 .9005 
 .9199 
 .9194 
 .9188 
 .9182 
 .9176 
 .9171 
 .9165 
 .9159 
 .9153 
 .9147 
 .9141 
 9135 
 
 : .4067 
 .4081 
 .4094 
 .4107 
 .4120 
 .4134 
 .4147 
 .4160 
 .4173 
 .4187 
 .4200 
 .4213 
 .4226 
 
 .9135 
 .9127 
 .9124 
 .9118 
 .9111 
 .9106 
 ..9100 
 .9094 
 .9087 
 .9081 
 .9075 
 .9069 
 .9063 
 
 60 
 55 
 50 
 45 
 40 
 35 
 30 
 25 
 20 
 15 
 10 
 5 
 
 
 vJosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine] Sine 
 
 Cosine 
 
 Sine 
 
 f 
 
 69° 
 
 68° 
 
 67° 
 
 66° 
 
 65° 
 
 \ 
 
 ~0 
 5 
 
 10 
 15 
 20 
 25 
 30 
 35 
 40 
 45 
 50 
 55 
 60 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 f 
 
 60 
 55 
 50 
 45 
 40 
 35 
 30 
 25 
 20 
 15 
 10 
 5 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 .4226 
 .4239 
 .4252 
 .4266 
 .4279 
 .4292 
 .4305 
 .4318 
 .4331 
 .4344 
 .4357 
 .4371 
 .4384 
 
 .9063 
 .9057 
 .9051 
 .9045 
 .9038 
 .9032 
 .9026 
 .9020 
 .9013 
 .9007 
 .9001 
 .8994 
 .8988 
 
 .4384 
 .4397 
 .4410 
 .4423 
 .4436 
 .4449 
 .4462 
 .4475 
 .4488 
 .4501 
 .4514 
 .4227 
 .4540 
 
 .8988 
 .8982 
 .8975 
 .8969 
 .8962 
 .8956 
 .8949 
 .8943 
 .8936 
 .8930 
 .8923 
 .8917 
 .8910 
 
 .4540 
 .4553 
 .4.566 
 .4579 
 .4592 
 .4605 
 .4617 
 .4630 
 .4643 
 .4656 
 .4669 
 .4682 
 .4695 
 
 .8910 
 .8903 
 .8897 
 .8890 
 .8883 
 .8877 
 .8870 
 .8863 
 .8857 
 .8850 
 .8843 
 .8836 
 .8829 
 
 .4695 
 .4708 
 .4720 
 .4733 
 .4746 
 .4759 
 .4772 
 .4784 
 .4797 
 .4810 
 .4823 
 .4835 
 .4848 
 
 .8829 
 .8823 
 .8816 
 .88u9 
 .8802 
 .8795 
 .8788 
 .8781 
 .8774 
 .8767 
 .8760 
 .8753 
 .8746 
 
 .4848 
 .4861 
 .4873 
 .4886 
 .4899 
 .4912 
 .4924 
 .4936 
 .4949 
 .4962 
 .4975 
 .4987 
 .5000 
 
 .8746 
 .8739 
 .8732 
 .8725 
 .8718 
 .8711 
 .8704 
 .8696 
 .8689 
 .8682 
 .8675 
 .8667 
 .8660 
 
 Cosine 
 
 Sine 
 
 Cosme 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 64° 1 
 
 63° 
 
 62° 1 61° 
 
 60° 
 

 T^BXjE II J^, 
 
 Natural Sines and Cosines — Continued. 
 
 "o 
 
 30" 
 
 31° 
 
 32^ 
 
 33^ 
 
 34° 
 
 60 
 
 Sine 
 .5000 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 v^osine 
 
 .8660 
 
 .5150 
 
 .8572 
 
 .5299 
 
 .8480 
 
 .5446 
 
 .8387 
 
 .5592 
 
 .8290 
 
 5 
 
 ,5013 
 
 .86.53 
 
 .5163 
 
 .8564 
 
 .5311 
 
 .8473 
 
 .5459 
 
 .8379 
 
 .5604 
 
 .8282 
 
 55 
 
 10 
 
 .5025 
 
 .8646 
 
 .5175 
 
 .8557 
 
 .5324 
 
 .8465 
 
 .5471 
 
 .8371 
 
 .5616 
 
 .8274 
 
 10 
 
 15 
 
 .5038 
 
 .8638 
 
 .5188 
 
 .8549 
 
 .5336 
 
 .8457 
 
 .5483 
 
 .8363 
 
 .5628 
 
 .8266 
 
 45 
 
 20 
 
 .6050 
 
 .8631 
 
 .5200 
 
 .8542 
 
 .5348 
 
 .8449 
 
 .5495 
 
 .83.55 
 
 ..5640 
 
 .8258 
 
 40 
 
 25 
 
 .5063 
 
 8624 
 
 .5213 
 
 .8534 
 
 .5361 
 
 .8442 
 
 ..5507 
 
 .8347 
 
 .5652 
 
 .8249 
 
 35 
 
 30 
 
 .5075 
 
 .8616 
 
 .5225 
 
 .8526 
 
 ..5373 
 
 .8434 
 
 ..5519 
 
 .8339 
 
 .5664 
 
 .8241 
 
 30 
 
 35 
 
 .5088 
 
 .8609 
 
 .5237 
 
 .8519 
 
 .5386 
 
 .8426 
 
 .5531 
 
 8331 
 
 .5676 
 
 .3233 
 
 25 
 
 40 
 
 .5100 
 
 .8601 
 
 .5250 
 
 .8511 
 
 .5397 
 
 .8418 
 
 .5544 
 
 .8323 
 
 .5688 
 
 .8225 
 
 20 
 
 45 
 
 .5113 
 
 .8594 
 
 .6262 
 
 .8503 
 
 .5410 
 
 .8410 
 
 .55.56 
 
 .8315 
 
 .5700 
 
 .8216 
 
 15 
 
 50 
 
 .5125 
 
 .8587 
 
 .5274 
 
 .8496 
 
 .5422 
 
 .8402 
 
 .5.568 
 
 .8307 
 
 .5712 
 
 .8208 
 
 10 
 
 65 
 
 .5138 
 
 .8579 
 
 .5287 
 
 .8488 
 
 .5434 
 
 .8395 
 
 .5580 
 
 .8298 
 
 .5724 
 
 .8200 
 
 5 
 
 60 
 
 .5150 
 
 8572 
 
 .5299 
 
 .8480 
 
 .5446 
 
 .8387 
 
 .5592 
 
 .8290 
 
 .5736 
 
 .8191 
 
 _0 
 
 / 
 
 ^ 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 59° 
 
 58° ! 5' 
 
 JO 
 
 56° 
 
 55° 
 
 ^ 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 ~^ 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cos ne 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine Cosine 
 
 
 
 .5736 
 
 .8191 
 
 .5878 
 
 .8090 
 
 .6018 
 
 .7986 
 
 .6157 
 
 .7880 
 
 .6293 
 
 .7771 
 
 60 
 
 5 
 
 .5747 
 
 .8183 
 
 .5890 
 
 .8082 
 
 .6030 
 
 .7978 
 
 .6168 
 
 .7871 
 
 .6304 
 
 .7762 
 
 55 
 
 10 
 
 .5760 
 
 .8175 
 
 .5901 
 
 .8073 
 
 .6041 
 
 .7969 .6179 
 
 .7862 
 
 .6316 
 
 .7753 
 
 50 
 
 15 
 
 .5771 
 
 .8166 
 
 ,5913 
 
 .8064 
 
 .6053 
 
 .7960 1 .6191 
 
 .7853 
 
 .6327 
 
 .7744 
 
 45 
 
 20 
 
 .5783 
 
 .8158 
 
 .5925 
 
 .8056 
 
 .6064 
 
 .7951 
 
 .6202 
 
 .7844 
 
 .6338 
 
 .7735 
 
 40 
 
 25 
 
 .5795 
 
 .8158 
 
 .5936 
 
 .8047 
 
 .6076 
 
 .7942 
 
 .6214 
 
 .7835 
 
 .6350 
 
 .7726 
 
 35 
 
 30 
 
 .5807 
 
 .8141 
 
 .5948 
 
 .8039 
 
 .6088 
 
 .7933 
 
 .6225 
 
 .7826 
 
 .6361 
 
 .7716 
 
 30 
 
 35 
 
 .5819 
 
 .8133 
 
 .5960 
 
 .8030 
 
 .6099 
 
 .7925 
 
 .6236 
 
 .7817 
 
 .6372 
 
 .7707 
 
 25 
 
 40 
 
 .5831 
 
 8124 
 
 .5972 
 
 .8021 
 
 .6111 
 
 .7916 
 
 .6248 
 
 7808 
 
 .6383 
 
 .7698 
 
 20 
 
 45 
 
 .5842 
 
 .8116 
 
 .5983 
 
 .8012 
 
 .6122 
 
 .7905 
 
 .6259 
 
 .7799 
 
 .6394 
 
 .7688 
 
 15 
 
 50 
 
 5854 
 
 .8107 
 
 .5995 
 
 .8004 
 
 .6134 
 
 .7898 
 
 .6271 
 
 .7790 
 
 .6406 
 
 .7679 
 
 10 
 
 55 
 
 .5866 
 
 .8099 
 
 .6006 
 
 .7995 
 
 .6145 
 
 .7889 
 
 .6282 
 
 .7781 
 
 .6417 
 
 .4670 
 
 5 
 
 60 
 
 .5878 
 
 .8090 
 
 .6018 
 
 .7986 
 
 .6157 
 
 .7880 
 
 .6293 
 
 .7771 
 
 .6428 
 
 .7660 
 Sine 
 
 _0 
 
 / 
 
 ^ 
 
 Cosine 
 
 Sine 
 
 v^Oaine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 54° 1 
 
 53° 
 
 52° 1 
 
 5] 
 
 ° 
 
 50° 
 
 ~0 
 
 40° 
 
 41° 
 
 42° 1 
 
 43° 1 
 
 44° 
 
 / 
 
 Sine 
 .6128 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 .7660 
 
 .6561 
 
 .7547 
 
 .6691 
 
 .7431 
 
 .6820 
 
 .7313 
 
 .6947 
 
 .7193 
 
 60 
 
 5 
 
 .6439 
 
 .7651 
 
 .6572 
 
 .7537 
 
 .6702 
 
 .7422 
 
 .6831 
 
 .7304 
 
 .6957 
 
 .5183 
 
 55 
 
 10 
 
 .6450 
 
 .7642 
 
 .6582 
 
 .7528 
 
 .6713 
 
 .7412 
 
 .6841 
 
 .7294 
 
 .6967 
 
 .7173 
 
 50 
 
 15 
 
 .6461 
 
 .7632 
 
 .6593 
 
 .7518 
 
 .6724 
 
 .7402 
 
 .6852 
 
 .7284 
 
 .6978 
 
 .7163 
 
 45 
 
 20 
 
 .6472 
 
 .7623 
 
 .6604 
 
 .7.509 
 
 .6734 
 
 .7392 
 
 .6862 
 
 .7274 
 
 .6988 
 
 .7153 
 
 40 
 
 25 
 
 .6483 
 
 .7613 
 
 .6615 
 
 .7499 
 
 .6745 
 
 .7383 
 
 .6873 
 
 .7364 
 
 .6999 
 
 .7143 
 
 35 
 
 30 
 
 .6494 
 
 .7604 
 
 .6626 
 
 .7490 
 
 .6756 
 
 .7363 
 
 .6883 
 
 .7254 
 
 .7009 
 
 .7132 
 
 30 
 
 35 
 
 .6505 
 
 .7595 
 
 .6637 
 
 .7480 
 
 .6766 
 
 .7373 
 
 .6894 
 
 .7244 
 
 .7019 
 
 .7122 
 
 25 
 
 40 
 
 .6.517 
 
 .7585 
 
 .6648 
 
 .7470 
 
 .6777 
 
 .7353 
 
 .6905 
 
 .7234 
 
 7030 
 
 .7112 
 
 20 
 
 45 
 
 .6528 
 
 .7576 
 
 .6659 
 
 .7461 
 
 .6788 
 
 .7343 
 
 .6915 
 
 .7224 
 
 .7040 
 
 .7092 
 
 15 
 
 50 
 
 .6.539 
 
 .7566 
 
 .6690 
 
 .7451 
 
 .6790 
 
 .7333 
 
 .6926 
 
 .7214 
 
 .7050 
 
 .7092 
 
 10 
 
 55 
 
 .6550 
 
 .7557 
 
 .6680 
 
 .7441 
 
 .6809 
 
 .7323 
 
 .6936 
 
 .72U3 
 
 .7061 
 
 .7081 
 
 5 
 
 60 
 
 .6561 
 
 .7547 
 
 .6691 
 
 .7431 
 
 .6820 
 
 .7313 
 
 .6947 
 
 .7193 
 
 .7071 
 
 .7071 
 
 
 
 Cosine 
 
 Sine 
 
 v^obine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 / 
 
 49° 1 
 
 48° 1 
 
 47° 1 
 
 46° 1 45° 1 
 
TJ^BXjE II B, 
 
 
 
 Natural Tangents and Cotangents. 
 
 
 ^ 
 
 0° 
 
 1° 
 
 2° 
 
 3° 
 
 4" 1 
 
 / 
 60 
 
 
 Tang. jCotan. 
 
 Tang. 
 
 Cotan . 
 
 Tang. 
 
 Cotan . 
 
 1 ang. 
 
 Cotan . 
 
 Tang. 
 
 Cotan. 
 
 
 
 
 .0000 Infinit. 
 
 .0175 
 
 57. 29 
 
 .0349 
 
 28.64 
 
 .0524 
 
 19.08 
 
 .0699 
 
 14.301 
 
 
 5 
 
 .0014 687.55 
 
 .0189 
 
 52.88 
 
 .0363 
 
 27.49 
 
 0589 
 
 18.56 
 
 .0714 
 
 14.008 
 
 55 
 
 
 10 
 
 .0029 ,343 77 
 
 .0204 
 
 49.10 
 
 .0378 
 
 26.43 
 
 .0533 
 
 18.07 
 
 .0728 
 
 13.727 
 
 50 
 
 
 15 
 
 .0044 
 
 229.18 
 
 .0218 
 
 45.83 
 
 .0393 
 
 25.45 
 
 .0568 
 
 17.61 
 
 .0743 
 
 13.456 
 
 45 
 
 
 20 
 
 0058 
 
 171.88 
 
 .0233 
 
 42 66 
 
 .0407 
 
 21.54 
 
 .0582 
 
 17.17 
 
 .0758 
 
 13.197 40 
 
 
 25 
 
 .0073 
 
 137.51 
 
 .0247 
 
 40.44 .0422 
 
 23.69 
 
 .0597 
 
 16.75 
 
 .0772 
 
 12.947 35 
 
 
 30 
 
 0087 
 
 114.59 
 
 .0262 
 
 38.19 
 
 0437 
 
 22.90 
 
 .0612 
 
 16.35 
 
 .0787 
 
 12.706 80 
 
 
 35 
 
 .0102 
 
 98.22 
 
 .0276 
 
 36.18 
 
 .0451 
 
 22.16 
 
 .0626 
 
 15.97 
 
 .0802 
 
 12.474 
 
 25 
 
 
 40 
 
 .0116 
 
 85.94 
 
 .0291 
 
 34.37 
 
 .0466 
 
 21.47 
 
 064J 
 
 15.60 I 
 
 .0816 
 
 12.251 
 
 20 
 
 
 45 
 
 .0131 
 
 76.39 
 
 .0305 
 
 32.73 
 
 .0480 
 
 20.82 
 
 .0655 
 
 15.26 
 
 .0831 
 
 12.035 15 
 
 
 50 
 
 .0145 
 
 68.75 
 
 .0320 
 
 31.24 
 
 .0495 
 
 20.21 
 
 .0670 
 
 14.92 
 
 .0846 
 
 11.826 
 
 10 
 
 
 55 
 
 .0160 
 
 62.50 
 
 .0335 
 
 29.88 
 
 .0509 
 
 19.63 
 
 .0685 
 
 14.61 
 
 .0860 
 
 11.625 
 
 5 
 
 
 60 
 
 .0175 
 
 27.29 
 
 .0349 
 
 28.64 
 
 .0524 19.08 
 
 .0699 
 
 14.30 
 
 .0875 
 
 11.430 
 
 
 
 
 Cotan 
 
 Tane. 
 
 Cotan, 
 
 rang i 
 
 V otan. 1 Tang. | 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 f 
 
 
 89° 
 
 88° i 87° 1 
 
 8t)° 
 
 85° 
 
 
 \ 
 
 5° 
 
 6° 
 
 7° 
 
 8° 
 
 9° 
 
 
 
 L'anid:. 
 
 Cotan. 
 
 1 ane. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. Cotan. 
 
 Tang 
 
 Cotan. 
 
 
 
 
 .0875 
 
 11.430 
 
 .1051 
 
 9.514 
 
 .1228 
 
 8.144 
 
 .1405 7.115 
 
 .1584 
 
 6.314 
 
 60 
 
 
 5 
 
 .0889 
 
 11.242 
 
 .1066 
 
 9.383 
 
 .1243 
 
 8.047 
 
 .1420 
 
 7.041 
 
 .1600 
 
 6.255 
 
 55 
 
 
 10 
 
 .0904 
 
 1.053 
 
 .1080 
 
 9.255 
 
 .1257 
 
 7.953 
 
 .1435 
 
 6.968 
 
 .1614 
 
 6.197 
 
 50 
 
 
 15 
 
 .0719 
 
 10.883 
 
 .1095 
 
 9.131 
 
 .1272 
 
 7.860 
 
 .1450 
 
 6.897 
 
 .1629 
 
 6.140 
 
 45 
 
 
 20 
 
 .0933 
 
 10.712 
 
 .1109 
 
 9.010 
 
 .1287 
 
 7.770 
 
 .1465 
 
 6 827 
 
 .1644 
 
 6.084 
 
 40 
 
 
 25 
 
 .0948 
 
 10.546 
 
 .1125 
 
 8.892 
 
 .1302 
 
 7.682 
 
 .1480 
 
 6.758 
 
 .1658 
 
 6.030 
 
 35 
 
 
 30 
 
 .0963 
 
 10.385 
 
 .1139 
 
 8.777 
 
 .1316 
 
 7.596 
 
 .1494 
 
 6.691 
 
 .1673 
 
 5.976 
 
 30 
 
 
 35 
 
 .0978 
 
 10.229 
 
 .1154 
 
 8.665 
 
 .1331 
 
 7.511 
 
 .1509 
 
 6.625 
 
 .1688 
 
 5.923 
 
 25 
 
 
 40 
 
 .0992 
 
 10.078 
 
 .1169 
 
 8.555 
 
 .1346 
 
 7.429 
 
 .1.524 
 
 6.561 
 
 .1703 
 
 5.871 
 
 20 
 
 
 45 
 
 .1007 
 
 9.931 
 
 .1184 
 
 8.449 
 
 .1361 
 
 7.348 
 
 .1539 
 
 6.497 
 
 .1718 
 
 5.820 
 
 15 
 
 
 50 
 
 .1022 
 
 9.788 
 
 .1198 
 
 8.345 
 
 .1376 
 
 7.269 
 
 .1554 
 
 5.435 
 
 .1733 
 
 5.769 
 
 10 
 
 
 55 
 
 .1036 
 
 9.649 
 
 .1213 
 
 9.243 
 
 .1391 
 
 7.191 
 
 .1569 
 
 6.374 
 
 .1748 
 
 5.720 
 
 5 
 
 
 60 
 
 .1051 
 
 9.514 
 
 .1228 
 
 8.144 
 
 1405 
 
 7.115 
 
 .1584 
 
 6.314 
 
 .1763 
 
 5.671 
 
 
 
 
 ^ 
 
 Cotan. 1 Tang;. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 lang. 
 
 Cotan. 
 
 ang 
 
 Cotan. 
 
 lang. 
 
 f 
 
 
 84° 
 
 83° 
 
 82° 
 
 81° 
 
 80° 
 
 
 ^ 
 
 10° 
 
 11° 
 
 12° 
 
 13° 
 
 14° 
 
 60 
 
 
 Tang. |»^otan . 
 
 Tang. |Cotan. 
 
 Tang 
 
 Cotan. 
 
 Tang. 
 
 Cota . 
 
 1 ang 
 
 Cotan . 
 
 
 
 
 .1763 
 
 5.671 
 
 .1944 15 144 1 .2126 
 
 4.705 
 
 .2308 
 
 4.331 
 
 .2493 
 
 4.011 
 
 
 5 
 
 .1778 
 
 5.623 
 
 .1959 
 
 5.104 
 
 .2141 
 
 4.671 
 
 .2324 
 
 4.303 
 
 .2509 
 
 3.986 
 
 55 
 
 
 10 
 
 .1799 
 
 5.576 
 
 .1974 
 
 5.066 
 
 .2156 
 
 4.638 
 
 .2339 
 
 4.275 
 
 .2524 
 
 3.962 
 
 50 
 
 
 15 
 
 .1808 
 
 5.530 
 
 .1989 
 
 5.097 
 
 .2171 
 
 4.606 
 
 .2355 
 
 4.247 
 
 .2540 
 
 3.937 
 
 45 
 
 
 20 
 
 .1823 
 
 5.484 
 
 .2004 
 
 4.989 
 
 .2186 
 
 4.574 
 
 .2370 
 
 4.219 
 
 .2555 
 
 3.914 
 
 40 
 
 
 25 
 
 .1838 
 
 5.140 
 
 .2019 
 
 4.952 
 
 .2202 
 
 4.542 
 
 .2385 
 
 4.192 
 
 .2571 
 
 3.890 
 
 35 
 
 
 30 
 
 .1853 5.395 
 
 .2034 
 
 4.915 
 
 .2217 
 
 4.511 
 
 .2401 
 
 4.165 
 
 .2586 
 
 3.867 
 
 30 
 
 
 35 
 
 .1868 
 
 5.352 
 
 .2050 
 
 4.879 
 
 .2232 
 
 4.480 
 
 .2416 
 
 4.139 
 
 .2602 
 
 3 844 
 
 25 
 
 
 40 
 
 .1883 
 
 5.309 
 
 .2065 
 
 4 843 
 
 .2247 
 
 4.449 
 
 .2432 
 
 4.113 
 
 .2617 
 
 3.821 
 
 20 
 
 
 45 
 
 .1899 
 
 5.267 
 
 .2080 
 
 4.808 
 
 .2263 
 
 4.419 
 
 .2447 
 
 4.087 
 
 .2633 
 
 3.798 
 
 15 
 
 
 50 
 
 .1914 
 
 5.226 
 
 .2095 
 
 4.773 
 
 .2278 i 4.390 
 
 .2462 
 
 4.061 
 
 2648 
 
 3.776 
 
 10 
 
 
 55 
 
 .1929 
 
 5.185 
 
 .2110 
 
 4.738 
 
 .2293 
 
 4.360 
 
 .2478 
 
 4.036 
 
 .2664 
 
 3.754 
 
 5 
 
 
 60 
 
 .1944 
 
 5.144 
 
 .2126 
 
 4.705 
 
 .2309 
 
 4.331 
 
 .2493 
 
 4.011 
 
 .2679 
 
 3.732 
 
 
 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang 
 
 Cotan. 
 
 lang 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 lang. 
 
 .1 
 
 79° 
 
 78° 
 
 77° 
 
 76° 
 
 75° ^ 
 
 
 I 
 
T.A.BILS II B, 
 
 
 Natural Tangents 
 
 and Cotangents — Cont'd. 
 
 ^ 
 
 15" 
 
 lb° i 
 
 17° 
 
 18° 1 
 
 19° 1 
 
 ' 
 
 Tan?. jCotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 fang. 
 
 Cotan . 
 
 Tang. 
 
 Cotan. 
 
 
 
 .2679 
 
 3.732 
 
 .2867 
 
 3.487 
 
 .3057 
 
 3.271 
 
 .3249 
 
 3.078 
 
 .3443 
 
 2.904 
 
 30 
 
 5 
 
 .2695 
 
 3 710 
 
 .2883 
 
 3.468 
 
 .3073 
 
 3 . 254 
 
 3265 
 
 3.062 
 
 .3460 
 
 2.890 
 
 55 
 
 1(1 
 
 .2711 
 
 3 689 
 
 .2899 
 
 3.449 
 
 .3089 
 
 3.237 
 
 .3281 
 
 3.017 
 
 .3476 
 
 2.877 
 
 50 
 
 15 
 
 .2726 
 
 3.6«8 
 
 .2915 
 
 3.431 
 
 .3105 
 
 3.220 
 
 .3297 
 
 3.033 
 
 .3492 
 
 2.864 
 
 45 
 
 •20 
 
 .2742 
 
 3.647 
 
 .2930 
 
 3 412 
 
 .3121 
 
 3.204 
 
 .3314 
 
 3.018 
 
 .3.508 
 
 2.850 
 
 40 
 
 25 
 
 .2758 
 
 3.626 
 
 .2946 
 
 3.394 
 
 .3137 
 
 3.188 
 
 .3330 
 
 3.003 
 
 .3525 
 
 2 . 837 
 
 15 
 
 30 
 
 .2773 
 
 3.606 
 
 .2962 
 
 3.376 
 
 .3153 
 
 3.172 
 
 .3346 
 
 2.989 
 
 . 3551 
 
 2.824 
 
 ^0 
 
 35 
 
 .2789 
 
 3.5«6 
 
 .2978 
 
 3.358 
 
 .3169 
 
 3.1.56 
 
 .3362 
 
 2.974 
 
 .35.58 
 
 2.811 
 
 25 
 
 40 
 
 .2805 
 
 3.566 
 
 .2994 3.340 
 
 .3185 
 
 3.140 
 
 3378 
 
 2.960 
 
 .3574 
 
 2.798 
 
 20 
 
 45 
 
 .2820 
 
 3.546 
 
 .3010 3.323 
 
 .3201 
 
 3.124 
 
 .3394 
 
 2.946 
 
 .3590 
 
 2.785 
 
 15 
 
 50 
 
 .2836 
 
 3.526 
 
 .3025 
 
 3.305 
 
 .3217 
 
 3.108 
 
 .3411 
 
 2.932 
 
 .3607 
 
 2.772 
 
 10 
 
 55 
 
 .2852 
 
 3.507 
 
 .3041 
 
 3.288 
 
 .3233 
 
 3.093 
 
 .3427 
 
 2.918 
 
 .3623 
 
 2.760 
 
 5 
 
 60 
 
 .2867 
 
 3.487 
 
 .3057 
 
 3.271 
 
 .3249 
 
 3.078 
 
 .3443 
 
 2.904 
 
 .3640 
 
 2.747 
 
 
 
 Cotan 
 
 i'an . 
 
 Cotan. 
 
 Tang 
 
 coian. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 I'ang. 
 
 / 
 
 74° 1 
 
 73° 
 
 72° 1 
 
 71° 
 
 70° 
 
 ~0 
 
 20° 1 
 
 21° 
 
 22° 
 
 23° 
 
 24° 
 
 ' 
 
 Tana. 
 
 Cotan. 
 
 Pane. 
 
 Cotan. 
 
 Tang. Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 .3640 
 
 2.747 
 
 .3839 
 
 2.605 
 
 .4040 2.475 
 
 .4245 
 
 2.356 
 
 .44.52 
 
 2.246 
 
 60 
 
 5 
 
 .3656 
 
 2.735 
 
 .3855 
 
 2.594 
 
 .4054 2.467 
 
 .4258 
 
 2.348 
 
 .4470 
 
 2.237 
 
 55 
 
 10 
 
 .3673 
 
 2.723 
 
 .3872 
 
 2.583 
 
 .4074 2.454 
 
 .4279 
 
 2.337 
 
 .4487 
 
 2.229 
 
 50 
 
 15 
 
 .3689 
 
 2.711 
 
 .3889 
 
 2.571 
 
 .4091 2.444 
 
 .4296 
 
 2.328 
 
 .4505 
 
 2.220 
 
 45 
 
 20 
 
 .3706 
 
 2.698 
 
 .3905 
 
 2.560 
 
 .4108 2.434 
 
 .4314 
 
 2.318 
 
 .4.522 
 
 2.211 
 
 40 
 
 25 
 
 .3722 
 
 2.686 
 
 .3922 
 
 2.549 
 
 .4125 2.424 
 
 .4331 
 
 2.309 
 
 .4540 
 
 2.201 
 
 35 
 
 30 
 
 .3739 
 
 2.675 
 
 .3939 
 
 2.539 
 
 .4142 2.414 
 
 .4348 
 
 2.300 
 
 .4557 
 
 2.194 
 
 30 
 
 35 
 
 .3755 
 
 2.663 
 
 .3956 
 
 2.528 
 
 .4159 
 
 2.404 
 
 .4365 
 
 2.291 
 
 .4575 
 
 2.186 
 
 25 
 
 40 
 
 .3772 
 
 2.651 
 
 .3973 
 
 2.517 
 
 .4176 
 
 2.394 1 .4383 
 
 2.282 
 
 .4592 2.177 
 
 20 
 
 45 
 
 .37e9 
 
 2.639 
 
 .3989 1 2.501 
 
 .4193 
 
 2.385 .4400 
 
 2.273 
 
 .4610 2.169 
 
 15 
 
 50 
 
 .3805 
 
 2.628 
 
 .4006 
 
 2.496 
 
 .4210 
 
 2.375 
 
 .4417 
 
 2.264 
 
 .4628 2.161 
 
 10 
 
 55 
 
 .3822 
 
 2.616 
 
 .4023 
 
 2.485 
 
 .4228 
 
 2.365 
 
 .4435 2.255 
 
 .4645 2.153 
 
 5 
 
 60 
 
 .3839 
 
 2.695 
 
 .4040 
 
 2.475 
 
 4245 
 
 2.3.56 
 
 .44.52 2.246 
 
 .4663 2.144 
 
 
 
 ^ 
 
 Cotan. i Tans. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 i ang, 
 
 Coian. I ans 
 
 Cotan. i'i'ang. 
 
 69° 
 
 68° 
 
 67° 
 
 66° 
 
 65° 
 
 
 ~0 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 / 
 
 60 
 
 Tang. 
 
 V otan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. Cotan. 
 
 i ang 
 
 Cotan. 
 
 .4663 
 
 2.144 
 
 .4877 
 
 2 050 
 
 .5095 
 
 1 962 
 
 .5317 
 
 1.881 
 
 ..5.543 
 
 1.804 
 
 5 
 
 .4681 
 
 2.136 
 
 .4895 
 
 2.043 
 
 .5114 
 
 1.956 
 
 ..5336 
 
 1.874 
 
 .5562 
 
 1.798 
 
 55 
 
 10 
 
 .4698 
 
 2.128 
 
 .4913 
 
 2.035 
 
 ..5132 
 
 1.949 
 
 .5354 
 
 1.868 
 
 .5.581 
 
 1.792 
 
 50 
 
 15 
 
 .4716 
 
 2.120 
 
 .4931 
 
 2.028 
 
 ,5150 
 
 1.942 
 
 .5373 
 
 1.861 
 
 .5600 
 
 1.786 
 
 45 
 
 20 
 
 .4734 
 
 2.112 
 
 .4949 
 
 2.020 
 
 .5169 1.935 
 
 .5392 
 
 1.855 
 
 .5619 
 
 1.779 
 
 40 
 
 25 
 
 .4752 
 
 2.104 
 
 .4968 
 
 2.013 
 
 .5187 
 
 1.928 
 
 .5411 
 
 1.848 
 
 .5638 
 
 1.773 
 
 35 
 
 30 
 
 .4770 
 
 2.096 
 
 .4986 
 
 2.005 
 
 .5206 
 
 1.921 
 
 .5429 
 
 1.842 
 
 .5658 
 
 1.767 
 
 30 
 
 35 
 
 .4788 
 
 2.089 
 
 .5004 1.998 
 
 ..5224 
 
 1.914 
 
 .5448 
 
 1.835 
 
 .5677 
 
 1 761 
 
 25 
 
 40 
 
 .4805 
 
 2.081 
 
 .5022 1 991 
 
 .5243 
 
 1 .907 ^ .5467 
 
 1.829 
 
 .5696 
 
 1.755 
 
 20 
 
 45 
 
 .4823 ' 2.073 
 
 .5040 1.984 
 
 .5261 
 
 1.901 
 
 .5486 
 
 1.823 
 
 .5715 
 
 1.750 
 
 15 
 
 50 
 
 .4841 2.065 
 
 ..50.59 1.977 
 
 .5280 
 
 1.894 
 
 .5505 
 
 1.816 
 
 5735 
 
 1.744 
 
 10 
 
 55 
 
 .4849 2.058 .5077 1.970 
 
 .5298 
 
 1.887 
 
 .5524 
 
 1.810 
 
 .5754 
 
 1.738 
 
 5 
 
 60 
 
 .4877 1 2.050 L -5095 | 1.963 
 
 .5317 
 
 1.881 
 
 .5543 
 
 1.804 
 
 .5773 
 
 1.732 
 
 
 
 Jotan. 1 Tang. 
 
 Cotan. 1 Tang 
 
 Cotan. I 1 ang 
 
 Cotan. 
 
 Tang, 
 
 Cotan 
 
 iang. 
 
 64° 
 
 63° 
 
 62° 
 
 61° 
 
 60° 
 
T^BILIE II B, 
 
 Natural Tangents and Cotangents — Cont'd. 
 
 > 
 
 "o 
 
 30° 
 
 3i° 
 
 3^° 
 
 33" 
 
 ii'^ 
 
 / 
 
 Tang. Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tans, 
 
 Cotan. 
 
 .5773 
 
 1.732 
 
 .6008 
 
 1.664 
 
 .62*9 
 
 1.600 
 
 .6494 
 
 1.540 
 
 .6745 
 
 1.483 
 
 60 
 
 5 
 
 .5793 
 
 1.726 
 
 .6028 
 
 1.659 
 
 .6269 
 
 1.595 
 
 6515 
 
 1.535 
 
 .6766 
 
 1.478 
 
 55 
 
 10 
 
 .5812 
 
 1,720 
 
 .6048 
 
 1.653 
 
 .6289 
 
 1.590 
 
 .6535 
 
 1.530 
 
 .6787 
 
 1.473 
 
 50 
 
 15 
 
 .5832 1.715 
 
 .6068 
 
 1.648 
 
 .6309 
 
 1.5S5 
 
 .6556 
 
 1.525 
 
 .6809 
 
 1.469 
 
 45 
 
 20 
 
 5851 '1.709 
 
 .6088 
 
 1 643 
 
 .6330 
 
 1.580 
 
 .6577 
 
 1.520 
 
 .6830 
 
 1.464 
 
 40 
 
 25 
 
 .5871 
 
 1.703 
 
 .6108 
 
 1.637 
 
 .63-30 
 
 1.575 
 
 .6598 
 
 1.516 
 
 .68.51 1.4601 
 
 35 
 
 30 
 
 5891 
 
 1.698 
 
 .6128 
 
 1.632 
 
 .6-^71 
 
 1.570 
 
 .6619 
 
 1.511 
 
 .6873 
 
 1.455 
 
 30 
 
 35 
 
 .5910 
 
 1.692 
 
 .6148 
 
 1.626 
 
 .6391 
 
 1.565 
 
 .6640 
 
 1.506 
 
 .6894 
 
 1.4.50 
 
 25 
 
 40 
 
 .5930 
 
 1.686 
 
 .6168 
 
 1.621 
 
 .6-12 
 
 1.560 
 
 6661 
 
 1.501 
 
 .6916 
 
 1.446 
 
 20 
 
 45 
 
 .5949 
 
 1.681 
 
 .6188 
 
 1.616 
 
 .64^*2 
 
 1.555 
 
 .6682 
 
 1.497 
 
 .6937 
 
 1.441 
 
 15 
 
 50 
 
 .5969 
 
 1.675 
 
 .6208 
 
 1.611 
 
 .6-^53 
 
 1.550 
 
 .6703 
 
 1.492 
 
 .6959 
 
 1.438 
 
 10 
 
 55 
 
 .5989 
 
 1.670 
 
 .6228 
 
 1.606 
 
 .6^73 
 
 1.545 .6724 
 
 1.487 
 
 .6980 
 
 1.433 
 
 5 
 
 60 
 
 .6099 
 
 1.664 
 
 .6249 
 
 1.600 
 
 .6494 
 
 1..540 .6745 
 
 1.483 
 
 .7002 
 
 1.428 
 
 _0 
 
 Cotan 
 
 lan^. 
 
 Cotan. 
 
 Tang 
 
 -,oian. 
 
 1 ang. Cotan. 
 
 Tang. 
 
 i^otan. 
 
 Tang. 
 
 
 59° 
 
 58° 
 
 57° 1 56° 
 
 55° 
 
 ^ 
 
 - 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 / 
 
 Tana. 
 
 Cotan. 
 
 Tane. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 
 
 .7002 
 
 1.428 
 
 .7265 
 
 1.376 
 
 .7535 
 
 1.327 
 
 .7813 
 
 1 288 
 
 .8098 
 
 1.235 
 
 60 
 
 5 
 
 .7024 
 
 1.424 
 
 .7288 
 
 1.372 
 
 .7558 
 
 1.323 
 
 .7836 
 
 1.276 
 
 .8122 
 
 1.231 
 
 55 
 
 10 
 
 .7045 
 
 1.419 
 
 .7310 
 
 1.368 
 
 .7581 
 
 1.319 
 
 .7860 
 
 1.272 
 
 .8146 
 
 1.228 
 
 50 
 
 15 
 
 .7067 
 
 1.415 
 
 .7332 
 
 1.364 
 
 .7604 
 
 1.315 
 
 .7883 
 
 1.268 
 
 .8170 
 
 1.224 
 
 45 
 
 20 
 
 .7089 
 
 1.411 
 
 .7355 
 
 1.360 
 
 .7627 
 
 1.311 
 
 .7907 
 
 1.265 
 
 .8195 
 
 1.220 
 
 40 
 
 25 
 
 .7111 
 
 1.406 
 
 .7377 
 
 1.355 
 
 .7650 
 
 1.307 
 
 .7931 
 
 1.261 
 
 .8219 
 
 1.217 
 
 35 
 
 3C 
 
 .7133 
 
 1.402 
 
 .7400 
 
 1.351 
 
 .7673 
 
 1.303 
 
 .7954 
 
 1.257 
 
 .8243 
 
 1.213 
 
 30 
 
 35 
 
 .7155 
 
 1.398 
 
 .7422 
 
 1.347 
 
 .7696 
 
 1.299 
 
 .7978 
 
 1.2.53 
 
 .8268 
 
 1.209 
 
 25 
 
 40 
 
 .7178 
 
 1.393 
 
 .7445 
 
 1.343 
 
 .7719 
 
 1.295 
 
 .8002 
 
 1.250 
 
 .8292 
 
 1.206 
 
 20 
 
 45 
 
 .7199 
 
 1.389 
 
 .7467 
 
 1.339 
 
 .7742 
 
 1.291 
 
 .e026 
 
 1.246 
 
 .8317 
 
 1.202 
 
 15 
 
 50 
 
 .7221 
 
 1.385 
 
 .7490 
 
 1.335 
 
 . 7766 
 
 1.287 
 
 .8050 
 
 1.242 
 
 .8341 
 
 1.199 
 
 10 
 
 55 
 
 .7243 
 
 1.381 
 
 .7513 
 
 1.331 
 
 .7789 1.284 
 
 .8074 
 
 1.239 
 
 .8366 
 
 1.195 
 
 5 
 
 60 
 
 .7265 
 
 1.376 
 
 .7535 
 
 1.327 
 
 7813 1.280 
 
 .8098 
 
 1.235 
 
 .8391 
 
 1.192 
 
 J) 
 
 > 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 1 ang, 
 
 Cotan. 
 
 1 ang 
 
 Cotan. 
 
 lang. 
 
 54° 
 
 53° 1 
 
 52° 
 
 5 
 
 o 
 
 50° 
 
 ^ 
 
 40° 
 
 41^ 
 
 42° 
 
 43° 
 
 44° 
 
 / 
 60 
 
 Tang. 
 
 otan. 
 
 Tang. 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang, 
 
 Cota . 
 
 1 ang 
 
 Cotan. 
 
 
 
 .8391 
 
 1.192 
 
 .8693 
 
 1 150 
 
 .9004 
 
 1 111 
 
 .9325 
 
 1.072 ^ .9657 
 
 1.036 
 
 5 
 
 .8416 
 
 1.188 
 
 .8718 
 
 1.147 
 
 .9030 
 
 1.107 
 
 .9352 
 
 1.069 
 
 .9685 
 
 1.032 
 
 55 
 
 10 
 
 .8441 
 
 1.185 
 
 .8744 
 
 1.144 
 
 .90.57 
 
 1.104 
 
 .9380 
 
 1.066 
 
 .9713 
 
 1.029 
 
 50 
 
 15 
 
 .8466 
 
 1.181 
 
 .8770 
 
 1.140 
 
 ,9083 
 
 1.101 
 
 .9407 
 
 1.063 
 
 .9742 
 
 1.026 
 
 45 
 
 20 
 
 .8491 
 
 1.178 
 
 .8795 
 
 1.137 
 
 .9101 
 
 1.098 
 
 .9434 
 
 1.060 
 
 .9770 
 
 1.023 
 
 40 
 
 25 
 
 .8516 
 
 1.174 
 
 .8821 
 
 1.134 
 
 .9137 
 
 1.094 
 
 .6462 
 
 1.057 
 
 .9798 
 
 1.021 
 
 35 
 
 'Sd 
 
 .8541 
 
 1.171 
 
 .8847 
 
 1.130 
 
 .9163 
 
 1.091 
 
 .9490 
 
 1.0.54 
 
 .9827 
 
 1.018 
 
 30 
 
 35 
 
 .8566 
 
 l.]67 
 
 .8873 
 
 1.127 
 
 .9190 
 
 1.088 
 
 .9517 
 
 1.051 
 
 .9856 
 
 1 015 
 
 25 
 
 40 
 
 .8591 
 
 1.164 
 
 .8899 
 
 1 124 
 
 .9217 
 
 1.085 
 
 .9545 
 
 1.048 
 
 .9884 
 
 1.012 
 
 20 
 
 45 
 
 .8617 
 
 1.161 
 
 .8925 
 
 1.120 
 
 .9244 
 
 1.082 
 
 .9573 
 
 1.045 
 
 .9913 
 
 1.009 
 
 15 
 
 50 
 
 .8642 
 
 1.157 
 
 .89.51 
 
 1.117 
 
 .9271 
 
 1.079 
 
 .9601 
 
 1.042 
 
 .9942 
 
 1.006 
 
 10 
 
 55 
 
 .8667 
 
 1.154 
 
 .8978 
 
 1.114 
 
 .9298 
 
 1.075 
 
 .9629 
 
 1.039 
 
 .9971 
 
 1.003 
 
 5 
 
 60 
 
 .8693 
 
 1.150 
 
 .9004 
 
 1.111 
 
 .9325 
 
 1.072 
 
 .9657 
 
 1.036 
 
 1.000 
 
 1.000 
 
 _0 
 / 
 
 Cotan. 
 
 Tang. 
 
 Cotan. 
 
 Tang 
 
 Cotan. 
 
 lang 
 
 Cotan. 
 
 rang. 
 
 Cotan 
 
 Tang. 
 
 49° 
 
 48° 
 
 47° 
 
 46° 
 
 45° 
 
T-i^BIjIE II C, 
 
 Natural Secants and Cosecants 
 
 1) 
 
 0- 
 
 1° 
 
 i" 
 
 3° 
 
 4° 
 
 ^ 
 
 Secant 
 
 v^osec. 
 
 Se^ani 
 
 Cosec. 
 
 Secant| Cosec. 
 
 Secant! Cosec. 
 
 Secantj Coj^ec. 
 
 1.0000 
 
 Infinit 
 
 1 0001 
 
 57.300 
 
 1.0006128.654 
 
 1.0014J19.107 
 
 1. 002414. 335 
 
 60 
 
 5 
 
 1.0000 
 
 687.55 
 
 1.00i)2 
 
 52.891 
 
 1.0007127.508 
 
 1.001418.591 
 
 1.002514.043 
 
 55 
 
 10 
 
 1.0000 
 
 343.77 
 
 1.0002 
 
 49.114 
 
 1.0(K)7]26.150 
 
 1.001518.103 
 
 1.O026 13.763 
 
 50 
 
 15 
 
 1.0000 
 
 229.18 
 
 1.0002 
 
 45.840 
 
 1.000825.471 
 
 1.001617.639 
 
 1.0027il3.494 
 
 45 
 
 •20 
 
 1 0000 
 
 171. «9 
 
 1.0003 
 
 42.976 
 
 1.000S24..562 
 
 1.0017:17.198 
 
 1.0029,13.255 
 
 40 
 
 25 
 
 1 0080 
 
 137.51 
 
 1.0003 
 
 40.448 
 
 1.0099 23.716 
 
 1.0018116.779 
 
 1.003012.985 
 
 35 
 
 30 
 
 1.0000 
 
 114 59 
 
 1.0003 
 
 38.201 
 
 1.000922.925 
 
 1.0019:16.380 
 
 1.003112.745 
 
 30 
 
 35 
 
 1.0000 
 
 98.22 
 
 1.0004 
 
 36.191 
 
 1.6010 22.186 
 
 1.00J9;i6.000 
 
 1.003212.514 
 
 25 
 
 40 
 
 1.0001 
 
 85.94 
 
 1.0004 
 
 34.382 
 
 1 001121.494 
 
 1.0020;15.637 
 
 1.003312.291 
 
 20 
 
 45 
 
 1.0001 
 
 76.39 
 
 1.9005 
 
 32. 745 
 
 1.001120 843 
 
 1.0021115 290 
 
 1.003412.076 
 
 15 
 
 50 
 
 1.0001 
 
 68.76 
 
 1.0005 
 
 31.257 
 
 1.0012 20.230 
 
 1.0022!l4.958 
 
 1.003611.868 
 
 10 
 
 55 
 
 1.0001 
 
 62.51 
 
 1.0005 
 
 29 899 
 
 1.0013 
 
 19.653 
 
 1.0023 
 
 14.«40 
 
 1.0037111.668 
 
 5 
 
 60 
 
 1.0601 
 
 57.30 
 
 1.0006 
 
 28.654 
 
 1.0014 
 
 19.107 
 
 1.0024 
 
 14.335 
 
 1 003811.474 
 
 
 
 Cosec. 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 -lecani 
 
 Cosec. S cant 
 
 ' 
 
 8S 
 
 o 
 
 88° 1 
 
 87° 
 
 86° 1 
 
 85° 
 
 ^ 
 
 5° 
 
 6° 
 
 7° 
 
 b° 1 
 
 9° 
 
 60 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec, 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 Cosine 
 
 Secant 
 
 Cosec 
 
 
 
 1.0038 
 
 11.474 
 
 1.0055 
 
 9.567 
 
 1.0075 
 
 8.205 
 
 1.0098 
 
 7.185 
 
 1 0125 
 
 6.392 
 
 5 
 
 1.0039 
 
 11.286 
 
 1.0057 
 
 9.436 
 
 1.0077 
 
 8.109 
 
 1.0100 
 
 7.112 
 
 1.0127 
 
 6 334 
 
 55 
 50 
 
 10 
 
 1.0041 
 
 11.104 
 
 1.0058 
 
 9.309 
 
 1.0079 
 
 8.016 
 
 1 0102 
 
 7 039 
 
 1.0129 
 
 6.277 
 
 15 
 
 1.0042 
 
 10.929 
 
 1.0060 
 
 9.185 
 
 1.0081' 
 
 7.924 
 
 1.0104 
 
 6.969 
 
 1.0132 
 
 6.221 
 
 45 
 
 20 
 
 1.0043 
 
 10.758 
 
 1.0061 
 
 9. 065 
 
 1.0082 
 
 7.«34 
 
 1.0107 
 
 6 900 
 
 1. 0134 
 
 6.166 
 
 40 
 
 25 
 
 1.0045 
 
 10 593 
 
 1.0063 
 
 8.948 
 
 1.0084 
 
 7.747 
 
 1.0109 
 
 6 832 
 
 1.0136 
 
 6.112 
 
 35 
 
 30 
 
 1.0046 
 
 10.433 
 
 1.0065 
 
 8.834 
 
 1.0086 
 
 7.661 
 
 1.0111 
 
 6.765 
 
 1.0l.i9 
 
 6.059 
 
 30 
 
 35 
 
 1.0048 
 
 10.278 
 
 1.0066 
 
 8.722 
 
 1.0088 
 
 7.577 
 
 1.0113 
 
 6.700 
 
 1.0141 
 
 6.007 
 
 25 
 
 4C 
 
 1.0049 
 
 10.127 
 
 1.0068 
 
 8.614 
 
 1.0090 
 
 7.496 
 
 1.0115 
 
 6.636 
 
 1.0144 
 
 5.955 
 
 20 
 
 45 
 
 1.0050 
 
 9.981 
 
 1.0070 
 
 8.508 
 
 1.0092 
 
 7.416 
 
 1.0118 
 
 6.574 
 
 1.0146 
 
 5.905 
 
 15 
 
 50\ 
 
 1.0052 
 
 9.839 
 
 1.0071 
 
 8.405 
 
 1.0094 
 
 7.337 
 
 1.0120 
 
 6.512 
 
 1 0149 
 
 5.855 
 
 10 
 
 55 
 
 1.0053 
 
 9.701 
 
 1.0073 
 
 8.304 
 
 1.0096 
 
 7.260 
 
 1.0122 
 
 6.452 
 
 1.0152 
 
 5.807 
 
 5 
 
 60 
 
 1.0055 
 
 9.567 
 
 1.0075 
 
 8.205 
 
 1.0098 
 
 7.185 
 
 1.0125 
 
 6 392 
 
 1.0154 
 
 5.7.59 
 
 
 
 ^ 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. iSecant 
 
 ' 
 
 84° 
 
 83° 
 
 82° 
 
 81° 
 
 80° 
 
 ^ 
 
 10° 
 
 11° 
 
 12° 1 13° 
 
 14° 
 
 60 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Co<ec 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 V ijsec. 
 
 
 
 1.0154 
 
 5.789 
 
 1.0187 
 
 5.241 
 
 1.0223 
 
 4.810 
 
 1.0263 
 
 4.445 
 
 1.0306 
 
 4.134 
 
 5 
 
 1.0157 
 
 5.712 
 
 1.0190 
 
 5. 202 
 
 1.0226 
 
 4.777 
 
 1.0266 
 
 4.417 
 
 1.0310 
 
 4.109 
 
 55 
 
 10 
 
 1.0159 
 
 5.665 
 
 1.0193 
 
 5.164 
 
 1.0230 
 
 4.745 
 
 1.0270 
 
 4 390 
 
 1.0314 
 
 4.086 
 
 50 
 
 15 
 
 1.0162 
 
 5.620 
 
 i.ort6 
 
 5.126 
 
 1.0233 
 
 4.713 
 
 1.0273 
 
 4.363 
 
 1.0317 
 
 4. 062 
 
 45 
 
 20 
 
 1.0165 
 
 5.574 
 
 1.0199 
 
 5.088 
 
 1.0236 
 
 4.682 
 
 1. 0277 
 
 4.336 
 
 1.0321 
 
 4.039 
 
 40 
 
 25 
 
 1.0167 
 
 5.531 
 
 1 0202 
 
 5.052 
 
 1.0239 
 
 4. 651 
 
 1.0280 
 
 4.310 
 
 1.0325 
 
 4.016 
 
 35 
 
 30 
 
 1.0170 
 
 5.487 
 
 1.0205 
 
 5.016 
 
 1.0243 
 
 4.620 
 
 1.0284 
 
 4.284 
 
 1.0329 
 
 3.994 
 
 30 
 
 35 
 
 1.0173 
 
 5.445 
 
 1.0208 
 
 4.980 
 
 1.0246 
 
 4.590 
 
 1.0288 
 
 4.258 
 
 1.0333 
 
 3.9^2 
 
 25 
 
 40 
 
 1. 0176 
 
 5.403 
 
 1.0211 
 
 4.945 
 
 1.0249 
 
 4.560 
 
 1.0291 
 
 4.232 
 
 1.0337 
 
 3.949 
 
 20 
 
 45 
 
 1.0179 
 
 5.361 
 
 1.0214 
 
 4.911 
 
 1.02.53 
 
 4 . 531 
 
 1.0295 
 
 4.207 
 
 1.0341 
 
 3.828 
 
 15 
 
 50 
 
 1.0181 
 
 5 320 
 
 1.0217 
 
 4.876 
 
 1.0256 
 
 4.502 
 
 1.0300 
 
 4.182 
 
 1.0345 
 
 3.906 
 
 10 
 
 55 
 
 1.0184 
 
 2.280 
 
 1.0220 
 
 4.843 
 
 1.0260 
 
 4.474 
 
 1. 0302 
 
 4.158 
 
 1.0349 
 
 3.845 
 
 5 
 
 60 
 
 1.0187 
 
 5.241 
 
 1.0223 
 
 4.810 
 
 1.0263 
 
 4.445 
 
 1 0306 
 
 4.134 
 
 1.0353 
 
 3.864 
 
 
 
 Cosec 
 
 S'^cant 
 
 Cosec. 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 Cose 
 
 '^ecant 
 
 Cos«c. 
 
 Seran' 
 
 79° 
 
 78° 
 
 77° 
 
 76° 1 75° 
 
TJ^SXjE II c. 
 
 Natural Secants and Cosecants — Continued. 
 
 > 
 
 16" 
 
 lb° 
 
 17° 
 
 18° 
 
 19^ 
 
 ' 
 
 Secant 
 
 Cosec. 
 
 Secant! Cosec. 
 
 Secant 
 
 Co^ec. 
 
 recant 
 
 Cosec. 
 
 ^secant 
 
 Cosec. 
 
 
 
 1.0353 
 
 3.864 
 
 1.0403 3.628 
 
 1.0457 
 
 3.420 
 
 1.0515 
 
 3.236 
 
 1.0576 
 
 3.071 
 
 60 
 
 5 
 
 1.0357 
 
 3.843 
 
 1.04'^"'; 3.610 
 
 1.0461 
 
 3 . 404 
 
 1.0520 
 
 3.222 
 
 1.U.581 
 
 3.059 
 
 55 
 
 10 
 
 1.0361 
 
 3.822 
 
 1.0412 
 
 3.591 
 
 1.0466 
 
 3.388 
 
 1.0525 
 
 3.207 
 
 1.0587 
 
 3.046 
 
 50 
 
 15 
 
 1.0365 
 
 3.802 
 
 1.0416 
 
 3.574 
 
 1..0471 
 
 3.372 
 
 1.0530 
 
 3.193 
 
 1.0592 
 
 3.033 45 1 
 
 20 
 
 ].0369 
 
 3.782 
 
 1.0420 
 
 3.. 5.56 
 
 1.0476 
 
 3.356 
 
 1.0535 
 
 3.179 
 
 l..C'J8 
 
 3.021 
 
 40 
 
 25 
 
 1.0374 
 
 3.762 
 
 1.0425 
 
 3.538 
 
 1.0480 
 
 3.341 
 
 1 . 0540 
 
 3.165 
 
 1 . o603 
 
 3.008 
 
 35 
 
 30 
 
 1.0377 
 
 3.742 
 
 1.0429 
 
 3.. 521 
 
 1.04^5 
 
 3.325 
 
 1.0545 
 
 3.151 
 
 1.0608 
 
 2.996 
 
 30 
 
 35 
 
 1.0382 
 
 3.722 
 
 1.0134 
 
 3.. 504 
 
 1.0490 
 
 3.310 
 
 1.0550 
 
 3.138 
 
 1.0614 
 
 2.983 
 
 25 
 
 40 
 
 1.0386 
 
 3.703 
 
 1.0438 
 
 3.487 
 
 1.0495 
 
 3.295 
 
 1.0555 
 
 3.124 
 
 1.0619 
 
 2.971 
 
 20 
 
 45 
 
 1.0390 
 
 3.684 
 
 1.0443 
 
 3.470 
 
 1.0500 
 
 3.280 
 
 1.0560 
 
 3.111 
 
 1.0625 
 
 2.959 
 
 15 
 
 50 
 
 1.0394 
 
 3.665 
 
 1.0448 
 
 3.453 
 
 1.0505 
 
 3.265 
 
 1.0566 
 
 3.098 
 
 1.0630 2.947 
 
 10 
 
 55 
 
 1.0399 
 
 3.646 
 
 1.0452 
 
 3.437 
 
 1.0510 
 
 3.251 
 
 1.0571 
 
 3.085 
 
 1.0636' 2.935 
 
 5 
 
 60 
 
 1.0403 
 
 3.628 
 
 1.0457 
 
 3.420 
 
 1.0515 
 
 3.236 
 
 1.0576 3.071 
 
 1.0642 2.924 
 
 
 
 \ 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. jSecant 
 
 Cosec. Ibecant 
 
 ' 
 
 7^ 
 
 1° 
 
 73° 
 
 72° 
 
 71° 
 
 70° 
 
 1) 
 
 20° 
 
 21° 
 
 22° 
 
 23° 1 24° 
 
 6(> 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec 
 
 becant 
 
 Cosec. 
 
 1.0642 
 
 2.924 
 
 1.0711 
 
 2.790 
 
 1.0785 
 
 2.669 
 
 1.0864 
 
 2.559 
 
 1.0946 
 
 2.459 
 
 5 
 
 1.0647 
 
 2.912 
 
 1.0717 
 
 2.780 '1.0792 
 
 2.660 
 
 1.0870 
 
 2.551 
 
 1.0953 
 
 2.451 
 
 55 
 
 10 
 
 1.0653 
 
 2.901 
 
 1.0723 
 
 2.769 1.0798 
 
 2.650 
 
 1.0877 
 
 2.542 
 
 1.0961 
 
 2.443 
 
 50 
 
 15 
 
 1.0659 
 
 2.889 
 
 1.0729 
 
 2.759 
 
 1.0804 
 
 2.641 
 
 1.0^84 
 
 2.533 
 
 1.0968 
 
 2.435 
 
 45 
 
 20 
 
 1.0664 
 
 2.878 
 
 1.0736 
 
 2.749 
 
 1.0811 
 
 2.632 
 
 1.0891 
 
 2.525 
 
 1.0975 
 
 2.427 
 
 40 
 
 25 
 
 1.0670 
 
 2.867 
 
 1.0742 
 
 2.738 
 
 1.0817 
 
 2.622 
 
 1.0897 
 
 2.516 
 
 1.0982 
 
 2.419 
 
 35 
 
 30 
 
 1.0676 
 
 2.855 
 
 1.0748 
 
 2.728 
 
 1.0824 
 
 2.613 
 
 1.0904 
 
 2.508 
 
 1.0989 
 
 2..411 
 
 30 
 
 35 
 
 1.0682 
 
 2.844 
 
 1.0754 
 
 2.718 
 
 1.0830 
 
 2.604 
 
 1.0911 
 
 2.499 
 
 1.0997 
 
 2.404 
 
 25 
 
 40 
 
 1.0688 
 
 2.833 
 
 1.0761 
 
 2.708 
 
 1.0837 
 
 2.595 
 
 1.0918 
 
 2.491 
 
 1.1004 
 
 2.396 
 
 20 
 
 45 
 
 1.0694 
 
 2.822 
 
 1.0766 
 
 2.698 
 
 1.0844 
 
 2.586 
 
 1.0925 
 
 2.483 
 
 1.1011 
 
 2.389 
 
 15 
 
 50 
 
 1.069912.812 
 
 1.0773 
 
 2.689 
 
 1.0850 
 
 2.577 
 
 1.0932 
 
 2.475 
 
 1.1019 
 
 2.381 
 
 10 
 
 55 
 
 1.0705 2.801 
 
 1.0779 
 
 2.679 
 
 1.0857 
 
 2.568 
 
 1.0939 
 
 2.467 
 
 1.1026 2.374 
 
 5 
 
 60 
 
 1.0711[ 2.790 
 
 1.07851 2.669 
 
 1.0864 2.5.59 
 
 1.0946 
 
 2 459 
 
 1.1034,2.366 
 
 
 
 Cosec. jSecant 
 
 Cosec. Secant 
 
 Cosec. iSecant 
 
 Cosec, 
 
 Secant 
 
 Cosec. ISecant 
 
 69^ 
 
 68° 
 
 67° 1 66° 
 
 65° 
 
 ^ 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 2t.° 
 
 / 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 Cosec 
 
 Secantj Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 
 
 1.1034 
 
 2.366 
 
 1.1126 
 
 2.281 
 
 1.1223 
 
 2.203 
 
 1.1326 
 
 2.130 
 
 1.1433 
 
 2.063 
 
 60 
 
 5 1.1041 
 
 2.359 
 
 1.1134 
 
 2.274 
 
 1.1231 
 
 2.196 
 
 1.1334 
 
 2.124 
 
 1.1443 
 
 2.057 
 
 55 
 
 10 1.1019 
 
 2 351 
 
 1.1142 
 
 2.267 
 
 1.1240 
 
 2.190 
 
 1.1343 
 
 2.118 
 
 1.1452 
 
 2.052 
 
 50 
 
 15 1.1056 
 
 2.344 
 
 1.1150 
 
 2.261 
 
 1.1248 
 
 2.184 
 
 1.1352 
 
 2.113 
 
 1.1461 
 
 2.047 
 
 45 
 
 2011.1064 
 
 2.337 
 
 1.1158 
 
 2.254 
 
 1.1257 
 
 2.178 
 
 1.1361 
 
 2.107 
 
 1.1471 
 
 2.041 
 
 40 
 
 25 1.1072 
 
 2.330 
 
 1.1166 
 
 2.248 
 
 1.1265 
 
 2.172 ,1.1370 
 
 2.101 : 1.1480 
 
 2.036 
 
 .35 
 
 30| 1.1079 
 
 2.323 
 
 1.1174 
 
 2.241 
 
 1.1274 
 
 2.166 
 
 1.1379 
 
 2.(96 i 1.1489 
 
 2.031 
 
 30 
 
 35 1.1087 
 
 2.316 
 
 1.1182 
 
 2.235 
 
 1.1282 
 
 2.160 
 
 1.1388 
 
 2.090 ; 1.1499 
 
 2.026 
 
 25 
 
 40 1.1095 
 
 2.309 
 
 L.1190 
 
 2.228 
 
 1.1291 
 
 2.154 
 
 1.1397 
 
 2.085 i 1.1508 
 
 2.020 
 
 90 
 
 45 1.1102 
 5rt! 1.1110 
 55 1.1118 
 
 2.302 
 
 1.1198 
 
 2.222 
 
 1.1299 
 
 2.148 
 
 1.1406 
 
 2.679 1.1518 
 
 2.015 
 
 15 
 
 2.295 
 
 1.1207 
 
 2.215 
 
 1.1308 
 
 2.142 
 
 1.1415 
 
 2.073 1.1528 
 
 2.010 
 
 10 
 
 2.288 
 
 1.1215 
 
 2.209 
 
 1.1317 
 
 2.136 
 
 1.1424 
 
 2.068 1 1.15371 2.005 
 
 5 
 
 60 
 
 1.1126 
 
 2.281 
 
 1.1223 
 
 2 203 
 
 1.1326 
 
 2.130 
 
 1.1433 
 
 2.063 1.1.547i 2.000 
 
 
 
 ^ 
 
 Cosec. 
 
 recant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec.lSecant 
 
 L 
 
 64° 1 63° 1 
 
 62° 
 
 6 ° 
 
 60° 1 
 
TJ^BIjE ii c. 
 
 Natural Secants and Cosecants — Continued. 
 
 1) 
 
 30" 
 
 31° 
 
 32^ 
 
 133° 
 
 34° 
 
 60 
 
 Secant 
 
 Cosec 
 
 SecantlCosec 
 
 Secant Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secantl Cosec. 
 
 1.1547 
 
 2.000 
 
 1 1666 1.942 
 
 1.1792 1.887 
 
 1.J924 
 
 1.836 
 
 1.2062 1.788 
 
 5 
 
 1 1557 
 
 1.995 
 
 1.167611.937 
 
 1.1802 1.883 
 
 1.1935 
 
 1.832 
 
 1.2074^ 1.784 
 
 55 
 
 10 
 
 1.1566 
 
 1.990 
 
 1.1687 
 
 1.932 1.1813 
 
 1.878 
 
 1.1946 
 
 1.8-28 
 
 1.2086: 1.781 
 
 50 
 
 15 
 
 ].1576 
 
 1.985 
 
 1.1697 
 
 1.928 1.1824 
 
 1.874 
 
 1.1958 
 
 1.824 
 
 1.2098 1.777 
 
 45 
 
 '20 
 
 1 1586 
 
 1.9^0 
 
 1.1707 
 
 1.923 
 
 1 1835 
 
 1.870 
 
 1.1969 
 
 1.820 
 
 1.2110 1.773 
 
 40 
 
 25 
 
 1 1596 
 
 1.975 
 
 1.1718 
 
 1.918 
 
 1.1846 
 
 1.865 
 
 1.1980 
 
 1.816 
 
 1.2122 1.769 
 
 35 
 
 80 
 
 1.1606 
 
 1 970 
 
 1.1728 
 
 1.914 
 
 1.1857 
 
 1.861 
 
 1.1992 
 
 1.812 
 
 1.2134; 1.765 
 
 30 
 
 35 
 
 1.1616 
 
 1.965 
 
 1.1739 
 
 1. 909 
 
 1.1868 
 
 1.857 
 
 1.2004 
 
 1.808 
 
 1.2146; 1.762 
 
 25 
 
 40 
 
 1.1626 
 
 1.9 1 
 
 1.1749 
 
 1.905 
 
 1 1879 
 
 1 853 
 
 1.2015 
 
 1.804 
 
 1.21.58 
 
 1.7.58 
 
 20 
 
 45 
 
 1.1636 
 
 1.956 
 
 1.1760 
 
 1.900 
 
 1.1890 
 
 1 848 
 
 1.2^)27 
 
 1 800 
 
 1.2171 
 
 1.754 
 
 1§ 
 
 50 
 
 1 . 1646 
 
 1.951 
 
 1.1770 
 
 1 896 
 
 1.1901 
 
 1.844 
 
 1.2039 
 
 1.796 
 
 1.2183 
 
 1.751 
 
 10 
 
 55 
 
 1 1656 
 
 1.946 
 
 1.1781 
 
 1.891 
 
 1.1912 
 
 1.940 
 
 1.2050 
 
 1.792 
 
 1.2195 
 
 1.747 
 
 5 
 
 6U 
 
 1.1666 
 
 1.942 
 
 1.1792 
 
 1.887 
 
 1.1924 
 
 1.836 
 
 1.2062 
 
 1.788 
 
 1 2208 
 
 1.743 
 
 ^ 
 
 Cosec. 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 < osec. 
 
 Secant 
 
 Cosec. 
 
 ■^ec.ini 
 
 Cosec. 
 
 S- cant 
 
 5S 
 
 58° 1 
 
 57° 1 56° 
 
 55° 
 
 "o 
 
 3o° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 60 
 
 S -cant 
 
 Cosec. 
 
 Secant 
 
 Cos-c, 
 
 Secant 
 
 Cosec 
 
 >ecant 
 
 Cospc 
 
 Secant 
 
 Cosec 
 
 1.2208 
 
 1.743 
 
 1.2361 
 
 1.701 
 
 1.2521 
 
 1.662 
 
 1.2690 
 
 1.624 
 
 1.2867 
 
 1.589 
 
 5 
 
 1.1^2 
 
 1.740 
 
 1 2374 
 
 1.698 
 
 1.2535 
 
 1 6.58 
 
 1.2705 
 
 1.621 
 
 1.2883 
 
 1 586 
 
 55 
 
 10 
 
 1.2233 
 
 1.736 
 
 1.2387 
 
 1 694 
 
 12549 
 
 1 655 
 
 1 2719 
 
 1 618 
 
 1.2898 
 
 1.583 
 
 50 
 
 15 
 
 1.2245 
 
 1.733 
 
 1.2400 
 
 1.691 
 
 1.2563 
 
 1 652 
 
 1.2734 
 
 1.615 
 
 1.2913 
 
 1.580 
 
 45 
 
 20 
 
 1.2258 
 
 1 . 72iJ 
 
 1.2413 
 
 1.688 
 
 1.2577 
 
 1 649 
 
 1.2748 
 
 1 612 
 
 1. 2929 
 
 1.578 
 
 40 
 
 25 
 
 1.2270 
 
 1 726 
 
 1.2427 
 
 1.684 
 
 1.2501 
 
 1.646 
 
 1.2763 
 
 1 609 
 
 1.2944 
 
 1 575 
 
 35 
 
 30 
 
 1.2283 
 
 1.722 
 
 1.2440 
 
 1.6^1 
 
 12005 
 
 1.643 
 
 1.277« 
 
 1.6r.6 
 
 1.2960 
 
 1.572 
 
 30 
 
 35 
 
 1.2296 
 
 1.718 
 
 1.2453 
 
 1.678 
 
 1.2619 
 
 1.640 
 
 1.2793 
 
 1.603 
 
 1.2975 
 
 1.569 
 
 25 
 
 40 
 
 1.2309 
 
 1715 
 
 1.2467 
 
 1 675 
 
 1 . 2633 
 
 1.636 
 
 1.2807 
 
 1.600 
 
 1.2991 
 
 1.567 
 
 20 
 
 45 
 
 1.2322 
 
 1.7J2 
 
 1.2480 
 
 1 671 
 
 1.2f^47 
 
 1.633 
 
 1.2822 
 
 1.598 
 
 1.3006i 1..564 
 
 15 
 
 50 
 
 1.2335 
 
 1.708 
 
 1 2494 
 
 1 668 
 
 1.2661 
 
 1.630 
 
 1.2837 
 
 1.595 
 
 1 3022 1.561 
 
 10 
 
 55 
 
 1.2348 
 
 1.705 
 
 1.2508 
 
 1.665 
 
 1.2676 
 
 1.627 
 
 1.2852 
 
 1.592 
 
 1.3038; 1 558 
 
 5 
 
 60 
 
 1 2364 
 
 1.701 
 
 1.2521 
 
 1.662 
 
 1.2690 
 
 1.624 
 
 1.2867 
 
 1.589 
 
 1.30.54 
 
 1..556 
 
 Secant 
 
 J) 
 
 Cosec. 
 
 Secant 
 
 Coser. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 Secant 
 
 Cosec. 
 
 5-1° 1 
 
 53° 1 
 
 52° 
 
 51° 
 
 50° 
 
 1, 
 
 40° 
 
 41° 
 
 42° 
 
 43° 
 
 44° 
 
 / 
 
 60 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 Co ec 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 Cosec 
 
 Secant 
 
 V osec. 
 
 1.3054 
 
 1.556 
 
 1.325 
 
 1.524 
 
 1.346 
 
 1.4^>4 
 
 1.367 
 
 1.466 
 
 1.390 
 
 1.439 
 
 5 
 
 1.3070 
 
 1.553 
 
 1.327 
 
 1 522 
 
 1.347 
 
 1 492 
 
 1.369 
 
 1.464 
 
 1.392 
 
 1.437 
 
 55 
 
 10 
 
 1.3086 
 
 1.550 
 
 1.328 
 
 1.519 
 
 1.349 
 
 1.490 
 
 1.371 
 
 1.462 1.394 
 
 1.435 
 
 50 
 
 15 
 
 1.3102 
 
 1.548 
 
 1 .330 
 
 1.517 
 
 1.351 
 
 1.487 
 
 1.373 
 
 1.459! 1.396 
 
 1.433 
 
 45 
 
 20 
 
 1.3118 
 
 1.545 
 
 1.332 
 
 1.514 
 
 1.353 
 
 1.485 
 
 1 375 
 
 1.457 
 
 1.398 
 
 1.431 
 
 ^0 
 
 25 
 
 1.3134 
 
 l.o42 
 
 1 333 
 
 1.512 
 
 1.354 
 
 1.483 
 
 1.377 
 
 1.455 
 
 1.400 
 
 1.429 
 
 35 
 
 30 
 
 1.3151 
 
 1.539 
 
 1.335 
 
 1.509 
 
 1.356 
 
 1.480 
 
 1.379 
 
 1.453 
 
 1 .402 
 
 1.427 
 
 30 
 
 3;- 
 
 1.3167 
 
 1.537 
 
 1.337 
 
 1.507 1.358 
 
 1.478 
 
 1.381 
 
 1. 450 
 
 1.404 
 
 1.425 
 
 25 
 
 40 
 
 1.3184 
 
 1.534 
 
 1.339 
 
 1.504 ! 1.360 
 
 1.476 
 
 1.382 
 
 1.448 
 
 1.406 
 
 1.423 
 
 20 
 
 45 
 
 1.3200 
 
 '1 532 
 
 1.340 
 
 1.502 1.362 
 
 1.473 
 
 1.384 
 
 1.446 
 
 1.408 
 
 1.420 
 
 15 
 
 50 
 
 1.3217 
 
 1 529 
 
 1.342 
 
 1 499 1.364 
 
 1.471 
 
 1.386 
 
 1.444 
 
 1.410 
 
 1.418 
 
 10 
 
 55 
 
 1.3233 
 
 1.527 
 
 1.344 
 
 1.497 1.365 
 
 1.469 
 
 1.388 
 
 1.442 
 
 1.412 , 1.416 
 
 5 
 
 60 
 
 1.3250 
 
 1.524 
 
 1.346 
 
 1.494 1.367 
 
 1 466 
 
 1 390 
 
 1.439 
 
 1.414 1.414 
 
 
 Cosec 
 
 Secant 
 
 Cosec. 
 
 -ecant 
 
 Cosec. 
 
 Secant 
 
 Cose . 
 
 Secant 
 
 Cos-c. Secant 
 
 49° 
 
 48° 
 
 47° 
 
 46° 
 
 45° 
 
Trigonometrical and Conversion Table. III. 
 
 BEING A TABLE OF RHUMBS. 
 
 Name 
 
 of 
 Course. 
 
 N. 
 and 
 S. 
 
 Points. 
 
 
 Degrees 
 and 
 
 Minutes, 
 
 1 
 
 49 
 13 
 
 373^ 
 
 2 
 26 
 50M 
 
 Sine 
 
 or 
 
 D eparture 
 
 .0000 
 .0246 
 .0491 
 .0735 
 .0980 
 .1224 
 .1467 
 .1709 
 
 Co Sine 
 
 or 
 Diff. Lat. 
 
 1.0000 
 .9997 
 .9988 
 .9973 
 .99;!;2 
 .9925 
 .9892 
 .9853 
 
 Tangent. 
 
 .0000 
 .0245 
 .0492 
 .0737 
 .0985 
 .1234 
 .1483 
 .1736 
 
 Secant. 
 
 0.0000 
 1.0003 
 1.0012 
 1.0027 
 1.0048 
 1.0076 
 1.0109 
 1.0150 
 
 and 
 3.by|E,. 
 
 N. N,{E-. 
 
 and 
 
 S. S. 
 
 (E. 
 
 iw. 
 
 fE. byN. 
 •tw.byN. 
 
 and 
 
 c fE. by S. 
 ^- 1 W. by S. 
 
 % 
 
 78 
 
 
 5/8 
 
 'A 
 
 11 15 
 
 12 395^ 
 
 14 4 
 
 15 28 
 
 16 52M 
 
 18 17 
 
 19 41 
 21 oK 
 
 .1951 
 .2191 
 .2430 
 .2667 
 .2903 
 .3137 
 .3368 
 .3599 
 
 .9757 
 .9700 
 .9638 
 .9569 
 .9495 
 .9417 
 .9330 
 
 22 30 
 
 23 54^ 
 
 25 19 
 
 26 43 
 
 28 VA 
 
 29 32 
 
 30 56 
 32 20^ 
 
 .3827 
 .4053 
 .4276 
 .4496 
 .4714 
 .4929 
 .5140 
 .5350 
 
 .9239 
 .9142 
 .9040 
 .8932 
 8819 
 .8701 
 .8578 
 .8448 
 
 33 
 
 45 
 
 .5556 
 
 35 
 
 9/2 
 
 .5758 
 
 36 
 
 34 
 
 .5958 
 
 37 
 
 58 
 
 .6152 
 
 39 
 
 12% 
 
 .6344 
 
 40 
 
 47 
 
 .6532 
 
 42 
 
 11 
 
 .6715 
 
 43 
 
 35K 
 
 .6895 
 
 .8315 
 .8176 
 .8032 
 .7884 
 .7730 
 .7572 
 .7410 
 .7243 
 
 .1989 
 .2246 
 .2506 
 .2767 
 .3034 
 .3304 
 .3577 
 .3857 
 
 .4142 
 .4433 
 .4730 
 .5033 
 .5345 
 .5665 
 . 5992 
 .6332 
 
 .6682 
 .7043 
 .7418 
 .7803 
 .8207 
 .8627 
 .9062 
 .9520 
 
 1.0196 
 1.0249 
 1.0309 
 1 0376 
 1.0450 
 1.0532 
 1.0629 
 1.0718 
 
 1.0824 
 1.0939 
 1.1062 
 1.1195 
 1.1339 
 1.1493 
 1.1658 
 1.1836 
 
 1.2025 
 1.2231 
 1.2451 
 1.2684 
 1 .2936 
 1.3207 
 1.3495 
 1.380T 
 
 122 
 
Trigonometrical and Conversion Table, ill. 
 
 BEING A TABLE OF RHUMBS. 
 
 Name 
 
 of 
 Course. 
 
 Points. 
 
 Degrees Sine 
 
 and or 
 INIinutes. Departure 
 
 Co Sine 
 
 or 
 DifF. Lat. 
 
 Tangent. 
 
 Secant. 
 
 X' f E. 
 and 
 
 4. 
 
 3/8 
 
 45 
 
 46 24^ 
 
 47 49 
 
 49 13 
 
 50 37K 
 
 52 2 
 
 53 26 
 
 54 50% 
 
 .7071 
 .7242 
 .7410 
 .7572 
 .7730 
 .7884 
 .8032 
 .8176 
 
 .7071 
 
 .6895 
 .6715 
 .6532 
 .6344 
 .6152 
 .5958 
 .5758 
 
 1.0000 
 1.0,504 
 1.1035 
 1.1592 
 1.2185 
 1.2815 
 1.3481 
 1.4198 
 
 1,4142 
 
 1.4502 
 1.4892 
 1.5.309 
 1..5763 
 1.6255 
 1.6785 
 1.7366 
 
 ^, fE.byE. 
 N'lw.byW. 
 
 and 
 
 ^ fE.byE. 
 ^•\W, byW. 
 
 5. 
 
 /8 
 
 56 15 
 
 57 39% 
 
 59 4 
 
 60 28 
 
 61 523^ 
 
 63 17 
 
 64 41 
 66 bV^ 
 
 .8315 
 .8449 
 .8578 
 .8701 
 .8819 
 .8932 
 .9040 
 .9142 
 
 .5556 
 .5350 
 .5i40 
 .4929 
 .4714 
 .4497 
 .4476 
 .4053 
 
 1.4966 
 1.5793 
 1.6687 
 1 . 7651 
 1.8708 
 1.9868 
 2.1139 
 2.2558 
 
 1.7999 
 1.8694 
 1.9454 
 2.0287 
 2.1214 
 2. 2243 
 2.3385 
 2.4675 
 
 E. N. E. 
 W. N. W. 
 
 and 
 
 E. S.E. 
 W. S. W. 
 
 6. 
 
 67 30 
 
 68 54^ 
 
 70 19 
 
 71 43 
 
 73 7^ 
 
 74 32 
 
 75 56 
 77 20!x^ 
 
 .9239 
 .9330 
 .9415 
 
 .9495 
 .9569 
 .9638 
 .9700 
 .9757 
 
 .3827 
 .3599 
 .3368 
 .3137 
 .2903 
 .2667 
 .2430 
 .2191 
 
 2.4142 
 2.5926 
 2.79.54 
 3.0267 
 3.2966 
 3.6140 
 3.9910 
 4.4525 
 
 2.6131 
 2.7788 
 2.9689 
 3.18'^6 
 3. 4449 
 3.7498 
 4.1144 
 4.5634 
 
 w.byN. 
 
 and 
 
 w! by s. 
 
 7. 
 
 3/8 
 
 /8 
 3/ 
 % 
 
 78 45 
 
 80 91 
 
 81 34 
 
 82 58 
 
 84 223^ 
 
 85 47 
 
 87 11 
 
 88 353^ 
 
 .9808 
 .9853 
 .9892 
 .9925 
 .9952 
 .9973 
 .9988 
 .9997 
 
 .1951 
 .1709 
 .1467 
 .1224 
 
 .0980 
 .0735 
 .0491 
 .0246 
 
 5.0273 5.1258 
 5.7647 i 5.8505 
 6.7448 6.8i86 
 8.1054 ' 8.1668 
 10.154 10.2023 
 13.563 13.6002 
 20.. 325 20.3809 
 40.688 40.6889 
 
 East— West. 8. 
 
 90 00 1.0000 
 
 .0000 
 
 Infinite. 
 
 123 
 
The Sun's Amplitude. IV. 
 
 With rate of change of Azimuth, from Sunrise to the time of his crossing the 
 Prime Vertical, and with the time of his Risinij and Setting. 
 
 Lat. 
 
 41° 
 
 42^ 
 
 43° 
 
 44° 
 
 45° 
 
 Dec. 
 
 ChHiige 
 
 of Azimuth 
 
 1^ in b^4 m. 
 
 Change 
 
 of Azimuth 
 
 1° in 6}i m. 
 
 Chancre 
 of Azimuth 
 1° in 6 m. 
 
 Change 
 of Azimuth 
 1° in 6 m. 
 
 Change 
 of Azimuth 
 1^ in 6 m. 
 
 
 AMP. 
 
 H. M. 
 
 AMP. 
 
 H. M. 
 
 AMP. 
 
 H. M. 
 
 AMP. 
 
 H. M. 
 
 AMP. 
 
 H. M. 
 
 1° 
 
 1°. 20' 
 
 R5.57 
 b 6.03 
 
 i°.2r 
 
 R5.56 
 
 S6.04 
 
 1°.22' 
 
 R5.56 
 =>6.04 
 
 1^.23' 
 
 R5.56 
 S6.04 
 
 1°.25' 
 
 R5.56 
 S6.04 
 
 2 
 
 2.39 
 
 5 53 
 6.07 
 
 2.42 
 
 5.53 
 6.07 
 
 2,44 
 
 5.53 
 6.07 
 
 2.47 
 
 5.52 
 6.08 
 
 2.50 
 
 5.52 
 6.08 
 
 3 
 
 3.59 
 
 5.50 
 6.10 
 
 4.02 
 
 5.49 
 6.11 
 
 4.06 
 
 5.49 
 6.11 
 
 4.10 
 
 5.48 
 6.12 
 
 4.15 
 
 5 48 
 6.12 
 
 4 
 
 5.18 
 
 5.46 
 6.14 
 
 5.23 
 
 5.46 
 6.14 
 
 5.28 
 
 5-45 
 6.15 
 
 5.34 
 
 5.45 
 6.15 
 
 5.40 
 
 5.44 
 6.16 
 
 5 
 
 6.38 
 
 5 43 
 6.17 
 
 6.44 
 
 5.42 
 6.J8 
 
 6.51 
 
 5.41 
 6.19 
 
 6.58 
 
 5.41 
 6.19 
 
 7.05 
 
 5.40 
 6.20 
 
 6 
 
 7.58 
 
 5.39 
 6.21 
 
 8.05 
 
 5.38 
 6.22 
 
 8.13 
 
 5.38 
 6.22 
 
 8.21 
 
 5.37 
 6.23 
 
 8.30 
 
 5 36 
 
 6.24 
 
 7 
 
 9.18 
 
 5.35 
 6.25 
 
 9.26 
 
 5.35 
 6.25 
 
 9.36 
 
 5.34 
 6.26 
 
 9.45 
 
 5.33 
 6.27 
 
 9.55 
 
 5.32 
 
 6.28 
 
 8 
 
 10.38 
 
 5.32 
 
 6.28 
 
 10.48 
 
 0.31 
 
 6.29 
 
 10.58 
 
 5.30 
 6.30 
 
 11.09 
 
 5.29 
 6 31 
 
 11.21 
 
 5.28 
 6.32 
 
 9 
 
 11.58 
 
 5.28 
 6.32 
 
 12.09 
 
 5.27 
 6.33 
 
 12 21 
 
 5.26 
 6.34 
 
 12.34 
 
 5.25 
 6.35 
 
 12.47 
 
 5.24 
 6.36 
 
 10 
 
 13.18 
 
 5.25 
 6.35 
 
 13.31 
 
 5.23 
 
 6 37 
 
 13.44 
 
 5.22 
 6.38 
 
 13.58 
 
 0.21 
 6.39 
 
 14.13 
 
 5.19 
 6.41 
 
 11 
 
 U.39 
 
 5.21 
 
 6.39 
 
 14.53 
 
 5.20 
 6.40 
 
 15.07 
 
 5.18 
 6.42 
 
 15.23 
 
 5.17 
 6.43 
 
 15.39 
 
 5.15 
 6.45 
 
 12 
 
 15.59 
 
 5.17 
 6.43 
 
 16.15 
 
 5 16 
 6.44 
 
 16.31 
 
 5.14 
 6.46 
 
 16.48 
 
 5.13 
 
 6.47 
 
 17.06 
 
 5 11 
 6.49 
 
 13 
 
 17.20 
 
 5.14 
 6.46 
 
 17.37 
 
 5.12 
 6.48 
 
 17.55 
 
 5.10 
 6.50 
 
 18.13 
 
 5.08 
 6.52 
 
 18.33 
 
 5.07 
 6.53 
 
 14 
 
 18.42 
 
 5.10 
 6.50 
 
 19.00 
 
 5.08 
 6.52 
 
 19.19 
 
 5.06 
 6.54 
 
 19.39 
 
 5.04 
 6.56 
 
 20.00 
 
 5.02 
 6.58 
 
 15 
 
 20.03 
 
 5.06 
 6.54 
 
 20.23 
 
 5.04 
 6.56 
 
 20.44 
 
 5.02 
 
 6.58 
 
 21.05 
 
 5.00 
 7.00 
 
 21.28 
 
 4.58 
 7.02 
 
 16 
 
 21.25 
 
 5.02 
 6.58 
 
 21.46 
 
 5.00 
 7.00 
 
 22.08 
 
 4.53 
 
 7.07 
 
 22.32 
 
 4.56 
 7.04 
 
 22.57 
 
 4.53 
 
 7.07 
 
 17 
 
 22.48 
 
 4.58 
 7.02 
 
 23.10 
 
 4.56 
 7.04 
 
 23.34 
 
 4.54 
 
 7.06 
 
 23.59 
 
 4.51 
 
 7.09 
 
 24.25 
 
 4.49 
 7.11 
 
 18 
 
 24.10 
 
 4.54 
 7.06 
 
 21.34 
 
 4.52 
 
 7.08 
 
 25.00 
 
 4.49 
 7.11 
 
 25.26 
 
 4.47 
 7.13 
 
 25.55 
 
 4.44 
 7.16 
 
 19 
 
 25.33 
 
 4.50 
 7.10 
 
 25.59 
 
 4.48 
 7.12 
 
 26.26 
 
 4.45 
 
 7.15 
 
 26.55 
 
 4.42 
 7.18 
 
 27.25 
 
 4.49 
 7.21 
 
 20 
 
 26.57 
 
 4.46 
 7.14 
 
 27.24 
 
 4.43 
 7.17 
 
 27.53 
 
 4.41 
 7.19 
 
 28.23 
 
 4.38 
 7.22 
 
 28.56 
 
 4.35 
 7.25 
 
 21 
 
 28.21 
 
 4.42 
 
 7.18 
 
 28.50 
 
 4.39 
 7.21 
 
 29.20 
 
 4.26 
 7.34 
 
 29.53 
 
 4.33 
 
 7.27 
 
 30.27 
 
 4.30 
 7.30 
 
 22 
 
 29.46 
 
 4.28 
 7.22 
 
 30.16 
 
 4.35 
 7.25 
 
 30.49 
 
 4.31 
 7.29 
 
 31.23 
 
 4.28 
 7.32 
 
 31.59 
 
 4.25 
 7.35 
 
 23 
 
 31.11 
 
 4.33 
 
 7.27 
 
 31.43 
 
 4 30 
 7.30 
 
 32.18 
 
 4.27 
 7.33 
 
 32.54 
 
 4.23 
 
 7.37 
 
 33.33 
 
 4.20 
 
 7.40 
 
 23^^ 
 
 31.51 
 
 4.31 
 
 7.29 
 
 32.24 
 
 4.28 
 7.32 
 
 32.59 
 
 4.24 
 7.36 
 
 33.37 
 
 4.21 
 7.39 
 
 34.16 
 
 4.17 
 7.43 
 
 R. and S. are applied for Lat. and Dec. of the same name. 
 
 124 
 
The Sun's Amplitude— Continued. 
 
 With rate of change of Azimuth, frc 
 Prime Vertical, and with th 
 
 m Sunrise to 
 e timj of his 
 
 the time of h 
 Rising and S 
 
 is crossing the 
 elting. 
 
 Lat. 
 
 46° 
 
 47 
 
 o 
 
 48 
 
 o 
 
 49 
 
 o 
 
 50° 
 
 Dec. 
 
 Change 
 of Azimuth 
 l°in sH m- 
 
 Chan ire 
 of Azimuth 
 1° in 5% m. 
 
 Chaii};e 
 of AziTiiuth 
 l^in 53^ m. 
 
 Cha 
 of Az 
 l°in 5 
 
 as;e 
 muth 
 V^ m. 
 
 Change 
 of Azimuth 
 1" in 5^^ m. 
 
 
 AMP. 
 
 y\. M. 
 
 AMP. 
 
 H. M. 
 
 AMP. 
 
 H. M. 
 
 AMP. 
 
 H. M. 
 
 AMP. 
 
 H. M. 
 
 r 
 
 1°.26' 
 
 R5.56 
 S6.04 
 
 1°.28' 
 
 R5.56 
 S 6.04 
 
 1°. 30' 
 
 R5.56 
 S6.04 
 
 1°. 31' 
 
 R5.55 
 
 S0.05 
 
 1°.33' 
 
 R5.55 
 S6.05 
 
 2 
 
 2.53 
 
 5.52 
 6.08 
 
 2.56 
 
 5.51 
 6.09 
 
 2.59 
 
 5.51 
 6.09 
 
 3.03 
 
 5.51 
 6.09 
 
 3.07 
 
 5.50 
 6.10 
 
 3 
 
 4.19 
 
 5.48 
 6.12 
 
 4.24 
 
 5.47 
 6.13 
 
 4.29 
 
 5.47 
 6.13 
 
 4.35 
 
 5.46 
 6.14 
 
 4.40 
 
 5.46 
 6.14 
 
 4 
 
 5.40 
 
 5.43 
 6.17 
 
 5.52 
 
 5.43 
 6.17 
 
 5.59 
 
 5.42 
 6.18 
 
 6.06 
 
 5.42 
 6.18 
 
 6.14 
 
 5.41 
 
 6.18 
 
 5 
 
 7.12 
 
 5.39 
 6.21 
 
 7.21 
 
 5.38 
 6.22 
 
 7.29 
 
 5.38 
 6.22 
 
 7..38 
 
 5.37 
 6.^3 
 
 7.48 
 
 5.36 
 6.24 
 
 6 
 
 8.39 
 
 5.35 
 6.25 
 
 8.49 
 
 5.34 
 6.26 
 
 8.59 
 
 5.33 
 
 6.27 
 
 9.10 
 
 5.32 
 6.28 
 
 9.22 
 
 5.31 
 
 6.29 
 
 7 
 
 10.06 
 
 5.31 
 6.29 
 
 10.18 
 
 5.30 
 6.30 
 
 10.30 
 
 5.29 
 6.31 
 
 10.42 
 
 5.28 
 6.32 
 
 10.56 
 
 5.26 
 6.34 
 
 8 
 
 11.33 
 
 5.27 
 6.33 
 
 11.46 
 
 5.25 
 6.35 
 
 12.00 
 
 5.24 
 6.36 
 
 12.15 
 
 5.23 
 6.37 
 
 12.30 
 
 5.21 
 6.39 
 
 9 
 
 13.01 
 
 5.22 
 6.38 
 
 13.16 
 
 5.21 
 6.39 
 
 13.31 
 
 5.19 
 6.41 
 
 13.48 
 
 5.18 
 6.42 
 
 14.05 
 
 5.16 
 6.44 
 
 10 
 
 14.29 
 
 5.18 
 6 42 
 
 14.45 
 
 5.16 
 6-44 
 
 15.02 
 
 5.15 
 0.45 
 
 15.21 
 
 5.13 
 
 6.47 
 
 15.40 
 
 5.11 
 6.49 
 
 11 
 
 15.57 
 
 5.14 
 6.46 
 
 16 15 
 
 5.12 
 
 6.48 
 
 16.34 
 
 5.10 
 6.50 
 
 16.54 
 
 5.08 
 6.52 
 
 17.16 
 
 5.06 
 6.54 
 
 12 
 
 17.25 
 
 5.09 
 6.51 
 
 17.45 
 
 5.07 
 6.53 
 
 18.06 
 
 5.05 
 6.55 
 
 18.29 
 
 5.03 
 6.57 
 
 18.52 
 
 5.01 
 6.59 
 
 13 
 
 18.54 
 
 5.05 
 6.55 
 
 19.16 
 
 5.03 
 6.57 
 
 19.39 
 
 5.01 
 6.59 
 
 20.03 
 
 4 58 
 7.02 
 
 20.29 
 
 4.56 
 
 7.04 
 
 14 
 
 20.23 
 
 5.00 
 
 7 00 
 
 20.47 
 
 4.58 
 7.02 
 
 21.12 
 
 4.56 
 7.04 
 
 21.38 
 
 4.53 
 
 7.07 
 
 22.07 
 
 4.51 
 
 7.09 
 
 15 
 
 21.53 
 
 4.56 
 
 7.04 
 
 22.18 
 
 4.53 
 
 7.07 
 
 22.45 
 
 4.51 
 
 7.09 
 
 23.14 
 
 4.48 
 
 7.ia 
 
 23.45 
 
 4.46 
 7.J4 
 
 16 
 
 23.23 
 
 4.5L 
 
 7.09 
 
 23.50 
 
 4.48 
 7.12 
 
 24.20 
 
 4.46 
 744 
 
 24.51 
 
 4.43 
 7.17 
 
 25.24 
 
 4.40 
 
 7.20 
 
 17 
 
 24.53 
 
 4.56 
 7.14 
 
 25.23 
 
 4.53 
 7.17 
 
 25.55 
 
 4.41 
 7.19 
 
 26.28 
 
 4.38 
 7.22 
 
 27.03 
 
 4.35 
 
 7.25 
 
 18 
 
 26.25 
 
 4.41 
 7.19 
 
 26.57 
 
 4.38 
 7.22 
 
 27.30 
 
 4.35 
 7.25 
 
 28.06 
 
 4.32 
 
 7.28 
 
 28.44 
 
 4.29 
 7.31 
 
 19 
 
 27.57 
 
 4.36 
 7.24 
 
 28.31 
 
 4.33 
 
 7.27 
 
 29.07 
 
 4.30 
 
 7 30 
 
 29.45 
 
 4.27 
 7.33 
 
 30.26 
 
 4.23 
 7.37 
 
 2U 
 
 29.30 
 
 4.31 
 
 7.29 
 
 30.06 
 
 4 28 
 7.32 
 
 30.44 
 
 4.25 
 7.35 
 
 31.25 
 
 4.21 
 
 7.39 
 
 32.09 
 
 4.17 
 7.43 
 
 21 
 
 31.03 
 
 4.26 
 7.34 
 
 31.42 
 
 4.23 
 7.37 
 
 32.23 
 
 4.19 
 7.41 
 
 33.07 
 
 4.15 
 7.45 
 
 33.53 
 
 4.11 
 7.49 
 
 22 
 
 32.38 
 
 4.21 
 7.39 
 
 33.19 
 
 4.17 
 7.43 
 
 34.03 
 
 4.13 
 
 7.47 
 
 34.49 
 
 4.09 
 7.51 
 
 35.39 
 
 4.05 
 
 7.55 
 
 23 
 
 34.14 
 
 4.16 
 7.44 
 
 34.57 
 
 4.12 
 
 7.48 
 
 35.44 
 
 4.07 
 7.53 
 
 36.33 
 
 4.03 
 
 7.57 
 
 37.26 
 
 3.58 
 8.02 
 
 233^ 
 
 34.59 
 
 4.13 
 
 7.47 
 
 35.43 
 
 4.09 
 7.51 
 
 36.31 
 
 4.05 
 7.55 
 
 37.22 
 
 4.00 
 8.00 
 
 38.17 
 
 3.55 
 
 8.05 
 
 When Lat. and Dec. are unlike, R. and S. must change places with each other. 
 
 125 
 
TABLE V. 
 
 
 Sun's Declination for every 2nd day, with 
 
 corresponding 
 
 Equation of Time, for 1891. 
 
 
 
 January. 
 
 February. 
 
 March. 
 
 April. 
 
 May. 
 
 June. 
 
 
 
 South. 
 
 South. 
 
 South. 
 
 North. 
 
 North. 
 
 North. 
 
 
 JNorih. 
 
 >^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 >. 
 
 
 Dec. 
 
 Eq. 
 
 Dec. 
 
 Kq. 
 
 Dec. 
 
 Kq. 
 
 Dec. 
 
 Eq. 
 
 Dec. 
 
 Eq. 
 
 Dec. 
 
 Eq. 
 
 
 
 S. 
 
 
 S. 
 
 
 S. 
 
 
 N. 
 
 
 N. 
 
 
 N. 
 
 
 
 
 o t 
 
 m. 
 
 o / 
 
 m. 
 
 o / 
 
 m. 
 
 o / 
 
 m. 
 
 o / 
 
 m. 
 
 c / 
 
 m. 
 
 
 1 
 
 23.01 
 
 — 4 
 
 17.06 
 
 —14 
 
 7.35 
 
 —13 
 
 4.33 
 
 —4 
 
 15.05 
 
 +3 
 
 22.04 
 
 +2 
 
 1 
 
 3 
 
 22.50 
 
 — 5 
 
 16.31 
 
 —14 
 
 6.49 
 
 —12 
 
 5.19 
 
 —3 
 
 15 40 
 
 +3 
 
 22.19 
 
 +2 
 
 A 
 
 5 
 
 22.37 
 
 — 6 
 
 15.55 
 
 —14 
 
 6.03 
 
 —12 
 
 6.04 
 
 -3 
 
 16.15 
 
 +4 
 
 22.33 
 
 ^2 
 
 5 
 
 7 
 
 22.23 
 
 — 6 
 
 15.18 
 
 —14 
 
 5.16 
 
 —11 
 
 6.50 
 
 -2 
 
 16.49 
 
 + 4 
 
 22.46 
 
 + 1 
 
 7 
 
 9 
 
 22.06 
 
 — 7 
 
 14.40 
 
 —14 
 
 4.29 
 
 —11 
 
 7.35 
 
 —2 
 
 17.21 
 
 +4 
 
 22.56 
 
 + 1 
 
 9 
 
 11 
 
 21.48 
 
 — 8 
 
 14.01 
 
 -14 
 
 3.42 
 
 —10 
 
 8.19 
 
 -1 
 
 17.53 
 
 +4 
 
 23.06 
 
 +1 
 
 11 
 
 13 
 
 21.29 
 
 — 9 
 
 13.21 
 
 —14 
 
 2.55 
 
 —10 
 
 9.03 
 
 —1 
 
 18.23 
 
 +4 
 
 23.13 
 
 + 
 
 13 
 
 15 
 
 21.07 
 
 —10 
 
 12.40 
 
 —14 
 
 2.08 
 
 — 9 
 
 9.46 
 
 —0 
 
 18.52 
 
 + 4 
 
 23.19 
 
 —0 
 
 15 
 
 17 
 
 20.44 
 
 —10 
 
 11.58 
 
 —14 
 
 1.20 
 
 — 9 
 
 10.29 
 
 +0 
 
 19.20 
 
 4-4 
 
 23.24 
 
 -1 
 
 17 
 
 19 
 21 
 
 20.20 
 19.54 
 
 —11 
 —12 
 
 11.16 
 10.33 
 
 —14 
 —14 
 
 S0.33 
 
 — 8 
 -7 
 
 11.10 
 11.52 
 
 + 1 
 -fl 
 
 19.46 
 20.11 
 
 + 4 
 +4 
 
 23.26 
 23.27 
 
 —1 
 —1 
 
 19! 
 21 
 
 NO. 15 
 
 23 
 
 19.26 
 
 — ]2 
 
 9.49 
 
 —14 
 
 1.02 
 
 — 7 
 
 12.32 
 
 +2 
 
 20.35 
 
 ^4 
 
 23.27 
 
 —2 
 
 23 
 
 25 
 
 18.57 
 
 -13 
 
 9.05 
 
 -13 
 
 1.49 
 
 — 6 
 
 13.11 
 
 +2 
 
 20.57 
 
 + 3 
 
 23.24 
 
 —2 
 
 25 
 
 27 
 
 18.27 
 
 —13 
 
 8.20 
 
 —13 
 
 2 36 
 
 — 6 
 
 13.50 
 
 + 2 
 
 21.18 
 
 + 3 
 
 23.20 
 
 —3 
 
 27 i 
 
 29 
 
 17.55 
 
 —13 
 
 7.35 
 
 -13 
 
 3.23 
 
 — 5 
 
 14.28 
 
 -f3 
 
 21.37 
 
 +3 
 
 23.15 
 
 —3 
 
 29; 
 
 31 
 
 17.23 
 
 —14 
 
 
 
 4.09 
 
 — 4 
 
 
 
 
 21.55 
 
 -r3 
 
 
 
 
 31 
 
 Note. — The sign prefixed to the Equ 
 
 ation of Time, 
 
 in the above table, is that for reducing n 
 
 lean time to 
 
 apparent time. 
 
 
 126 
 
TABLE V.-Continued. 
 
 Sun's Declination for every 2nd day, with corresponding 
 
 Equation of Time, for 1891. 
 
 
 July. 
 
 August. 
 
 September. 
 
 October. 
 
 November. 
 
 December. 
 
 
 
 North. 
 
 North. 
 
 North. 
 
 South. 
 
 South. 
 
 South. 
 
 
 South. 
 
 >» 
 
 
 
 
 
 
 
 
 
 
 
 
 
 >. 
 
 
 Dec. 
 
 Eq. 
 
 Dec. 
 
 Eq. 
 
 Dec. 
 
 Eq. 
 
 Dec. 
 
 Eq. 
 
 Dec. 
 
 Eq. 
 
 Dec. 
 
 Eq. 
 
 
 
 N. 
 
 
 N. 
 
 
 N. 
 
 
 S. 
 
 
 S. 
 
 
 S. 
 
 
 
 
 o / 
 
 m. 
 
 o / 
 
 m. 
 
 o / 
 
 m. 
 
 o r 
 
 m. 
 
 o / 
 
 m. 
 
 o / 
 
 m. 
 
 
 1 
 
 23.08 
 
 -4 
 
 18.03 
 
 —6 
 
 8.18 
 
 + 
 
 3.11 
 
 + 10 
 
 14.26 
 
 +16 
 
 21.49 
 
 +11 
 
 1 
 
 3 
 
 22.59 
 
 —4 
 
 17.32 
 
 -6 
 
 7.34 
 
 + 1 
 
 3.57 
 
 + 11 
 
 15.04 
 
 +16 
 
 22.07 
 
 +10 
 
 3 
 
 5 
 
 22.48 
 
 -4 
 
 17.00 
 
 —6 
 
 6.50 
 
 + 1 
 
 4.44 
 
 + 12 
 
 15.41 
 
 +16 
 
 22.^3 
 
 + 9 
 
 5 
 
 7 
 
 22.36 
 
 -5 
 
 16 27 
 
 —6 
 
 6.05 
 
 + 2 
 
 5.30 
 
 + 12 
 
 16.17 
 
 +16 
 
 22.37 
 
 + 8 
 
 7 
 
 9 
 
 22.23 
 
 —5 
 
 15.53 
 
 —5 
 
 5.20 
 
 + 3 
 
 6.16 
 
 + 13 
 
 16.52 
 
 +16 '22.50 
 
 + 8 
 
 9 
 
 11 
 
 22.08 
 
 —5 
 
 15.18 
 
 —5 
 
 4.35 
 
 + 3 
 
 7.01 
 
 + 13 
 
 17.26 
 
 + 16 23.01 
 
 + 7 
 
 11 
 
 13 
 
 21.51 
 
 —5 
 
 14.42 
 
 - 5 
 
 3.49 
 
 + 4 
 
 7.46 
 
 + 14 
 
 17.59 
 
 + 16 23.10 
 
 + 6 
 
 13 
 
 15 
 
 21.33 
 
 —6 
 
 14.05 
 
 —4 
 
 3 03 
 
 + 5 
 
 8.31 
 
 + 14 
 
 18.30 
 
 + 15.23.17 
 
 + 5 
 
 15 
 
 17 
 
 21.13 
 
 -6 
 
 13 27 
 
 -4 
 
 2.16 
 
 + 5 
 
 9.15 
 
 + 15 
 
 19.00 
 
 + 15 23.22 
 
 + 4 
 
 17 
 
 19 
 
 20.52 
 
 —6 
 
 12.48 
 
 —4 
 
 1.30 
 
 + 6 
 
 9.59 
 
 + 15 
 
 19.29 
 
 + 15 23.25 
 
 + 3 
 
 19 
 
 21 
 23 
 
 20.30 
 20.06 
 
 -6 
 -6 
 
 12.08 
 11.28 
 
 -3 
 -3 
 
 N.43 
 
 + 7 
 + 7 
 
 10.42 
 11.24 
 
 + 15 
 + 16 
 
 19.56 
 20.21 
 
 + 14|23.27 
 + 14:23.27 
 
 + 2 
 
 + 1 
 
 21 
 23 
 
 S. 4 
 
 25 
 
 19.41 
 
 —6 
 
 10.47 
 
 —2 
 
 .50 
 
 + 8 
 
 12 06 
 
 +16 
 
 20.46 
 
 +13 23.25 
 
 — 
 
 25 
 
 27 
 
 19.14 
 
 —6 
 
 10.05 
 
 —2 
 
 1.37 
 
 + 9 
 
 12.47 
 
 + 16 
 
 21.08 
 
 + 12 23.20 
 
 — 1 
 
 27 
 
 29 
 
 18.47 
 
 —6 
 
 9.23 
 
 —1 
 
 2.24 
 
 +10 
 
 13 27 
 
 +16 
 
 21.30 
 
 + 12 23.14 
 
 — 2 
 
 29 
 
 31 
 
 18.18 
 
 —6 
 
 8.40 
 
 -0 
 
 
 
 14.07 
 
 +16 
 
 
 23.07 
 
 — 3 
 
 31 
 
 Note. — The sign prefixed to the Equation of Time, 
 
 in the above table, is that for reducing mean time to 
 
 apparent time. . 
 
 127 
 
Azimuth and Hour Angle, for Latitude 
 
 
 
 and Declination.— Table VI. 
 
 LATITUDE 42^. 
 
 
 
 ^ZIiyCTJTBCS. 
 
 jJeciiiiiiLJiuii. 
 
 15° 
 
 20° 
 
 25° 
 
 30^ 
 
 35° 
 
 40° 
 
 45° 
 
 50° 
 
 55° 
 
 60° 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 
 +24° 
 
 .21 
 
 .28 
 
 .35 
 
 .43 
 
 .52 
 
 1.01 
 
 1.11 
 
 1.22 
 
 1.35 
 
 1.49 
 
 
 22 
 
 .22 
 
 .30 
 
 .29 
 
 .47 
 
 .57 
 
 1.06 
 
 1.17 
 
 1.29 
 
 1.43 
 
 1.57 
 
 
 20 
 
 0.24 
 
 0.33 
 
 0.42 
 
 0.51 
 
 1.01 
 
 1.12 
 
 1.23 
 
 1.30 
 
 1.50 
 
 2.06 
 
 TJ 
 
 18 
 
 .26 
 
 .35 
 
 .45 
 
 .55 
 
 1.05 
 
 1.17 
 
 1.29 
 
 1.42 
 
 1.57 
 
 2.13 
 
 3 
 
 16 
 
 .28 
 
 .37 
 
 .48 
 
 .58 
 
 1.09 
 
 1.21 
 
 1.34 
 
 1.48 
 
 2.04 
 
 2.21 
 
 *rt 
 
 14 
 
 .29 
 
 .40 
 
 .50 
 
 1.02 
 
 1.13 
 
 1.26 
 
 1.40 
 
 1.55 
 
 2.11 
 
 2.28 
 
 -1 
 
 \l 
 
 .31 
 
 .42 
 
 .53 
 
 1.05 
 
 1.17 
 
 1.31 
 
 1.45 
 
 2.00 
 
 2.17 
 
 2.36 
 
 0) 
 
 0.33 
 
 0.44 
 
 0.56 
 
 J. 08 
 
 1.21 
 
 1.35 
 
 1.50 
 
 2.06 
 
 2.24 
 
 2.43 
 
 ^ 
 
 8 
 
 .34 
 
 .46 
 
 .59 
 
 1.12 
 
 1.25 
 
 1.40 
 
 1.55 
 
 2.12 
 
 2.30 
 
 2.50 
 
 '^ 
 
 6 
 
 .36 
 
 .48 
 
 1.01 
 
 1.15 
 
 1.29 
 
 1.44 
 
 2.00 
 
 2.18 
 
 2.36 
 
 2.57 
 
 
 4 
 
 .38 
 
 .51 
 
 1.04 
 
 1.18 
 
 1.33 
 
 1.49 
 
 2.05 
 
 2.23 
 
 2.42 
 
 3.03 
 
 
 + 2 
 
 .39 
 
 .53 
 
 1.07 
 
 1.21 
 
 1.37 
 
 1.53 
 
 2.10 
 
 2.29 
 
 2.49 
 
 3.10 
 
 
 
 0.41 
 
 0.55 
 
 1.09 
 
 1.24 
 
 1.40 
 
 1 57 
 
 2.15 
 
 2.34 
 
 2.55 
 
 3.17 
 
 
 — 2 
 
 .42 
 
 .57 
 
 1.12 
 
 1.28 
 
 1.44 
 
 2.02 
 
 2.20 
 
 2.40 
 
 3.01 
 
 3.24 
 
 
 4 
 
 .44 
 
 .59 
 
 1.15 
 
 1.31 
 
 1.48 
 
 2.06 
 
 2.25 
 
 2.45 
 
 3.07 
 
 3.30 
 
 V 
 
 6 
 
 .45 
 
 1.01 
 
 1.17 
 
 1.34 
 
 1.52 
 
 2.10 
 
 2.30 
 
 2.51 
 
 3.13 
 
 3.37 
 
 3 
 
 8 
 
 .47 
 
 1.03 
 
 1.20 
 
 1.37 
 
 1.56 
 
 2.15 
 
 2.35 
 
 2.57 
 
 3.19 
 
 3.44 
 
 
 10 
 
 0.49 
 
 1.05 
 
 1.23 
 
 1.41 
 
 1.59 
 
 2.19 
 
 2.40 
 
 3.02 
 
 3.26 
 
 3.51 
 
 ^ 
 
 12 
 
 .50 
 
 1.08 
 
 1.25 
 
 1.44 
 
 2.03 
 
 2.24 
 
 2.45 
 
 3.08 
 
 3.32 
 
 3.58 
 
 14 
 
 .52 
 
 1.10 
 
 1.28 
 
 1.47 
 
 2.07 
 
 2.28 
 
 2.51 
 
 3.14 
 
 3.39 
 
 4.05 
 
 J^ 
 
 16 
 
 .54 
 
 1.12 
 
 1.31 
 
 1.51 
 
 2.11 
 
 2.33 
 
 2.56 
 
 3.20 
 
 3.46 
 
 4.13 
 
 •-= 
 
 18 
 
 .55 
 
 1.14 
 
 1.34 
 
 1.54 
 
 2.16 
 
 2.38 
 
 3.01 
 
 3.26 
 
 3.53 
 
 4.20 
 
 ^ 
 
 20 
 22 
 
 0.57 
 ,59 
 
 1.17 
 1.19 
 
 1.37 
 1.40 
 
 1.58 
 2.02 
 
 2.20 
 2.24 
 
 2.43 
 2.48 
 
 3.07 
 3.13 
 
 3.33 
 3.39 
 
 4.00 
 4.07 
 
 4.28 
 
 4.36 
 
 
 —24 
 
 1.01 
 
 1.22 
 
 1.43 
 
 2.06 
 
 2.29 
 
 2.53 
 
 3.19 
 
 3.46 
 
 4.1^^ 
 
 4.45 
 
 
 
 
 
 
 
 
 
 
 125° 
 
 120° 
 
 128 
 

 Table VI - 
 
 -Continued. 
 
 LATITUDE 42^. 
 
 
 .A.ZiX3y[:TJTSLS. 1 
 
 Declination. 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 ea^' 
 
 66° 
 
 69° 
 
 72° 
 
 75° 
 
 78° 
 
 81° 
 
 84° 
 
 87° 
 
 90° 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H. M. 
 
 H.M. 
 
 H. M. 
 
 24° 
 
 1 58 
 
 2.08 
 
 2.19 
 
 2.3i 
 
 2.42 
 
 2.57 
 
 3.12 
 
 3.27 
 
 3.44 
 
 4.01 
 
 22 
 
 2.07 
 
 2.18 
 
 2.29 
 
 2.41 
 
 2.54 
 
 3.08 
 
 3.23 
 
 3.39 
 
 3.56 
 
 4.13 
 
 20 
 
 2.16 
 
 2.27 
 
 2.39 
 
 2.51 
 
 3.05 
 
 3.19 
 
 3.34 
 
 3 .50 
 
 4.07 
 
 4.25 
 
 18 
 
 2.24 
 
 2 36 
 
 2.48 
 
 3.01 
 
 3.14 
 
 3.29 
 
 3.44 
 
 4.01 
 
 4.18 
 
 4.35 
 
 16 
 
 2.32 
 
 2.44 
 
 2.56 
 
 3.10 
 
 3.24 
 
 3.39 
 
 3.55 
 
 4.11 
 
 4.28 
 
 4.46 
 
 14 
 
 2.40 
 
 2 52 
 
 3.05 
 
 3.19 
 
 3.33 
 
 3.48 
 
 4.04 
 
 4.21 
 
 4.38 
 
 4.56 
 
 12 
 
 2.47 
 
 3 00 
 
 3.13 
 
 3.27 
 
 3.42 
 
 3.57 
 
 4.14 
 
 4.30 
 
 4 48 
 
 5.05 
 
 10 
 
 2.55 
 
 3.08 
 
 3.22 
 
 3.36 
 
 3.51 
 
 4.06 
 
 4.23 
 
 4.40 
 
 4.57 
 
 5.15 
 
 8 
 
 3.02 
 
 3.16 
 
 3.30 
 
 3.44 
 
 3.59 
 
 4.15 
 
 4.32 
 
 4.49 
 
 5.06 
 
 5.24 
 
 6 
 
 3.10 
 
 3.23 
 
 3.37 
 
 3.52 
 
 4.08 
 
 4.24 
 
 4.41 
 
 4.58 
 
 5 15 
 
 5.33 
 
 4 
 
 3.17 
 
 3.31 
 
 3.45 
 
 4.00 
 
 4.16 
 
 4.33 
 
 4.49 
 
 5.07 
 
 5.24 
 
 5.42 
 
 + 2 
 
 3.24 
 
 3.38 
 
 3.53 
 
 4.08 
 
 4.24 
 
 4.41 
 
 4.58 
 
 5.16 
 
 5.33 
 
 5.51 
 
 
 
 3.31 
 
 3.45 
 
 4.01 
 
 4.16 
 
 4.33 
 
 4.50 
 
 5.07 
 
 5.24 
 
 5.42 
 
 6.00 
 
 
 — 2 
 4 
 6 
 8 
 10 
 12 
 14 
 
 3.38 
 3.45 
 3.52 
 3.59 
 4.07 
 4 14 
 4.22 
 
 3.53 
 
 4.00 
 4.08 
 4.15 
 4.23 
 4.31 
 4.39 
 
 4 08 
 4.16 
 4.24 
 4.32 
 4.40 
 4 48 
 4.56 
 
 4.24 
 4.32 
 4.41 
 
 4.49 
 4.57 
 5.05 
 
 4.41 
 4.49 
 4.58 
 5.06 
 5.15 
 
 4.58 
 5.06 
 5.15 
 5.24 
 
 5.15 
 5.24 
 5.33 
 
 5.33 
 5.42 
 
 5.51 
 
 6.09 
 6.18 
 6.27 
 6.36 
 6.45 
 6.55 
 7.04 
 
 6.00 
 6.09 
 6.18 
 6.27 
 6.37 
 6.46 
 
 5.51 
 
 6.00 
 6.09 
 6.18 
 6.28 
 
 5.42 
 5.51 
 6 00 
 6.09 
 
 5.33 
 5.42 
 5.51 
 
 5.23 
 5,32 
 
 5.14 
 
 16 
 
 18 
 
 4.30 
 4.38 
 
 4.47 
 
 5.05 
 5.14 
 
 5.23 
 5.32 
 
 5.41 
 
 5.51 
 
 6.00 
 6.10 
 
 6.19 
 6.29 
 
 6.38 
 6.48 
 
 6.56 
 7.06 
 
 7.14 
 7.25 
 
 4.58 
 
 20 
 
 4.46 
 
 5.04 
 
 5.23 
 
 5.42 
 
 6.01 
 
 6.20 
 
 6.39 
 
 6.58 
 
 7.17 
 
 7.35 
 
 22 
 
 4.55 
 
 5.13 
 
 5.32 
 
 5.52 
 
 6.11 
 
 6 31 
 
 6.50 
 
 7.10 
 
 7.28 
 
 7.47 
 
 24 
 
 5.04 
 
 5.23 
 
 5. 42 
 
 6.02 
 
 6.23 
 
 6.42 
 
 7.02 
 
 7.21 
 
 7.40 
 
 7.59 
 
 
 117° 
 
 114^ 
 
 111° 
 
 108° 
 
 105° 
 
 102° 
 
 99° 
 
 96° 
 
 93° 
 
 90° 
 
 129 
 
Azimuth and Hour Angle, for Latitude 
 
 
 
 and Declination —Table VI. 
 
 LATITUDE 430. 
 
 
 
 ^zinynTJTiHiS- 
 
 
 15° 
 
 20° 
 
 25° 
 
 30^ 
 
 35° 40° 
 
 45° 
 
 50° 
 
 55° 
 
 60° 
 
 
 H.M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. H. M. 
 
 H. M, 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 
 +24° 
 
 .22 
 
 .29 
 
 .37 
 
 .46 
 
 .55 
 
 1.04 
 
 1.15 
 
 1.26 1.39 
 
 1.54 
 
 
 22 
 
 .24 
 
 .32 
 
 .40 
 
 .49 
 
 .59 
 
 1.09 
 
 1.21 
 
 1.33 i 1.47 
 
 2.02 
 
 
 20 
 
 0.25 
 
 0.34 
 
 0.43 
 
 0.53 
 
 1.03 
 
 1.14 
 
 1.26 
 
 1.39 i 1.54 
 
 2.10 
 
 TJ 
 
 18 
 
 .27 
 
 .37 
 
 .46 
 
 .57 
 
 1.08 
 
 1.19 
 
 1.32 
 
 1.40 1 2.01 
 
 2.17 
 
 3 
 
 16 
 
 .29 
 
 .39 
 
 .49 
 
 1.00 
 
 1.12 
 
 1.24 
 
 1.37 
 
 1.52 
 
 2.07 
 
 2.23 
 
 "5 
 
 14 
 
 .30 
 
 .41 
 
 .52 
 
 1.03 
 
 1.16 
 
 1.29 
 
 1.43 
 
 1.58 
 
 2.14 
 
 2.32 
 
 hJ 
 
 12 
 
 .32 
 
 .43 
 
 .55 
 
 1.07 
 
 1.20 
 
 1.33 
 
 1.48 
 
 2.03 
 
 2.20 
 
 2.39 
 
 0) 
 
 10 
 
 0.34 
 
 0.45 
 
 0.57 
 
 1.10 
 
 1.23 
 
 1.38 
 
 1.52 
 
 2.09 
 
 2.27 
 
 2.46 
 
 .id 
 
 8 
 
 .35 
 
 .47 
 
 1.00 
 
 1.13 
 
 1.27 
 
 1.42 
 
 1.58 
 
 2.15 
 
 2.33 
 
 2.52 
 
 'h^ 
 
 6 
 
 .37 
 
 .50 
 
 1.03 
 
 1.17 
 
 1.31 
 
 1.46 
 
 2.03 
 
 2.20 
 
 2.39 
 
 2.59 
 
 
 4 
 
 .38 
 
 .52 
 
 1.05 
 
 1.20 
 
 1.35 
 
 1.51 
 
 2.07 
 
 2.26 
 
 2.45 
 
 3.06 
 
 
 + 2 
 
 .40 
 
 .54 
 
 1.08 
 
 1.23 
 
 1.38 
 
 1.55 
 
 2.12 
 
 2.31 
 
 2.51 
 
 3.12 
 
 
 
 0.41 
 
 0.56 
 
 1.11 
 
 1.26 
 
 1.42 
 
 1,59 
 
 2.17 
 
 2.36 
 
 2.57 
 
 3.19 
 
 
 — 2 
 
 .43 
 
 .58 
 
 1.13 
 
 1.29 
 
 1.46 
 
 2.03 
 
 2.22 
 
 2.42 
 
 3.03 
 
 3.26 
 
 
 4 
 
 .45 
 
 1.00 
 
 1.16 
 
 1.32 
 
 1.50 
 
 2.08 
 
 2.27 
 
 2.47 3.09 
 
 3.32 
 
 u 
 
 6 
 
 .46 
 
 1.02 
 
 1.18 
 
 1.35 
 
 1.53 
 
 2.12 
 
 2.32 
 
 2.53 
 
 3.15 
 
 3.39 
 
 3 
 
 8 
 
 .48 
 
 1.04 
 
 1.21 
 
 1.39 
 
 1.57 
 
 2.16 
 
 2.37 
 
 2.58 
 
 3.21 
 
 3.46 
 
 
 10 
 
 0.49 
 
 1.06 
 
 1.24 
 
 1.42 
 
 2.01 
 
 2.21 
 
 2.42 
 
 3.04 
 
 3.27 
 
 3.52 
 
 
 12 
 
 .51 
 
 1.08 
 
 1.26 
 
 1.45 
 
 2.05 
 
 2.25 
 
 2.46 
 
 3.09 
 
 3.34 
 
 3.59 
 
 14 
 
 .52 
 
 1.11 
 
 1.29 
 
 1.48 
 
 2.09 
 
 2.30 
 
 2.52 
 
 3.15 
 
 3.40 
 
 4.06 
 
 ^ 
 
 16 
 
 .54 
 
 1.13 
 
 1.32 
 
 1.52 
 
 2.13 
 
 2.34 
 
 2.57 
 
 3.21 
 
 3.47 
 
 4.13 
 
 — 
 
 18 
 
 .56 
 
 1.15 
 
 1.35 
 
 1.55 
 
 2.17 
 
 2.39 
 
 3.03 
 
 3.27 
 
 3.53 
 
 4.21 
 
 :::> 
 
 20 
 22 
 
 0.58 
 .59 
 
 1.17 
 1.20 
 
 1.38 
 1.41 
 
 1.59 
 2.03 
 
 2.21 
 2.25 
 
 2.44 
 2.49 
 
 3. 08 
 3.14 
 
 3.33 
 3.40 
 
 4.00 
 4.07 
 
 4.28 
 
 4.36 
 
 
 —24 
 
 1.01 
 
 1.22 
 
 1.44 
 
 2.06 
 
 2.30 
 
 2.54 
 
 3.20 
 
 3.47 
 
 4.1'> 
 
 4.44 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 130 
 
Table VI.— Continued. 
 
 LATITUDE 43''. 
 
 
 ^ZXl^TJTHIS. 
 
 Declination. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 63° 
 
 66° 
 
 69° 
 
 72° 
 
 75° 
 
 78° 
 
 81° 
 
 84° 
 
 87° 
 
 90° 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H. M. 
 
 +24° 
 
 2 03 
 
 2.13 
 
 2.24 
 
 2.36 
 
 2.49 
 
 3.02 
 
 3.17 
 
 3.32 
 
 3.49 
 
 4.06 
 
 22 
 
 2.12 
 
 2.22 
 
 2.34 
 
 2.46 
 
 2.59 
 
 3.13 
 
 3.28 
 
 3.44 
 
 4.00 
 
 4.17 
 
 20 
 
 2.20 
 
 2.31 
 
 2.43 
 
 2.56 
 
 3.09 
 
 3.23 
 
 3.38 
 
 3 54 
 
 4.11 
 
 4.28 
 
 18 
 
 2.28 
 
 2 40 
 
 2.52 
 
 3.05 
 
 3.19 
 
 3.33 
 
 3.48 
 
 4.04 
 
 4.21 
 
 4.38 
 
 16 
 
 2.36 
 
 2.48 
 
 3.00 
 
 3.14 
 
 3.28 
 
 3.43 
 
 3.58 
 
 4.14 
 
 4.31 
 
 4.48 
 
 14 
 
 2.43 
 
 2 56 
 
 3.09 
 
 3.22 
 
 3.37 
 
 3.52 
 
 4.07 
 
 4.24 
 
 4.41 
 
 4.58 
 
 IG 
 
 2.51 
 
 3 03 
 
 3.17 
 
 3.31 
 
 3.45 
 
 4.00 
 
 4.16 
 
 4.33 
 
 4 50 
 
 5.07 
 
 10 
 
 2.58 
 
 3.11 
 
 3.25 
 
 3.39 
 
 3.54 
 
 4.09 
 
 4.25 
 
 4.42 
 
 4.59 
 
 5.16 
 
 8 
 
 3.05 
 
 3.18 
 
 3.32 
 
 3.47 
 
 4.02 
 
 4.18 
 
 4.34 
 
 4.51 
 
 5.08 
 
 5.25 
 
 6 
 
 3.12 
 
 3.26 
 
 3.40 
 
 3.55 
 
 4.]0 
 
 4.26 
 
 4.42 
 
 4.59 
 
 5 17 
 
 5.34 
 
 4 
 
 3.19 
 
 3.33 
 
 3.48 
 
 4.03 
 
 4.18 
 
 4.34 
 
 4.51 
 
 5.08 
 
 5.25 
 
 5.43 
 
 + 2 
 
 3.26 
 
 3.40 
 
 3.55 
 
 4.10 
 
 4.26 
 
 4.43 
 
 4.59 
 
 5.16 
 
 5.34 
 
 5.51 
 
 
 
 3.33 
 
 3.47 
 
 4.03 
 
 4.18 
 
 4.34 
 
 4.51 
 
 5.08 
 
 5.25 
 
 5.42 
 
 6.00 
 
 
 — 2 
 4 
 
 3.40 
 3.47 
 
 3.55 
 4.02 
 
 4 10 
 4.17 
 
 4.26 
 4.34 
 
 4.42 
 4.50 
 
 4.59 
 5.07 
 
 5.16 
 5.24 
 
 5.33 
 5.42 
 
 5.51 
 
 6.09 
 6.17 
 
 6.00 
 
 6 
 
 8 
 
 10 
 
 12 
 
 3.54 
 4.01 
 4.08 
 4.15 
 
 4.09 
 4.16 
 4.24 
 4.31 
 
 4.25 
 4.33 
 4.40 
 4.48 
 
 4.41 
 4.49 
 4.57 
 5.06 
 
 4.58 
 5.06 
 5.15 
 
 5.15 
 5.24 
 
 5.33 
 
 5.51 
 5.59 
 6.08 
 6.17 
 
 6.08 
 6.17 
 6.26 
 6.35 
 
 6.26 
 6.35 
 6.44 
 6.53 
 
 5.41 
 5.50 
 5.59 
 
 5.32 
 5.41 
 
 5.23 
 
 14 
 
 4.22 
 
 4.39 
 
 4.56 
 
 5.14 
 
 5.32 
 
 5.50 
 
 6.08 
 
 6.26 
 
 6.44 
 
 7.02 
 
 16 
 18 
 
 4.30 
 4.38 
 
 4.47 
 
 5.05 
 5.13 
 
 5.23 
 5.31 
 
 5.41 
 5.50 
 
 5.59 
 6.08 
 
 6.17 
 6.27 
 
 6.36 
 6.46 
 
 6.54 
 7.04 
 
 7.12 
 
 7.22 
 
 4.55 
 
 
 
 
 
 
 
 
 
 
 
 
 20 
 
 4.46 
 
 5.04 
 
 5.22 
 
 5.41 
 
 5.59 
 
 6.18 
 
 6.34 
 
 6.56 
 
 7.14 
 
 7.32 
 
 22 
 
 4.54 
 
 5.13 
 
 5.31 
 
 5.50 
 
 6.09 6.28 
 
 6.48 
 
 7.06 
 
 7.25 
 
 7 43 
 
 —24 
 
 5.03 
 
 5.22 
 
 5.41 6.00 
 
 6.20 6.39 
 
 6.59 
 
 7.1§ 
 
 7.36 
 
 7.54 
 
 
 117° 
 
 114° 
 
 111° 
 
 108° 
 
 105° 102° 
 
 99° 
 
 96° 
 
 93° 
 
 90° 
 
 131 
 
Azimuth and Hour Angle, for Latitude 
 
 
 and Dec ination.— Tab e Vl. 
 
 LATITUDE 44^. 
 
 Declination. 
 
 j^zi:]ynTJTH:s_ 
 
 15° 
 
 20^ 
 
 2C° 33^ 1 35° 40° 
 
 45° 
 
 50° 
 
 55° 
 
 60° 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H . M . 
 
 H . M . H . M . 
 
 H. M. 
 
 H. M. j H. M. 
 
 H. M. 
 
 -f 24° 
 
 .23 
 
 .31 
 
 .39 
 
 .48 
 
 .57 1.07 
 
 1.18 
 
 1.30 ! 1.43 
 
 1.58 
 
 22 
 
 .24 
 
 .33 
 
 .42 
 
 .52 
 
 1 02 , 1.12 
 
 1 24 
 
 1.36 
 
 1.51 
 
 2.06 
 
 ^ 20 
 
 0.26 
 
 0.36 
 
 0.45 
 
 0.55 
 
 1.06 
 
 1.17 
 
 1.29 
 
 1 43 
 
 1.58 
 
 2.14 
 
 -o 18 
 
 .28 
 
 .38 
 
 .48 
 
 .59 
 
 1.10 
 
 1 22 ( 1 35 
 
 1.49 
 
 2.04 
 
 2.21 
 
 B 16 
 
 .30 
 
 .40 
 
 .51 
 
 1.02 
 
 1.14 
 
 1 27 
 
 1.40 
 
 1.55 
 
 2.11 
 
 2.28 
 
 rt 14 
 
 .31 
 
 .42 
 
 .54 
 
 1.05 
 
 1.18 
 
 1.31 
 
 1.45 
 
 2.00 
 
 2.17 
 
 2.35 
 
 2 12 
 
 .33 
 
 .44 
 
 .56 
 
 1.08 
 
 1.22 ' 1 35 
 
 1.50 
 
 2.06 
 
 2.23 
 
 2.42 
 
 a; 10 
 
 0.35 
 
 0.46 
 
 0.59 
 
 1.12 
 
 1.25 1.40 
 
 1 55 
 
 2.12 
 
 2.30 
 
 2.49 
 
 -. 8 
 
 .36 
 
 .49 
 
 1.02 
 
 1.15 
 
 1.29 t 1.44 
 
 2.00 
 
 2.17 
 
 2.36 
 
 2.55 
 
 6 
 
 .38 
 
 .51 
 
 1.04 
 
 1.18 
 
 1.33 1 1.48 
 
 2 05 
 
 2.22 
 
 2.42 
 
 3.02 
 
 4 
 
 .39 
 
 .53 
 
 1.07 
 
 1.21 
 
 1.36 1 1.53 
 
 2.10 
 
 2.28 
 
 2.47 
 
 3.08 
 
 + 2 
 
 .41 
 
 .55 
 
 1.09 
 
 1.24 
 
 1.40 j 1.57 
 
 2.14 
 
 2.33 
 
 2.53 
 
 3.15 
 
 
 
 0.42 
 
 0.57 
 
 1.12 
 
 1.27 
 
 1 
 1.44 1 2 01 
 
 1 
 
 2.19 
 
 2.38 
 
 2.59 
 
 3.21 
 
 — 2 
 
 .44 
 
 .59 
 
 1.14 
 
 1.31 
 
 1.47 
 
 2.05 
 
 2 24 
 
 2.44 
 
 3.05 
 
 3.27 
 
 4 
 
 .45 
 
 1.01 
 
 117 
 
 1.34 
 
 1.51 
 
 2.09 
 
 2.29 
 
 2.49 
 
 3.11 
 
 3.34 
 
 i? 6 
 
 .47 
 
 1.03 
 
 1.19 
 
 1.37 
 
 1.55 
 
 2.14 
 
 2.33 
 
 2-54 
 
 3.17 
 
 3.40 
 
 5 8 
 
 .48 
 
 1.05 
 
 1.22 
 
 1.40 
 
 1.58 
 
 2.18 
 
 2.38 
 
 3.00 
 
 3.23 
 
 3 47 
 
 •S 10 
 
 0.50 
 
 1.07 
 
 1.25 
 
 1.43 
 
 2.02 
 
 2 . 22 
 
 2.43 
 
 3 05 
 
 3.29 
 
 3.53 
 
 ^ 12 
 
 .51 
 
 1.09 
 
 1.27 
 
 1.46 
 
 2.06 
 
 2.26 
 
 2.48 
 
 3 11 
 
 3.35 
 
 4.00 
 
 Z 14 
 
 .53 
 
 1.11 
 
 1,30 
 
 1.49 
 
 2.10 
 
 2.31 
 
 2 53 
 
 3.16 
 
 3.41 
 
 4.07 
 
 ^ 16 
 
 .55 
 
 1.13 
 
 1.33 
 
 1.53 
 
 2.14 
 
 2.35 
 
 2.58 
 
 3.22 
 
 3.47 
 
 4.14 
 
 ■■H 18 
 
 .56 
 
 1.16 
 
 1.36 
 
 1.56 
 
 2.18 
 
 2.40 
 
 3.03 
 
 3.28 
 
 3.54 
 
 4.21 
 
 ;d 20 
 22 
 
 0.58 
 1.00 
 
 1.18 
 1.20 
 
 1.38 
 1.41 
 
 2.00 
 2 03 
 
 2.22 
 2 26 
 
 2.45 
 2.50 
 
 3.09 
 3.14 
 
 3.34 
 
 4.01 
 
 4 28 
 
 3.40 4.08 
 
 4.36 
 
 —24 
 
 1.02 
 
 1 . 23 [ 1 . 44 
 
 2.07 
 
 2.30 
 
 2.55 
 
 3.20 
 
 3.47 
 
 4.1=> 
 
 4.44 
 
 
 
 
 
 i 
 
 
 
 
 
 125° 
 
 120° 
 
 132 
 

 Table VL- 
 
 -Continued. 
 
 
 LATITUDE 
 
 440, 
 
 Declination. 
 
 .i^ZinVCTJTS-S- 
 
 
 
 
 
 
 
 
 
 ! 
 
 
 63^ 
 
 66° 
 
 69° 
 
 72° 
 
 75° 
 
 78° 
 
 81° 
 
 84° 
 
 87° 
 
 90° 
 
 
 H. !\1. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H . .M . 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 + 21^ 
 
 2 08 2.18 
 
 2.29 
 
 2.41 
 
 2.54 
 
 3.07 
 
 3.22 
 
 3.37 
 
 3.53 
 
 4.10 
 
 22 
 
 2.16 1 2.27 
 
 2.38 
 
 2.51 
 
 3.04 
 
 3.18 
 
 3.32 
 
 3.48 
 
 4.04 
 
 4.21 
 
 20 
 
 2.24 
 
 2.35 
 
 2.47 
 
 3.00 
 
 3.13 
 
 3.27 
 
 3.42 
 
 3 58 
 
 4.14 
 
 4.31 
 
 18 
 
 2 32 
 
 2 44 
 
 2.56 
 
 3.09 
 
 3.22 
 
 3.37 
 
 3.52 
 
 4.08 
 
 4.24 
 
 4.41 
 
 16 
 
 2.40 
 
 2.51 
 
 3.04 
 
 3.17 
 
 3.31 
 
 3.46 
 
 4.01 
 
 4.17 
 
 4.34 
 
 4.51 
 
 14 
 
 2.47 
 
 2 59 
 
 3.12 
 
 3 26 
 
 3.40 
 
 3.55 
 
 4.10 
 
 4.26 
 
 4.43 
 
 5.00 
 
 12 
 
 2.54 
 
 3 07 
 
 3.20 
 
 3.34 
 
 3 48 
 
 4.03 
 
 4.19 
 
 4.35 
 
 4 52 
 
 5.09 
 
 10 
 
 3.01 
 
 3.14 
 
 3.28 
 
 3 42 
 
 3.56 
 
 4.12 
 
 4.28 
 
 4.44 
 
 5.01 
 
 5.18 
 
 8 
 
 3.08 
 
 3.21 
 
 3.35 
 
 3.49 
 
 4.04 
 
 4.20 
 
 4.36 
 
 4.52 
 
 5.09 
 
 5.26 
 
 6 
 
 3.15 
 
 3.28 
 
 3.42 
 
 3.57 
 
 4.]2 
 
 4.28 
 
 4.44 
 
 5.01 
 
 5 18 
 
 5.35 
 
 4 
 
 3.22 
 
 3.35 
 
 3.50 
 
 4.05 
 
 4.20 
 
 4.36 
 
 4.52 
 
 5.09 
 
 5.26 
 
 5.43 
 
 + 2 
 
 3.28 
 
 3.42 
 
 3.57 
 
 4.12 
 
 4.28 
 
 4.44 
 
 5.01 
 
 5.17 
 
 5.34 
 
 5.52 
 
 
 
 3.35 
 
 3.49 
 
 4.04 
 
 4.20 
 
 4.36 
 
 4.52 
 
 5.09 
 
 5.26 
 
 5.43 
 
 6.00 
 
 
 — 2 
 4 
 
 3.42 
 
 3.48 
 
 3.56 \ 4 12 
 
 4.27 
 4.35 
 
 4.43 
 4.51 
 
 5.00 
 5.08 
 
 5.17 
 5.25 
 
 5.34 
 5.42 
 
 5.51 
 
 6.08 
 6.17 
 
 4.03 
 
 4.19 
 
 5 , 59 
 
 6 
 
 3.55 
 
 4.10 
 
 4.26 
 
 4.42 
 
 4.59 
 
 5.16 
 
 5.33 
 
 5.50 
 
 6.08 
 
 6.25 
 
 8 
 10 
 12 
 14 
 
 4.02 
 4.09 
 4.16 
 4.23 
 
 4.18 
 4.25 
 4.32 
 4.40 
 
 4.34 
 4.41 
 4.49 
 4.57 
 
 4.50 
 4.58 
 5.06 
 
 5.07 
 5.15 
 
 5.24 
 
 5.41 
 5.50 
 5.58 
 6.07 
 
 5.59 
 6.07 
 6.16 
 6.25 
 
 6.16 
 6.25 
 6.33 
 6.42 
 
 6.34 
 6.42 
 6.51 
 
 7.00 
 
 5 32 
 5.41 
 5.49 
 
 5.23 
 5,31 
 
 5.14 
 
 16 
 18 
 
 4.30 
 4.38 
 
 4.47 
 
 5.05 
 5.13 
 
 5.22 
 5.31 
 
 5.40 
 5 49 
 
 5.58 
 6.07 
 
 6.16 
 6.25 
 
 6.34 
 6.43 
 
 6.52 
 7.01 
 
 7.09 
 7.19 
 
 4.55 
 
 20 
 
 4.46 
 
 5.03 
 
 5.21 
 
 5.40 
 
 5.58 
 
 6.I0 
 
 6.35 
 
 6.53 
 
 7.11 
 
 7.29 
 
 22 
 
 4.54 
 
 5.12 
 
 5.30 
 
 5.49 
 
 6.08 
 
 6 26 
 
 6.45 
 
 7.03 
 
 7.21 
 
 7.39 
 
 —24 
 
 5.02 
 
 5.21 
 
 5 40 
 
 5.58 
 
 6.18 
 
 6.37 
 
 6.55 
 
 7.14 
 
 7.32 
 
 7.50 
 
 
 117° 
 
 lir 
 
 111° 
 
 108° 
 
 105° 
 
 102° 
 
 99° 
 
 96° 
 
 93° 
 
 90° 
 
 133 
 
Azimuth and Hour Angle, for Latitude 
 
 
 and Dec ination— Table VL 
 
 LATITUDE 45^. 
 
 Declination. 
 
 ^ZXnVCTJTHIS. 
 
 15° 
 
 20° 
 
 25° 
 
 30^ 
 
 35° 
 
 40° 
 
 45° 
 
 50° 
 
 55° 
 
 60° 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 +24° 
 
 .24 
 
 .32 
 
 .41 
 
 .50 
 
 1.00 
 
 1.10 
 
 1.21 
 
 1.34 
 
 1.47 
 
 2.02 
 
 22 
 
 .26 
 
 .35 
 
 .44 
 
 .54 
 
 1.04 
 
 1.15 
 
 1.27 
 
 1.40 
 
 1.54 
 
 2.10 
 
 «5 20 
 
 0.27 
 
 0.37 
 
 0.47 
 
 0.57 
 
 1.08 
 
 1.20 
 
 1.33 
 
 1.46 
 
 2.01 
 
 2.18 
 
 -S 18 
 
 .29 
 
 .39 
 
 .50 
 
 1.01 
 
 1.12 
 
 1.25 
 
 1.38 
 
 1.52 
 
 2.08 
 
 2.25 
 
 B 16 
 
 .31 
 
 .41 
 
 .52 
 
 1.04 
 
 1.16 
 
 1.29 
 
 1.43 
 
 1.58 
 
 2.14 
 
 2.32 
 
 rt 14 
 
 .32 
 
 .43 
 
 .55 
 
 1.07 
 
 1.20 
 
 1.33 
 
 1.48 
 
 2.03 
 
 2.20 
 
 2.39 
 
 2 12 
 
 .34 
 
 .46 
 
 .58 
 
 1.10 
 
 1.24 
 
 1.38 
 
 1.53 
 
 2.09 
 
 2.26 
 
 2.45 
 
 V 10 
 
 0.35 
 
 0.48 
 
 1.00 
 
 1.14 
 
 1.27 
 
 1.42 
 
 1.58 
 
 2.14 
 
 2.32 
 
 2.52 
 
 ^ 8 
 
 .37 
 
 .50 
 
 1.03 
 
 1.17 
 
 1.31 
 
 1.46 
 
 2.02 
 
 2 20 
 
 2.38 
 
 2.58 
 
 ^ 6 
 
 .38 
 
 .52 
 
 1.05 
 
 1.20 
 
 1.35 
 
 1.50 
 
 2.07 
 
 2.25 
 
 2.44 
 
 3.04 
 
 4 
 
 .40 
 
 .54 
 
 1.08 
 
 1.23 
 
 1.38 
 
 1.55 
 
 2.12 
 
 2.30 
 
 2.50 
 
 3.11 
 
 + 2 
 
 .41 
 
 .56 
 
 1.10 
 
 1.26 
 
 1.42 
 
 1.59 
 
 2.16 
 
 2.35 
 
 2.55 
 
 3.17 
 
 
 
 0.43 
 
 0.58 
 
 1.13 
 
 1.29 
 
 1.45 
 
 2.03 
 
 2.21 
 
 2.40 
 
 3.01 
 
 3.23 
 
 — 2 
 
 .44 
 
 1.00 
 
 1.16 
 
 1.32 
 
 1.49 
 
 2.07 
 
 2 26 
 
 2.46 
 
 3.07 
 
 3.29 
 
 4 
 
 .46 
 
 1.02 
 
 1.18 
 
 1.35 
 
 1.52 
 
 2.11 
 
 2.30 
 
 2.51 
 
 3.13 
 
 3.35 
 
 ^ 6 
 
 .47 
 
 1.04 
 
 1.21 
 
 1.38 
 
 1.56 
 
 2.15 
 
 2.35 
 
 2.56 
 
 3.18 
 
 3.42 
 
 3 8 
 
 .49 
 
 1.06 
 
 1.23 
 
 1 41 
 
 2.00 
 
 2.19 
 
 2.40 
 
 3.01 
 
 3.24 
 
 3.48 
 
 •S 10 
 
 0.50 
 
 1.08 
 
 1.26 
 
 1.44 
 
 2.03 
 
 2.23 
 
 2.44 
 
 3.07 
 
 3.30 
 
 3.54 
 
 i 11 
 
 .52 
 
 1.10 
 
 1.28 
 
 1.47 
 
 2.07 
 
 2.28 
 
 2.49 
 
 3.12 
 
 3.36 
 
 4.01 
 
 .54 
 
 1.12 
 
 1.31 
 
 1.50 
 
 2.11 
 
 2.32 
 
 2 54 
 
 3.17 
 
 3.42 
 
 4.08 
 
 ^ 16 
 
 .55 
 
 1.14 
 
 1.34 
 
 1.54 
 
 2.15 
 
 2.36 
 
 2.59 
 
 3.23 
 
 3.48 
 
 4.14 
 
 "H 18 
 
 .57 
 
 1.16 
 
 1.36 
 
 1.57 
 
 2.19 
 
 2.41 
 
 3.04 
 
 3.29 
 
 3.55 
 
 4.21 
 
 ^ 20 
 
 22 
 
 —24 
 
 .58 
 1.00 
 1.02 
 
 1.19 
 1.21 
 1.23 
 
 1.39 
 1.42 
 1.45 
 
 2.00 
 2.04 
 2.08 
 
 2.23 
 2.27 
 2.31 
 
 2.46 
 2.50 
 2.55 
 
 3.10 
 3.15 
 3.21 
 
 3.35 
 3.41 
 3.47 
 
 4.01 
 4.08 
 
 4.29 
 
 4.36 
 4.44 
 
 4.15 
 
 
 
 
 
 
 
 
 
 
 125° 
 
 120° 
 
 134 
 
Table VI.— Continued. 
 
 LATITUDE 45^. 
 
 
 ^ZIDVCTJTSIS. 
 
 Declination. 
 
 
 
 
 
 
 
 
 
 
 
 
 63" 
 
 66° 
 
 69° 
 
 72° 
 
 75° 
 
 78° 
 
 81° 
 
 84° 
 
 87° 
 
 90° 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H.M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 +24° 
 
 2.12 
 
 2.23 
 
 2.34 
 
 2.46 
 
 2.59 
 
 3.12 
 
 3.26 
 
 3,42 
 
 3.58 
 
 4.14 
 
 22 
 
 2.20 
 
 2.31 
 
 2.43 
 
 2.55 
 
 3.08 
 
 3.22 
 
 3.37 
 
 3.52 
 
 4.08 
 
 4.25 
 
 20 
 
 2.28 
 
 2.39 
 
 2.51 
 
 3.04 
 
 3.17 
 
 3.31 
 
 3.46 
 
 4.02 
 
 4.18 
 
 4.35 
 
 18 
 
 2 36 
 
 2.47 
 
 3,00 
 
 3.13 
 
 3.26 
 
 3.41 
 
 3.56 
 
 4.11 
 
 4.27 
 
 4.44 
 
 16 
 
 2.43 
 
 2.55 
 
 3.08 
 
 3.21 
 
 3.35 
 
 3.49 
 
 4.05 
 
 4.20 
 
 4.37 
 
 4.53 
 
 14 
 
 2.50 
 
 3 02 
 
 3.15 
 
 3 29 
 
 3.43 
 
 3.58 
 
 4.13 
 
 4.29 
 
 4.45 
 
 5.02 
 
 12 
 
 2.57 
 
 3 10 
 
 3.23 
 
 3.37 
 
 3.51 
 
 4.06 
 
 4.22 
 
 4.38 
 
 4 54 
 
 5.11 
 
 10 
 
 3.04 
 
 3.17 
 
 3.30 
 
 3.44 
 
 3.59 
 
 4.14 
 
 4.30 
 
 4.46 
 
 5.03 
 
 5.19 
 
 8 
 
 3.11 
 
 3.24 
 
 3.38 
 
 3.52 
 
 4.07 
 
 4.22 
 
 4.38 
 
 4.54 
 
 5.11 
 
 5.28 
 
 6 
 
 3.17 
 
 3.31 
 
 3.45 
 
 3.59 
 
 4.34 
 
 4.30 
 
 4.46 
 
 5.02 
 
 5.19 
 
 5.36 
 
 4 
 
 3.24 
 
 3.38 
 
 3.52 
 
 4.07 
 
 4.22 
 
 4.38 
 
 4.54 
 
 5.10 
 
 5.27 
 
 5.44 
 
 + 2 
 
 3.30 
 
 3.44 
 
 3.59 
 
 4.14 
 
 4.30 
 
 4.45 
 
 5.02 
 
 5.18 
 
 5.35 
 
 5.52 
 
 
 
 3.37 
 
 3.52 
 
 4.06 
 
 4.21 
 
 4.37 
 
 4.53 
 
 5.09 
 
 5.26 
 
 5.43 
 
 6.00 
 
 
 — 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 3.43 
 3.50 
 3.56 
 4.03 
 4.10 
 4.17 
 4.24 
 4.31 
 4.38 
 
 3.58 
 4.05 
 4.12 
 4.19 
 4.26 
 4.33 
 4.40 
 4.47 
 
 4.13 
 4.20 
 
 4.27 
 4.34 
 4.42 
 4. 49 
 4.57 
 
 4.29 
 4.36 
 4.43 
 4.51 
 4.58 
 5.06 
 
 4.44 
 4.52 
 5.00 
 5.07 
 5.15 
 
 5.01 
 5.08 
 5.16 
 5.24 
 
 5.17 
 5.25 
 5.33 
 
 5.34 
 5.42 
 
 5.51 
 
 6.08 
 6.16 
 6.24 
 6.32 
 6.41 
 6.49 
 6.58 
 7.07 
 7.16 
 
 5.59 
 6.07 
 6.15 
 6.24 
 6.32 
 6.41 
 6.50 
 6.59 
 
 5.50 
 5.58 
 6.06 
 6.15 
 6.23 
 6.32 
 6.41 
 
 5.41 
 5.49 
 5.57 
 6.06 
 6.14 
 6.23 
 
 5.32 
 5.40 
 5.48 
 5.57 
 6.06 
 
 5.23 
 5.31 
 5.39 
 
 5.48 
 
 5.14 
 5.22 
 5.30 
 
 5.04 
 5.12 
 
 4.55 
 
 20 
 
 4.46 
 
 5.03 
 
 5.21 
 
 5.39 
 
 5.57 
 
 6.15 
 
 6.33 
 
 6.51 
 
 7.08 
 
 7.25 
 
 22 
 
 4.53 
 
 5.11 
 
 5.29 
 
 5.47 
 
 6.06 
 
 6 24 
 
 6.42 
 
 7.00 
 
 7.18 
 
 7.35 
 
 —24 
 
 5.02 
 
 5.20 
 
 5. 38 
 
 5.57 
 
 6.15 
 
 6.34 
 
 6.53 
 
 7.11 
 
 7.28 
 
 7.46 
 
 
 117° 
 
 114° 
 
 111° 
 
 108° 
 
 105° 
 
 102° 
 
 99° 
 
 96° 
 
 93° 
 
 90° 
 
 135 
 
Azimuth and Hour Angle, for Latitude 
 
 
 and Declination— Table VI. 
 
 LATITUDE 46"^. 
 
 Declination. 
 
 -A^ZZDVCTTTIHIS- 
 
 15° 
 
 20° 
 
 25° 
 
 30^ 
 
 35° 40° 
 
 45° 
 
 50° 
 
 55° 
 
 60° 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 1 
 
 H. M. j H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 +24° 
 
 .25 
 
 .34 
 
 .43 
 
 .52 
 
 1.02 
 
 1.13 
 
 1.25 
 
 1.38 
 
 1.51 
 
 2.07 
 
 22 
 
 .27 
 
 .36 
 
 .46 
 
 .56 
 
 1.07 
 
 1.18 
 
 1.30 
 
 1.44 
 
 1.58 
 
 2.14 
 
 .\ 20 
 
 0.28 
 
 0.38 
 
 0.49 
 
 0.59 
 
 1.11 
 
 1.23 
 
 1.36 
 
 1.50 
 
 2.05 
 
 2.21 
 
 -S 18 
 
 .30 
 
 .40 
 
 .51 
 
 1.03 
 
 1.14 
 
 1.27 
 
 1.41 
 
 1.56 
 
 2.11 
 
 2.28 
 
 2 16 
 
 .32 
 
 .43 
 
 .54 
 
 1.06 
 
 1.18 
 
 1.32 
 
 1.46 2.01 
 
 2.17 
 
 2.35 
 
 1i 14 
 
 .33 
 
 .45 
 
 .57 
 
 1.09 
 
 1.22 
 
 1.36 
 
 1.51 
 
 2.07 
 
 2.23 
 
 2.42 
 
 ^ 12 
 
 .35 
 
 .47 
 
 .59 
 
 1.12 
 
 1.26 
 
 1.40 
 
 1.55 
 
 2.12 
 
 2.29 
 
 2.48 
 
 0) 10 
 
 0.36 
 
 0.49 
 
 1.02 
 
 1.15 
 
 1.29 
 
 1.41 
 
 2.00 
 
 2.17 
 
 2.35 
 
 2.54 
 
 ^ 8 
 
 .38 
 
 .51 
 
 1.04 
 
 1.18 
 
 1.33 
 
 1.48 
 
 2.05 
 
 2 22 
 
 2.41 
 
 3.01 
 
 ^ 6 
 
 .39 
 
 .53 
 
 1.07 
 
 1.21 
 
 1.36 
 
 1.52 
 
 2.09 
 
 2.28 
 
 2.46 
 
 3.08 
 
 4 
 
 .41 
 
 .55 
 
 1.09 
 
 1.24 
 
 1.40 
 
 1.56 
 
 2.14 
 
 2.33 
 
 2.52 
 
 3.13 
 
 + 2 
 
 .42 
 
 .57 
 
 1.12 
 
 1.27 
 
 1.43 
 
 2.00 
 
 2.18 
 
 2.38 
 
 2.58 
 
 3.19 
 
 
 
 0.44 
 
 0.59 
 
 1.14 
 
 1.30 
 
 1.47 2.04 
 
 2.23 
 
 2.43 
 
 3.03 
 
 3.25 
 
 — 2 
 
 .45 
 
 1.01 
 
 1.17 
 
 1.33 
 
 1.50 2.03 
 
 2.27 
 
 2.48 
 
 3.09 
 
 3.31 
 
 4 
 
 .47 
 
 1.03 
 
 1.19 
 
 1.36 
 
 1.54 I 2.12 
 
 2.32 
 
 2.53 
 
 3.14 
 
 3.37 
 
 V 6 
 
 .48 
 
 1.05 
 
 1.22 
 
 1.39 
 
 1.57 
 
 2.16 
 
 2.37 
 
 2.58 
 
 3.20 
 
 3.42 
 
 ^ 8 
 
 .50 
 
 1.07 
 
 1.24 
 
 1.42 
 
 2.01 
 
 2.21 
 
 2.41 
 
 3.03 
 
 3.25 
 
 3.49 
 
 •S 10 
 
 0.51 
 
 1.09 
 
 1.27 
 
 1.45 
 
 2.05 
 
 2.25 
 
 2.46 
 
 3.08 
 
 3.31 
 
 3.56 
 
 5 12 
 ^ 14 
 
 .53 
 
 1.11 
 
 1.29 
 
 1.48 
 
 2.08 
 
 2.29 
 
 2.50 
 
 3.13 
 
 3.37 
 
 4.02 
 
 .54 
 
 1.13 
 
 1.32 
 
 1.51 
 
 2.12 
 
 2.33 
 
 2. 55 
 
 3.19 
 
 3.43' 
 
 4.08 
 
 J 16 
 
 .56 
 
 1.15 
 
 1.34 
 
 1.55 
 
 2.16 
 
 2.37 
 
 3.00 
 
 3.24 
 
 3.49 
 
 4.15 
 
 •-§ 18 
 
 .57 
 
 1,17 
 
 1.37 
 
 1.58 
 
 2.19 
 
 2.42 
 
 3.05 
 
 3.30 
 
 3.55 
 
 4.22 
 
 ^ 20 
 
 22 
 
 —24 
 
 0.59 
 1.01 
 1.02 
 
 1.19 
 1.21 
 1.24 
 
 1.40 
 1.43 
 1.46 
 
 2.01 
 2.04 
 2.08 
 
 2.23 
 2.27 
 2.32 
 
 2.46 
 2.51 
 2.56 
 
 3.10 
 3.16 
 3.21 
 
 3.36 
 3.42 
 3.48 
 
 4.01 
 4.08 ' 
 
 4.29 
 
 4.36 
 4.43 
 
 4.15 
 
 
 
 
 
 
 
 
 
 
 125° 
 
 120° 
 
 136 
 
Table VI —Continued. 
 
 LATITUDE 46^. 
 
 
 ^ZZnSdlTJTHIS. 
 
 Declination. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 63° 
 
 66° 
 
 69° 
 
 72° 
 
 75° 
 
 78° 
 
 81° 
 
 84° 
 
 87° 
 
 90° 
 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H. M. 
 
 H.M. 
 
 H.M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M . 
 
 + 24° 
 
 2 16 
 
 2.27 
 
 2.38 
 
 2.50 
 
 3.03 
 
 3.17 
 
 3.31 
 
 3.46 
 
 4.02 
 
 4 18 
 
 22 
 
 2.25 
 
 2.35 
 
 2.47 
 
 3.00 
 
 3.12 
 
 3.27 
 
 3.41 
 
 3.56 
 
 4.12 
 
 4.28 
 
 20 
 
 2.32 
 
 2.43 
 
 2.55 
 
 3.08 
 
 3.21 
 
 3.36 
 
 3.50 
 
 4 05 
 
 4.21 
 
 4.38 
 
 18 
 
 2 39 
 
 2 51 
 
 3.03 
 
 3.16 
 
 3.30 
 
 3.44 
 
 3.59 
 
 4.14 
 
 4.30 
 
 4.47 
 
 16 
 
 2.47 
 
 2.58 
 
 3.11 
 
 3.24 
 
 3.38 
 
 3.53 
 
 4.08 
 
 4.23 
 
 4.39 
 
 4. 56 
 
 14 
 
 2.53 
 
 3 06 
 
 3. 19 
 
 3.32 
 
 3.46 
 
 4.01 
 
 4.16 
 
 4.32 
 
 4.48 
 
 5 04 
 
 12 
 
 3.00 
 
 3 13 
 
 3.26 
 
 3.40 
 
 3 54 
 
 4.09 
 
 4.24 
 
 4.40 
 
 4 56 
 
 5.13 
 
 10 
 
 3.07 
 
 3.20 
 
 3.33 
 
 3.47 
 
 4.02 
 
 4.17 
 
 4.32 
 
 4.48 
 
 5. 04 
 
 5.21 
 
 8 
 
 3.13 
 
 3.26 
 
 3.40 
 
 3.54 
 
 4.09 
 
 4.25 
 
 4.40 
 
 4.56 
 
 5.1-2 
 
 5.29 
 
 6 
 
 3.20 
 
 3.33 
 
 3.47 
 
 4.02 
 
 4.]6 
 
 4.32 
 
 4.48 
 
 5.04 
 
 5 20 
 
 5.37 
 
 4 
 
 3.26 
 
 3.40 
 
 3.54 
 
 4.09 
 
 4.24 
 
 4.40 
 
 4.55 
 
 5.11 
 
 5.28 
 
 5.45 
 
 + 2 
 
 3.32 
 
 3.46 
 
 4.01 
 
 4.16 
 
 4.31 
 
 4.47 
 
 5.03 
 
 5.19 
 
 5.36 
 
 5.52 
 
 
 
 3.39 
 
 3.53 
 
 4.03 
 
 4.23 
 
 4.38 
 
 4.55 
 
 5.10 
 
 5.27 
 
 5.43 
 
 6.00 
 
 
 — 2 
 4 
 
 3.45 
 3.51 
 
 4.00 
 4 06 
 
 4 14 
 4.21 
 
 4.30 
 4.37 
 
 4.46 
 4.53 
 
 5.02 
 5.09 
 
 5.18 
 5.25 
 
 5.34 
 5.42 
 
 5.51 
 
 6.08 
 6.15 
 
 5.59 
 
 6 
 8 
 10 
 12 
 14 
 16 
 
 3.58 
 4.04 
 4.11 
 4.17 
 4.24 
 4.31 
 
 4.13 
 4.19 
 4.26 
 4.33 
 4.40 
 4.47 
 
 4.28 
 4.35 
 4.42 
 4.49 
 4.57 
 
 4.44 
 4.51 
 
 4.58 
 5.06 
 
 5.00 
 5.08 
 5.15 
 
 5.17 
 5.24 
 
 5.33 
 
 5.50 
 5.58 
 6.06 
 6.14 
 6.22 
 6.30 
 
 6.07 
 6.14 
 6.22 
 6.30 
 6.39 
 6.47 
 
 6.23 
 6.31 
 6.39 
 6.47 
 6 56 
 7.04 
 
 5.41 
 5.49 
 5.57 
 6.05 
 6.13 
 
 5.32 
 5.40 
 5.48 
 5.56 
 
 5.23 
 5.30 
 5.38 
 
 5.13 
 5.21 
 
 5.04 
 
 18 
 
 4,38 
 
 4.55 
 
 5.12 
 
 5.29 
 
 5.47 
 
 6.05 
 
 6.22 
 
 6.39 
 
 6.56 
 
 7.13 
 
 20 
 
 4.45 
 
 5.03 
 
 5.20 
 
 5.37 
 
 5.55 
 
 6.13 
 
 6.31 
 
 6.48 
 
 7.05 
 
 7 22 
 
 22 
 
 4.53 
 
 5.10 
 
 5.28 
 
 5.46 
 
 6.04 
 
 6 22 
 
 6.40 
 
 6.58 
 
 7.15 
 
 7.32 
 
 -24 
 
 5.01 
 
 5.19 
 
 5.37 
 
 5.55 
 
 6.13 
 
 6.32 
 
 6.50 
 
 7.07 
 
 7.25 
 
 7.42 
 
 
 117° 
 
 lir 
 
 111° 
 
 108° 
 
 105° 
 
 102° 
 
 99° 96° 
 
 93° 
 
 90° 
 
 137 
 
Azimuth and Hour Angle, for Latitude 
 
 
 and Declination— Table VI. 
 
 LATITUDE 47^. 
 
 Declination. 
 
 .A^ZIDyCTTTHES. 
 
 
 
 
 
 
 
 
 
 
 
 
 15° 
 
 20° 
 
 25° 
 
 30^ 
 
 35^ 
 
 40° 
 
 45° 
 
 50° 
 
 55° 
 
 60° 
 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H.M. 
 
 H, M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H. M. 
 
 +24° 
 
 .26 
 
 .35 
 
 .45 
 
 .54 
 
 1.05 
 
 1.16 
 
 1.28 
 
 1.41 
 
 1.55 
 
 2.11 
 
 22 
 
 .28 
 
 .37 
 
 .47 
 
 .58 
 
 1 09 
 
 1.21 
 
 1.33 
 
 1.47 
 
 2.02 
 
 2.18 
 
 .; 20 
 
 0.29 
 
 0.40 
 
 0.50 
 
 1.01 
 
 1.13 
 
 1.25 
 
 1.38 
 
 1.53 
 
 2.08 
 
 2.25 
 
 -S 18 
 
 .31 
 
 .42 
 
 .53 
 
 1.04 
 
 1.17 
 
 1.30 
 
 1.44 
 
 1.58 
 
 2.14 
 
 2.32 
 
 2 16 
 
 .33 
 
 .44 
 
 .56 
 
 1.08 
 
 1.20 
 
 1 34 
 
 1.48 
 
 2.04 
 
 2.20 
 
 2.38 
 
 '5 14 
 
 .34 
 
 .46 
 
 .58 
 
 1.11 
 
 1.24 
 
 1.38 
 
 1.53 
 
 2.09 
 
 2.26 
 
 2.45 
 
 2 12 
 
 .36 
 
 .48 
 
 1.01 
 
 1.14 
 
 1.28 
 
 1.42 
 
 1.58 
 
 2.14 
 
 2.32 
 
 2.51 
 
 0) 10 
 
 0.37 
 
 0.50 
 
 1.03 
 
 1.17 
 
 1.31 
 
 1.46 
 
 2.02 
 
 2.19 
 
 2.38 
 
 2.57 
 
 •i 8 
 
 .38 
 
 .52 
 
 1.06 
 
 1.20 
 
 1.35 
 
 1.50 
 
 2.07 
 
 2 25 
 
 2.43 
 
 3.03 
 
 ^ 6 
 
 .40 
 
 .54 
 
 1.08 
 
 1.23 
 
 1.38 
 
 1.54 
 
 2.11 
 
 2 30 
 
 2.49 
 
 3.09 
 
 4 
 
 .41 
 
 .56 
 
 1.11 
 
 1.26 
 
 1.42 
 
 1 58 
 
 2.16 
 
 2.34 
 
 2.54 
 
 3.15 
 
 + 2 
 
 .43 
 
 .58 
 
 1.13 
 
 1.29 
 
 1.45 
 
 2.02 
 
 2.20 
 
 2.39 
 
 3.00 
 
 3.21 
 
 
 
 0.44 
 
 1.00 
 
 1.15 
 
 1.32 
 
 1.48 
 
 2.06 
 
 2.25 
 
 2.44 
 
 3.05 
 
 3.27 
 
 — 2 
 
 .46 
 
 1.02 
 
 1.18 
 
 1 34 
 
 1.52 
 
 2.10 
 
 2.29 
 
 2.49 
 
 3.10 
 
 3.33 
 
 4 
 
 .47 
 
 1.03 
 
 1.20 
 
 1.37 
 
 1.55 
 
 2.11 
 
 2.34 
 
 2.54 
 
 3.16 
 
 3.39 
 
 ii 6 
 
 .49 
 
 1.06 
 
 1.23 
 
 1.40 
 
 1.59 
 
 2.18 
 
 2.38 
 
 2.59 
 
 3.21 
 
 3.44 
 
 1 8 
 
 .50 
 
 1.07 
 
 1.25 
 
 1.43 
 
 2.02 
 
 2.22 
 
 2.42 
 
 3.04 
 
 3.27 
 
 3.50 
 
 •2 10 
 
 0.52 
 
 1.09 
 
 1.27 
 
 1.46 
 
 2.06 
 
 2.26 
 
 2.47 
 
 3 09 
 
 3.32 
 
 3.57 
 
 ^ 12 
 "^ 14 
 
 .53 
 
 1.11 
 
 1.30 
 
 1.49 
 
 2.09 
 
 2.30 
 
 2.52 
 
 3 14 
 
 3.38 
 
 4.03 
 
 .55 
 
 1.13 
 
 1.83 
 
 1.52 
 
 2.13 
 
 2.34 
 
 2 56 
 
 3.19 
 
 3.44 
 
 4.09 
 
 i^ 16 
 
 .56 
 
 1.15 
 
 1.35 
 
 1.55 
 
 2.16 
 
 2.38 
 
 3.01 
 
 3.25 
 
 3.50 
 
 4.15 
 
 '■^ 18 
 l:^ 20 
 
 .58 
 0.59 
 
 1.18 
 1.20 
 
 1.38 
 1.40 
 
 1.59 
 2.02 
 
 2.20 
 2.24 
 
 2.43 
 2.47 
 
 3.06 
 3.11 
 
 3.30 
 3.3d 
 
 3.56 
 4.02 
 
 4.22 
 
 4.29 
 
 22 
 —24 
 
 1.01 
 1.03 
 
 1.22 
 1.24 
 
 1.43 
 1.46 
 
 2.05 
 2.09 
 
 2 28 
 2.32 
 
 2.52 
 2.56 
 
 3.16 
 3.31 
 
 3.42 
 
 3.48 
 
 4.08 
 
 4.36 
 4.43 
 
 4.15 
 
 
 
 
 
 
 
 
 
 
 125° 
 
 120° 
 
 138 
 
Tabe VI— Continued. 
 
 LATITUDE 47^. 
 
 Declination. 
 
 .A-ZinVwdlTTTSIS- 
 
 
 
 
 
 
 
 
 
 
 
 
 63" 
 
 66° 
 
 69° 
 
 72° 
 
 75° 
 
 78° 
 
 81° 
 
 84° 
 
 87° 
 
 90° 
 
 
 H.M. 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H.M. 
 
 H.M. 
 
 H.M. 
 
 H. M. 
 
 H. M. 
 
 H, M. 
 
 +24° 
 
 2 21 
 
 2.32 
 
 2.43 
 
 2.55 
 
 3.08 
 
 3.21 
 
 3.35 
 
 3,50 
 
 4.06 
 
 4.22 
 
 22 
 
 2.29 
 
 2.40 
 
 2.51 
 
 3.04 
 
 3.17 
 
 3.30 
 
 3.45 
 
 4.00 
 
 4.15 
 
 4.31 
 
 20 
 
 2.86 
 
 2.47 
 
 2.59 
 
 3.12 
 
 3.25 
 
 3.39 
 
 3.54 
 
 4 09 
 
 4.24 
 
 4.41 
 
 18 
 
 2 43 
 
 2 55 
 
 3.07 
 
 3.20 
 
 3.33 
 
 3.48 
 
 4.02 
 
 4.17 
 
 4.83 
 
 4.49 
 
 16 
 
 2.50 
 
 3.02 
 
 3.14 
 
 3.28 
 
 3.41 
 
 3.56 
 
 4 10 
 
 4.26 
 
 4.42 
 
 4.58 
 
 14 
 
 2.57 
 
 3 09 
 
 3.22 
 
 3.35 
 
 3.49 
 
 4.04 
 
 4.18 
 
 4.84 
 
 4.50 
 
 5.06 
 
 12 
 
 3.03 
 
 3.16 
 
 3.29 
 
 3,42 
 
 3 57 
 
 4.11 
 
 4.26 
 
 4.42 
 
 4 58 
 
 5.14 
 
 10 
 
 3.10 
 
 3.22 
 
 3.36 
 
 3.49 
 
 4.04 
 
 4.19 
 
 4.34 
 
 4.50 
 
 5.06 
 
 5.22 
 
 8 
 
 3.16 
 
 3.29 
 
 3.43 
 
 3.57 
 
 4.11 
 
 4.26 
 
 4.42 
 
 4.57 
 
 5.14 
 
 5.30 
 
 6 
 
 3.22 
 
 3.35 
 
 3.49 
 
 4.04 
 
 4.]8 
 
 4.84 
 
 4.49 
 
 5.05 
 
 5 21 
 
 5.37 
 
 4 
 
 3.28 
 
 3.42 
 
 3.56 
 
 4.11 
 
 4.25 
 
 4.41 
 
 4.56 
 
 5.12 
 
 5.29 
 
 5.45 
 
 + 2 
 
 3.34 
 
 3.48 
 
 4.03 
 
 4.17 
 
 4.33 
 
 4.48 
 
 5.04 
 
 5.20 
 
 5.36 
 
 5.53 
 
 
 
 3.41 
 
 3.55 
 
 4.09 
 
 4.24 
 
 4.40 
 
 4.55 
 
 5.11 
 
 5.27 
 
 5.44 
 
 6.00 
 
 
 — 2 
 4 
 
 3.47 
 3.53 
 
 4.01 
 
 4.07 
 
 4 16 
 4.22 
 
 4.31 
 
 4.38 
 
 4.47 
 4.54 
 
 5.02 
 5.10 
 
 5.18 
 5.26 
 
 5.35 
 5.42 
 
 5.51 
 
 6.07 
 6.15 
 
 5.59 
 
 6 
 
 8 
 10 
 12 
 14 
 
 3.59 
 4.05 
 4.12 
 4.18 
 4.25 
 
 4.14 
 4.20 
 4.27 
 4.34 
 
 4.40 
 
 4.20 
 4.86 
 4.43 
 4.50 
 4.57 
 
 4.45 
 4.52 
 4.59 
 5.06 
 
 5.01 
 5.08 
 5 15 
 
 5.17 
 5. 24 
 
 5.33 
 
 5.50 
 5.57 
 6.05 
 6.13 
 6.21 
 
 6.06 
 6.14 
 6.21 
 6.29 
 6.37 
 
 6.28 
 6.30 
 6.38 
 6.46 
 6.54 
 
 5.40 
 5.48 
 5.56 
 6.04 
 
 5 82 
 5.39 
 5.47 
 
 5.22 
 5,30 
 
 5.18 
 
 16 
 
 4.31 
 
 4.47 
 
 5.04 
 
 5.21 
 
 5.38 
 
 5.55 
 
 6.12 
 
 6.29 
 
 6.45 
 
 7.02 
 
 18 
 
 4.38 
 
 4.55 
 
 5.11 
 
 5.28 
 
 5 46 
 
 6.03 
 
 6.20 
 
 6.37 
 
 6.54 
 
 7.11 
 
 20 
 
 4.45 
 
 5.02 
 
 5.19 
 
 5.36 
 
 5.54 
 
 6.11 
 
 6.29 
 
 6.46 
 
 7,03 
 
 7-19 
 
 22 
 
 4.53 
 
 5.10 
 
 5.27 
 
 5.45 
 
 6.02 
 
 6 20 
 
 6.37 
 
 6.55 
 
 7.12 
 
 7,29 
 
 —24 
 
 5.00 
 
 5.18 
 
 5. 36 
 
 5.53 
 
 6,11 
 
 6.29 
 
 6.47 
 
 7.04 
 
 7.21 
 
 7.38 
 
 
 117° 
 
 iir 
 
 111° 
 
 108° 
 
 105° 
 
 102° 
 
 99° 
 
 96° 
 
 93° 
 
 90° 
 
 139 
 
Azimuth and Hour Angle, for Latitude 
 
 
 and Declination —Table Vl. 
 
 LATITUDE 48^. 
 
 Declination. 
 
 ^ziiynTJTHis. 
 
 15° 
 
 20° 
 
 25° 
 
 30° 
 
 35° 40° 
 
 45° 
 
 50° 
 
 55° 
 
 60° 
 
 
 H.M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H.M. H.M. 
 
 H. M. 
 
 H. M. 
 
 H.M. 
 
 H. M. 
 
 +24° 
 
 .27 
 
 .37 
 
 .46 
 
 .57 
 
 1.07 1.19 
 
 1.31 
 
 1.44 
 
 1.59 
 
 2.15 
 
 22 
 
 .29 
 
 .39 
 
 .49 
 
 1.00 
 
 1.11 , 1.24 
 
 1.35 
 
 1.50 
 
 2.05 
 
 2.22 
 
 .] 20 
 
 0.30 
 
 0.41 
 
 0.52 
 
 1.03 
 
 1.15 
 
 1.28 
 
 1.41 
 
 1 56 
 
 2.12 
 
 2.29 
 
 -S 18 
 
 .32 
 
 .43 
 
 .54 
 
 1.06 
 
 1.19 
 
 1.32 
 
 1 46 
 
 2.01 
 
 2.18 
 
 2.35 
 
 B 16 
 
 .33 
 
 .45 
 
 .57 
 
 l.]0 
 
 1.23 
 
 1 37 
 
 1.51 
 
 2.07 
 
 2.23 
 
 2.42 
 
 ■? 14 
 
 .35 
 
 .47 
 
 1.00 
 
 1.13 
 
 1.26 
 
 1.41 
 
 1.56 
 
 2.12 
 
 2 29 
 
 2.48 
 
 2 12 
 
 .37 
 
 .49 
 
 1.02 
 
 1.16 
 
 1.30 
 
 1.45 
 
 2.00 
 
 2.17 
 
 2.35 
 
 2.54 
 
 ii 10 
 
 .38 
 
 0.51 
 
 1.05 
 
 1.19 
 
 1.33 
 
 1.48 
 
 2.05 
 
 2.22 
 
 2 40 
 
 3.00 
 
 •1 8 
 
 .39 
 
 .53 
 
 1.07 
 
 1.21 
 
 1.37 
 
 1.52 
 
 2.09 
 
 2 27 
 
 2.46 
 
 3.06 
 
 ^ 6 
 
 .41 
 
 .55 
 
 1.09 
 
 1.24 
 
 1.40 
 
 1.56 
 
 2.U 
 
 2 32 
 
 2.. 51 
 
 3.11 
 
 4 
 
 .42 
 
 .57 
 
 ].12 
 
 1.27 
 
 1.43 
 
 2 00 
 
 2.18 
 
 2.37 
 
 2.56 
 
 3.17 
 
 + 2 
 
 .44 
 
 .59 
 
 1.14 
 
 1.30 
 
 1.47 
 
 2.04 
 
 2.22 
 
 2.41 
 
 3.02 
 
 3.23 
 
 
 
 0.45 
 
 1.01 
 
 1.16 
 
 1.33 
 
 1.50 2.08 
 
 2.26 
 
 2 46 
 
 3.07 
 
 3.29 
 
 — 2 
 
 .46 
 
 1.02 
 
 1.19 
 
 1.36 
 
 1.53 
 
 2.12 
 
 2 31 
 
 2. .51 
 
 3.12 
 
 3.34 
 
 4 
 
 .48 
 
 1.04 
 
 121 
 
 1.39 
 
 1.57 
 
 2.15 
 
 2.35 
 
 2.56 
 
 3.17 
 
 3.40 
 
 o 6 
 
 .49 
 
 1.06 
 
 1.24 
 
 1.41 
 
 2.00 
 
 2 19 
 
 2.39 
 
 3.01 
 
 3 23 
 
 3.46 
 
 ^ 8 
 
 .51 
 
 1.08 
 
 1.26 
 
 1.44 
 
 2.03 
 
 2.23 
 
 2.44 
 
 3.05 
 
 3.28 
 
 3 52 
 
 •2 10 
 
 0.52 
 
 1.10 
 
 1.28 
 
 1.47 
 
 2.07 
 
 2 27 
 
 2.48 
 
 3 10 
 
 3.33 
 
 3.57 
 
 i u 
 
 .53 
 
 1.12 
 
 1.31 
 
 1.50 
 
 2.10 
 
 2.31 
 
 2.53 
 
 3 15 
 
 3.39 
 
 4.03 
 
 .55 
 
 1.14 
 
 1.33 
 
 1.53 
 
 2.14 
 
 2.35 
 
 2 57 
 
 3.20 
 
 3.44 
 
 4.09 
 
 ^ 16 
 
 .57 
 
 1.16 
 
 1.36 
 
 1..56 
 
 2.17 
 
 2.39 
 
 3.02 
 
 3.26 
 
 3.50 
 
 4.16 
 
 ^ 18 
 'P 20 
 
 .58 
 1.00 
 
 1.18 
 1.20 
 
 1.38 
 1.41 
 
 1.59 
 2.03 
 
 2.21 
 2.25 
 
 2.43 
 
 2.48 
 
 3.07 
 3 12 
 
 3.31 
 3.36 
 
 3.56 
 4. 02 
 
 4.22 
 
 4 29 
 
 22 
 
 1.01 
 
 1.22 
 
 1.44 
 
 2.06 
 
 2 29 
 
 2.52 
 
 3.18 
 
 3.42 
 
 4.0^ 
 
 4 35 
 
 —24 
 
 1.03 
 
 1.25 
 
 1.47 
 
 2.09 
 
 2.33 
 
 2.56 
 
 3.22 
 
 3.48 
 
 4.10 
 
 4.42 
 
 
 
 
 
 
 
 
 
 130° 
 
 125° 
 
 120° 
 
 140 
 
Table VI.— Continued. 
 
 LATITUDE 48^. 
 
 
 
 .A-Zinv^TJTHIS- 
 
 Declination. 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 63^ 
 
 66° 
 
 69° 
 
 72° : 75° 
 
 78° 
 
 81° 
 
 84° 
 
 87° 
 
 90° 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. ; H. M. 
 
 1 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H . .M . 
 
 H. .M. 
 
 +24° 
 
 2.25 
 
 2.36 
 
 2.47 
 
 2.59 3.12 
 
 3.25 
 
 3.40 
 
 3.54 
 
 4.10 
 
 4.25 
 
 22 
 
 2.32 
 
 2.44 
 
 2.55 
 
 3.08 3.20 
 
 3.34 
 
 3.48 
 
 4.03 
 
 4.19 
 
 4.35 
 
 20 
 
 2.40 
 
 2.51 
 
 3.03 ! 3.16 3.28 
 
 3.43 
 
 3.57 
 
 4.12 
 
 4.28 
 
 4.43 
 
 18 
 
 2.46 
 
 2.58 
 
 3.10 
 
 3.23 3.36 
 
 3.51 
 
 4.05 
 
 4.20 
 
 4.36 
 
 4.52 
 
 16 
 
 2.53 ; 3.05 
 
 3.18 
 
 3.31 3.44 
 
 3.59 
 
 4.13 
 
 4.29 
 
 4.44 
 
 5.00 
 
 14 
 
 3.00 
 
 3.12 
 
 3.25 
 
 3.38 3.51 
 
 4.06 
 
 4.21 
 
 4.36 
 
 4.52 
 
 5.08 
 
 12 
 
 3.03 
 
 3 18 
 
 3.32 
 
 3.45 j 3.59 
 
 4.14 
 
 4.29 
 
 4.44 
 
 5 00 
 
 5.16 
 
 10 
 
 3.12 
 
 3.25 
 
 3.38 
 
 3.52 : 4.06 
 
 4.21 
 
 4.36 
 
 4.52 
 
 5.07 
 
 5.23 
 
 8 
 
 3.18 
 
 3.31 
 
 3.45 
 
 3.59 ; 4.13 
 
 4.28 
 
 4.43 
 
 4. 59 
 
 5.15 
 
 5.31 
 
 6 
 
 3.24 
 
 3.38 
 
 3.51 
 
 4.06 1 4.20 
 
 4.35 
 
 4.51 
 
 5.06 
 
 5 22 
 
 5.38 
 
 4 
 
 3.30 
 
 3.44 
 
 3.58 
 
 4.12 
 
 4.27 
 
 4.42 
 
 4.58 
 
 5.14 
 
 5.29 
 
 5.46 
 
 + 2 
 
 3.. 36 
 
 3.50 
 
 4.04 
 
 4.13 
 
 4.33 
 
 4.49 
 
 5.05 
 
 5.21 
 
 5.37 
 
 5.53 
 
 
 
 3.42 
 
 3.56 
 
 4.11 
 
 4.26 
 
 4.40 
 
 4.56 
 
 5.12 
 
 5.28 
 
 5.44 
 
 6.00 
 
 
 — 2 
 4 
 
 3.48 
 3.54 
 
 4.02 
 4.09 
 
 4 17 
 4. 24 
 
 4.32 
 4.39 
 
 4.47 
 4.54 
 
 5.03 
 5.10 
 
 5.19 
 5.26 
 
 5.35 
 5.42 
 
 5.51 
 
 6.07 
 6.14 
 
 5.58 
 
 6 
 8 
 10 
 12 
 14 
 16 
 
 4.00 
 4.06 
 4.12 
 4.19 
 4.25 
 4.31 
 
 4.15 
 4.2] 
 4.28 
 4.34 
 4.41 
 
 4.30 
 4.37 
 4.43 
 4.50 
 
 4.45 
 4.52 
 4.59 
 
 5.01 
 
 5.08 
 5.15 
 
 5.17 
 5.24 
 
 5.33 
 
 5.49 
 5.57 
 6.04 
 6.11 
 6.19 
 6.27 
 
 6.06 
 6.13 
 6.20 
 6.28 
 6.36 
 6.44 
 
 6.22 
 6.29 
 6.37 
 6.44 
 6.52 
 7.00 
 
 5.40 
 5.48 
 5.55 
 6.03 
 6.10 
 
 5.31 
 5.39 
 5.46 
 5.54 
 
 5.06 
 5.13 
 5.20 
 
 5.22 
 5.29 
 5.36 
 
 4.57 
 5.04 
 
 4.47 
 
 
 
 
 
 
 
 
 
 
 
 
 18 
 
 4.38 
 
 4.54 
 
 5.11 
 
 5.28 
 
 5.44 
 
 6.01 
 
 6.18 
 
 6.35 
 
 6.52 
 
 7.08 
 
 20 
 
 4.45 
 
 5.02 
 
 5.18 
 
 5.35 
 
 5.52 
 
 6.10 
 
 6.27 
 
 6.44 
 
 7.00 
 
 7 17 
 
 22 
 
 4.. 52 
 
 5.09 
 
 5.26 
 
 5.43 
 
 6.00 
 
 6 18 
 
 6.35 
 
 6.52 
 
 7.09 
 
 7.25 
 
 —24 
 
 4.59 
 
 5.17 
 
 5.34 
 
 5.52 
 
 6.09 
 
 6.27 
 
 6.44 
 
 7.01 
 
 7.18 
 
 7.35 
 
 
 117° 
 
 114° 
 
 111° 
 
 108° 
 
 105° 
 
 102° 
 
 99° 
 
 96° 
 
 93° 
 
 90° 
 
 141 
 
TABLE VII. 
 
 A table of differences of local and standard time, on the 
 great lakes, for reducing standard tinie to the mean local time 
 of the places mentioned. The sign prefixed, indicates the 
 manner of applying the correction, to the standard time, 
 the + sign meaning addition to, and the — sign meaning 
 subtraction from standard time to get local mean time. 
 
 LAKE ONTARIO. 
 
 75=^ Long. 
 
 EASTERN STANDARD. 
 
 M. S. 
 
 Sackett's Harbor — 4.36 
 
 Stony Point — 5.12 
 
 Fair Haven. — 6.52 
 
 Big Sodus — 7.56 
 
 Genesee —10.24 
 
 Oak Orchard —12 .48 
 
 30 Mile Point —13.56 
 
 Fort Niagara —16.16 
 
 Kingston.. — 5.54 
 
 Desoronto , — 8.08 
 
 BellviUe — 9.32 
 
 Weller's Bay —10.44 
 
 Coburg —12.56 
 
 Port Hope —13 . 20 
 
 Frenchman's Bay — 16 . 08 
 
 Toronto —17.56 
 
 Fort Dalhousie —17.24 
 
 LAKE ERIE. 
 
 75° Long. 
 
 EASTERN STANDARD. 
 
 M. S. 
 
 Buffalo —15 .36 
 
 Dunkirk —17.24 
 
 Erie —20.16 
 
 Ashtabula —23.12 
 
 Fairport —25.08 
 
 Cleveland —26.48 
 
 Black River —28.44 
 
 Cedar Point —30.48 
 
 Sandusky Bay —30.52 
 
 Green Island —31.28 
 
 Turtle Island —33.32 
 
 Detroit River (mouth) —33 . 32 
 
 Detroit City —27.52 
 
 Fort Colbourne — 17.16 
 
 Fort Maitland —18.40 
 
 Long Point (east end) —20 36 
 
 Port Burwell —23.36 
 
 Fort Stanley. —24.52 
 
 Pelee Spit —30.32 
 
 Middle Island —30.40 
 
 Kingsville —30.56 
 
 Amherst Bay — 32.52 
 
 142 
 
TABLE VII.- 
 
 -Continued. 
 
 
 LAKE HURON. 
 
 90° Long. 
 
 CENTRAL STANDARD. 
 
 M. S. 
 
 BeUe Isle (Saitit Clair) +28. 12 
 
 Fort Gratiot +30.20 
 
 Grand Traverse 
 
 .. +17.52 
 
 South Manitou 
 
 .. +15.40 
 
 Manistee 
 
 White River 
 
 .. +14.36 
 . +14.20 
 
 Aluskegon 
 
 Grand Haven. 
 
 .. +14.40 
 .. +15.00 
 
 Fort Sanilac +29 .52 
 
 Sand Beach +29.40 
 
 St. Joseph 
 
 Michigan City 
 
 Chicago 
 
 .. +14.04 
 .. +12.24 
 .. + 9.32 
 
 Point of Barques +29.00 
 
 Charity Island +26 . 16 
 
 Saginaw River (mouth) +24.36 
 
 Sturgeon Point +26.56 
 
 Thunder Bay Island +27 .24 
 
 Presque Isle +26.08 
 
 Kenosha 
 
 .. + 8.44 
 
 Racine.. 
 
 .. +8.52 
 
 Milwaukee 
 
 Sheboygan. 
 
 .. + 8.32 
 + 9.12 
 
 Sturgeon Bay Canal 
 
 Pilot Island 
 
 .. + 9.44 
 .. +12.20 
 
 Spectacle Reef +23.28 
 
 Cheboygan +22.20 
 
 Detour +24.24 
 
 Escanaba 
 
 + 11 52 
 
 Chamber's Island. 
 
 . +10 32 
 
 CANADIAN SIDE. 
 
 75° Long. 
 
 M. S. 
 
 Goderich —26.12 
 
 Menominee 
 
 .. + 9.40 
 
 
 OR. 
 
 mD. 
 
 M. S. 
 
 .. +22.32 
 
 .. +20.12 
 
 LAKE SUPER 
 
 00^ Long. 
 CENTRAL stand; 
 
 St. Mary's Falls 
 
 White Fish Point 
 
 South Hampton — 25.32 
 
 Michael's Point .'. —27 . 44 
 
 Great Duck Island —31.44 
 
 Owen Sound —23.44 
 
 CoUingwood — 20 . 08 
 
 Grand Island 
 
 .. +13.16 
 
 Whiskey Island —19 . 32 
 
 FrenchRiver —23.40 
 
 Manitowaning — 27.12 
 
 Thessdon River —23.40 
 
 Marquette 
 
 .. +10.28 
 
 Stanard's Rock 
 
 Copper Harbor 
 
 Ontonagon 
 
 .. +11.08 
 
 .. + 8.32 
 
 . +2.40 
 
 
 Devil's Island 
 
 St. Louis River ... . 
 
 .. —3.12 
 
 .. — 8.04 
 
 LAKE MICHIGAN. 
 
 90® Long. 
 
 CENTRAL STANDARD. 
 
 M. S. 
 
 McGulpin's Point..,. +20.56 
 
 Skilligallee +19.20 
 
 Duluth 
 
 ... — 8.20 
 
 Grand Marias 
 
 .. — 1.20 
 
 Isle Royal 
 
 CANADA. 
 Agate Island 
 
 .. 4- 5.00 
 
 M. S. 
 
 .. +15.52 
 
 Lamb Island 
 
 .. + 6.32 
 
 Beaver Island + 17 . 44 
 
 Little Traverse +20 08 
 
 Thunder Cape 
 
 .. + 3.20 
 
 Port Arthur 
 
 .. + 3.08 
 
 South Fox Island +16.40 
 
 Victoria Island 
 
 .. + 2.36 
 
 143 
 
Table 
 
 Of 
 
 Chords to 
 
 Radius Unity, for 1 
 
 
 
 
 Protracting- 
 
 -VII 
 
 ■ 
 
 
 
 
 
 
 
 1° 
 
 2° 
 
 3° 
 
 4° 
 
 5° 
 
 6° 
 
 7° 
 
 8° 
 
 9° 
 
 
 0' 
 
 .jooo 
 
 .0175 
 
 .0349 
 
 .0.524 
 
 .0698 
 
 .0872 
 
 .0147 
 
 .0 
 
 .1395 
 
 .1.569 
 
 0' 
 
 10 
 
 .0027 
 
 .0204 
 
 .0378 
 
 .0553 
 
 .0727 
 
 .0901 
 
 .1076 
 
 .1250 
 
 .1424 
 
 .1598 
 
 10 
 
 20 
 
 .0058 
 
 .0233 
 
 .0407 
 
 .0582 
 
 .0756 
 
 .0931 
 
 .1105 
 
 .1279 
 
 .14.53 
 
 .1627 
 
 20 
 
 30 
 
 .0087 
 
 .0262 
 
 .0436 
 
 .0611 
 
 .0785 
 
 .0960 
 
 .1134 
 
 .1308 
 
 .1482 
 
 .16.56 
 
 30 
 
 40 
 
 .0116 
 
 .0291 
 
 .0465 
 
 .0640 
 
 .0814 
 
 .0989 
 
 .1163 
 
 .1337 
 
 .1511 
 
 .1685 
 
 40 
 
 50 
 
 .0145 
 
 .0320 
 
 .0494 
 
 .0669 
 
 .0843 
 
 .1018 
 
 .1192 
 
 .1366 
 
 .1540 
 
 .1714 
 
 50 
 
 
 10° 
 
 11° 
 
 12° 
 
 13° 
 
 14° 
 
 15° 
 
 16° 
 
 17° 
 
 18° 
 
 1S° 
 
 
 0' 
 
 .1743 
 
 .1917 
 
 .2091 
 
 .2264 
 
 .2437 
 
 .2611 
 
 .2783 
 
 .29.56 
 
 .3129 
 
 .3301 
 
 0' 
 
 10 
 
 .1772 
 
 .1946 
 
 .2119 
 
 .2293 
 
 .2466 
 
 .2639 
 
 .2812 
 
 .2985 
 
 .3157 
 
 . 3330 
 
 10 
 
 20 
 
 .1801 
 
 .1975 
 
 .2148 
 
 .2322 
 
 .2495 
 
 .2668 
 
 .2841 
 
 .3014 
 
 .3186 
 
 .3358 
 
 20 
 
 30 
 
 .1830 
 
 .2004 
 
 .2177 
 
 .2351 
 
 .2524 
 
 .2697 
 
 .2870 
 
 .3042 
 
 .3215 
 
 .3387 
 
 30 
 
 40 
 
 .1859 
 
 .2033 
 
 .2206 
 
 .2380 
 
 .2.5.53 
 
 .2726 
 
 .2899 
 
 .3071 
 
 .3244 
 
 .3416 
 
 40 
 
 60 
 
 .1888 
 
 .2062 
 
 .2235 
 
 .2409 
 
 .2582 
 
 .2755 
 
 .2927 
 
 .3100 
 
 .3272 
 
 .3444 
 
 50 
 
 
 20° 
 
 21° 
 
 22° 
 
 23° 
 
 24° 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 
 0' 
 
 .3473 
 
 .3645 
 
 .3816 
 
 .3987 
 
 .4158 
 
 .4329 
 
 .4499 
 
 .4669 
 
 .4838 
 
 ..5008 
 
 0' 
 
 10 
 
 .3502 
 
 .3673 
 
 .3845 
 
 .4016 
 
 .4187 
 
 .4357 
 
 .4527 
 
 .4697 
 
 .4867 
 
 ..5036 
 
 10 
 
 20 
 
 .3.530 
 
 .3702 
 
 .3873 
 
 .4044 
 
 .4215 
 
 .4386 
 
 .5556 
 
 .4725 
 
 .4895 
 
 .5064 
 
 20 
 
 30 
 
 .3559 
 
 .3730 
 
 .3902 
 
 .4073 
 
 .4244 
 
 .4414 
 
 .4584 
 
 .4754 
 
 .4923 
 
 .5092 
 
 30 
 
 40 
 
 .3587 
 
 .3759 
 
 .3930 
 
 .4101 
 
 .4272 
 
 .4442 
 
 .4612 
 
 .4782 
 
 .4951 
 
 .5120 
 
 40 
 
 50 
 
 .3616 
 
 .3788 
 
 .3959 
 
 .4130 
 
 .4300 
 
 .4471 
 
 .4641 
 
 .4810 
 
 .4979 
 
 .5148 
 
 50 
 
 
 30° 
 
 31° 
 
 32° 
 
 33° 
 
 34° 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 
 0' 
 
 .5176 
 
 .5345 
 
 .5513 
 
 .5680 
 
 .5847 
 
 .6014 
 
 .6180 
 
 .6346 
 
 .6511 
 
 .6676 
 
 0' 
 
 10 
 
 .5204 
 
 .5373 
 
 .5541 
 
 .5708 
 
 .5875 
 
 .6042 
 
 .6208 
 
 .6374 
 
 .6539 
 
 .6704 
 
 10 
 
 20 
 
 .5233 
 
 .5401 
 
 .5569 
 
 .5736 
 
 ..5903 
 
 .6070 
 
 .6236 
 
 .6401 
 
 .6566 
 
 6731 
 
 20 
 
 30 
 
 .5261 
 
 .5429 
 
 .5597 
 
 .5764 
 
 .5931 
 
 .6097 
 
 .6263 
 
 .6429 
 
 .6594 
 
 .6778 
 
 30 
 
 40 
 
 ..5289 
 
 .5457 
 
 ..5625 
 
 .5792 
 
 .5959 
 
 .6125 
 
 .6291 
 
 .6456 
 
 .6621 
 
 .6786 
 
 40 
 
 50 
 
 .5317 
 
 .5485 
 
 .5652 
 
 .5820 
 
 .5986 
 
 .6153 
 
 .6318 
 
 .6484 
 
 .6649 
 
 .6813 
 
 50 
 
 
 40° 
 
 41° 
 
 42° 
 
 43° 
 
 44° 
 
 45° 
 
 46° 
 
 47° 
 
 48° 
 
 49° 
 
 
 0' 
 
 .6840 
 
 .7004 
 
 .7167 
 
 .7330 
 
 .7492 
 
 .7654 
 
 .7815 
 
 .7975 
 
 .8135 
 
 .8294 
 
 0' 
 
 10 
 
 .6868 
 
 .7031 
 
 . 7195 
 
 . 7357 
 
 .7519 
 
 .7681 
 
 .7841 
 
 .8002 
 
 .8161 
 
 .8320 
 
 10 
 
 20 
 
 .6895 
 
 .7059 
 
 .7222 
 
 .7384 
 
 .7.546 
 
 .7707 
 
 .7868 
 
 .8028 
 
 .8188 
 
 .8347 
 
 20 
 
 30 
 
 .6922 
 
 .7086 
 
 .7249 
 
 .74J1 
 
 .7573 
 
 .7734 
 
 .7895 
 
 .8055 
 
 .8214 
 
 .8373 
 
 30 
 
 40 
 
 .6950 
 
 .7113 
 
 .7276 
 
 .7438 
 
 7600 
 
 7761 
 
 .7922 
 
 .8082 
 
 .8241 
 
 .840') 
 
 40 
 
 50 
 
 .6977 
 
 .7140 
 
 .7303 
 
 .7465 .7627 
 
 .7788 
 
 .7948 
 
 .8108 
 
 .8267 
 
 .8426 
 
 50 
 
 144 
 
Table VIII.— Continued. 
 
 
 50° 
 
 51° 
 
 52° 
 
 53° 
 
 54° 
 
 55° 
 
 56° 
 
 57° 
 
 58° 
 
 59° 
 
 
 0' 
 
 .8452 
 
 .8610 
 
 .8767 
 
 .8924 
 
 .9080 
 
 .9235 
 
 .9389 
 
 .9543 
 
 9606 
 
 .9848 
 
 0' 
 
 10 
 
 .8479 
 
 .8606 
 
 .8794 
 
 .89r.O 
 
 .9106 
 
 .9261 
 
 .9415 
 
 .9569 
 
 .9722 
 
 .9874 
 
 10 
 
 20 
 
 .•8505 
 
 .8663 
 
 .8820 
 
 .8976 
 
 .9132 
 
 .9287 
 
 .9441 
 
 .9594 
 
 .9747 
 
 .9899 
 
 20 
 
 30 
 
 .8531 
 
 .'8689 
 
 .8846 
 
 .9002 
 
 .9157 
 
 .9312 
 
 .9466 
 
 .9620 
 
 .9772 
 
 .9924 
 
 30 
 
 40 
 
 .8558 
 
 .8715 
 
 .8872 
 
 .9028 
 
 .9183 
 
 .9338 
 
 .9192 
 
 .9645 
 
 .<-798 
 
 .9950 
 
 40 
 
 50 
 
 .8584 
 
 .8741 
 
 .8898 
 
 .9054 
 
 .9209 
 
 .9364 
 
 .9518 
 
 .9671 
 
 .9823 
 
 .9975 
 
 50 
 
 
 60° 
 
 61° 
 
 62° 
 
 63° 
 
 64° 
 
 65° 
 
 66° 
 
 67° 
 
 68° 
 
 6S° 
 
 
 0' 
 
 1.0000 
 
 1.0157 
 
 1.0,301 
 
 1.0450 
 
 1.0598 
 
 1.0746 
 
 1.0893 
 
 1.1C39 
 
 1.1184 
 
 1.1328 
 
 0' 
 
 10 
 
 1.0025 
 
 1.0176 
 
 1.032611.04751.06231.0771 
 
 1.09171.1063 
 
 1 . 1208 
 
 i.l352 
 
 10 
 
 20 
 
 1.0050 
 
 1.0201 
 
 1.03.51! 1.0500 1.06481.0795 
 
 1 094211.1087 
 
 1 1232 
 
 1.1376 
 
 20 
 
 30 
 
 1 0075 
 
 1.0226 
 
 1 .0375 '1 .0.524 1 .0672 1 .0820 
 
 1.09661.1111 
 
 1.1256 
 
 1.1400 
 
 30 
 
 40 
 
 i.oini 
 
 1.0251 
 
 1.0400 1.0549 1.06971 0844 
 
 1.09901.1136 
 
 1.1280 
 
 1.1424 
 
 40 
 
 60 
 
 1.0126 
 
 1.0.!76 
 
 1.0425 
 
 1.05741.07211.0868 
 
 1.1014 
 
 1.1160 
 
 1.1304 
 
 L.1448 
 
 50 
 
 
 70° 
 
 71° 
 
 72° 
 
 73° 74° 
 
 75° 
 
 76° 
 
 77° 
 
 78^ 
 
 79° 
 
 
 0' 
 
 1.1472 
 
 1.1614 
 
 1.1756 
 
 l.i896'l. 20361. 2175 
 
 1.2313 
 
 1.2450 
 
 1.2.586 
 
 1.2722 
 
 0' 
 
 10 
 
 1.1495 
 
 1,1638 
 
 1.1779 
 
 1.19201.20601.2198 
 
 1.23361 2473 
 
 1.2609 
 
 1.2744 
 
 10 
 
 20 
 
 1.1519 
 
 1.1661 
 
 1.1803 
 
 1.194311.20831.2221 
 
 1.23591 2496 
 
 1 2632 
 
 1.2766 
 
 20 
 
 30 
 
 1.1543 
 
 1.1685 
 
 1.1826 
 
 1.1966;i.2106 
 
 1.2244 
 
 1.2382 
 
 1.2518 
 
 1.2654 
 
 1.2789 
 
 30 
 
 40 
 
 1.1.567 
 
 1.1709 
 
 1.1850 
 
 1.19901.2129 
 
 1.2267 
 
 1.2405 
 
 1.2541 
 
 1.2677 
 
 1.2811 
 
 40 
 
 50 
 
 1.1590 
 
 1-1732 
 
 1.1873 
 
 1.20131.2152 
 
 1.2290 
 
 1.2428 
 
 1.2564 
 
 1.2699 
 
 1.2833 
 
 50 
 
 
 80° 
 
 81° 
 
 82° 
 
 83° 
 
 84° 
 
 85° 
 
 86° 
 
 87° 
 
 88° 
 
 89° 
 
 
 0' 
 
 1.2856 
 
 1.2989 
 
 1.3121 
 
 1.3252 
 
 1.3383 
 
 1.3512 
 
 1 3640 
 
 1.3767 
 
 1.3893 
 
 1.4018 
 
 0' 
 
 10 
 
 1.2878 
 
 1.3011 
 
 1.3143 
 
 1. 3274 :x. 3404 
 
 1.3533 
 
 1 3661 
 
 1.3788 
 
 1.3914 
 
 1.4039 
 
 10 
 
 20 
 
 1.2900 
 
 1.3033 
 
 1.3165 
 
 1. 3296 !x. 3426 
 
 1.3555 
 
 1.3682 
 
 1.3809 
 
 1.3935 
 
 1.4060 
 
 20 
 
 30 
 
 1.2922 
 
 1.3055 
 
 1.3187 
 
 1. 3318 il. 3447 
 
 1.3.576 
 
 1.3704 
 
 1 3830 
 
 1.3956 
 
 1.4080 
 
 30 
 
 40 
 
 1.2945 
 
 1.3077 
 
 1.3209 
 
 1 33391.3469 
 
 1.3597 
 
 1.3725 
 
 1.3851 
 
 1.3977 
 
 1.4101 
 
 40 
 
 50 
 
 1.2967 
 
 1.3099 
 
 1.3231 
 
 1.33611.3490 
 
 1.3619 
 
 1.3746 
 
 1.3872 
 
 1.3997 
 
 1.4122 
 
 50 
 
 
 90° 
 
 91° 
 
 92° 
 
 9o° 
 
 1.4508 
 
 94° 
 
 95° 
 
 96° 
 
 97° 
 
 98° 
 
 990 
 
 
 0' 
 
 1.4142 
 
 • 
 1.4265 
 
 1.4386 
 
 1.4627 
 
 1.4745^1.4863 
 
 1.4979,1.5094 
 
 1.5208 
 
 0' 
 
 10 
 
 1.4162 
 
 1 .4285 1. 4406 1 .4528 1 .464.7 1 .4765 1 .4883 1 .4998 
 
 1.51131.5227 
 
 10 
 
 20 
 
 1.41831.43051.44261.45481.46671.47851.49021.5017 
 
 1.51321.5246 
 
 20 
 
 30 
 
 1 .4203 1 .4325 1 .4446 1 .4568 1 .46861 .4804 1 .4921 1 .5037 
 
 1.. 51.51 1.5265 
 
 30 
 
 40 
 
 1 .4221 1 .4346 1 .4467 1 .4588 1 .4706 1 .4824 1 .4940 1 .5056 
 
 1.51701.5283 
 
 40 
 
 50 
 
 1 .4245 1 .4366 1 .4487 1 .4608 1 .4726 1 .4843 1 .4960 1 .5074 1 .5189 1 .5302 
 
 ! i ! ' 1 1 1 1 1 
 
 50 
 
 145 
 

 
 
 
 TABLE 
 
 IX 
 
 ■ 
 
 
 
 
 
 MERIDIANAL PARTS. 
 
 
 
 M. 
 
 0° 
 
 P 
 
 2° 
 
 2P 
 
 40 
 
 5° 
 
 Qo 
 
 70 
 
 8° 
 
 9° 
 
 10° 
 
 IP 
 
 12° 
 
 M. 
 
 
 
 
 
 60 
 
 120 
 
 180 
 
 240 
 
 300 
 
 361 
 
 421 
 
 482 
 
 542 
 
 603 
 
 664 
 
 725 
 
 
 
 2 
 
 2 
 
 62 
 
 122 
 
 182 
 
 242 
 
 302 
 
 363 
 
 423 
 
 4S4 
 
 544 
 
 (•(5 
 
 666 
 
 727 
 
 2 
 
 4 
 
 4 
 
 64 
 
 124 
 
 184 
 
 244 
 
 304 
 
 365 
 
 425 
 
 486 
 
 546 
 
 607 
 
 66^ 
 
 729 
 
 4 
 
 6 
 
 6 
 
 66 
 
 126 
 
 1% 
 
 246 
 
 306 
 
 367 
 
 427 
 
 488 
 
 548 
 
 609 
 
 670 
 
 731 
 
 6 
 
 8 
 
 8 
 
 68 
 
 128 
 
 188 
 
 248 
 
 308 
 
 369 
 
 4^9 
 
 490 
 
 550 
 
 611 
 
 672 
 
 734 
 
 8 
 
 10 
 
 10 
 
 70 
 
 130 
 
 190 
 
 250 
 
 310 
 
 371 
 
 431 
 
 492 
 
 552 
 
 613 
 
 674 
 
 736 
 
 10 
 
 12 12 
 
 72 
 
 132 
 
 192 
 
 252 
 
 312 
 
 373 
 
 4M3 
 
 4 4 
 
 554 
 
 615 
 
 676 
 
 7:^8 
 
 12 
 
 14 
 
 14 
 
 74 
 
 134 
 
 194 
 
 254 
 
 3] 4 
 
 375 
 
 435 
 
 496 
 
 556 
 
 617 
 
 678 
 
 740 
 
 14 
 
 16 
 
 16 
 
 76 
 
 136 
 
 196 
 
 256 
 
 316 
 
 377 
 
 437 
 
 498 
 
 558 
 
 619 
 
 6S0 
 
 742 
 
 16 
 
 18 
 
 18 
 
 78 
 
 138 
 
 198 
 
 258 
 
 318 
 
 379 
 
 439 
 
 500 
 
 560 
 
 621 
 
 682 
 
 744 
 
 18 
 
 20 
 
 20 
 
 80 
 
 140 
 
 200 
 
 260 
 
 320 
 
 381 
 
 441 
 
 502 
 
 562 
 
 623 
 
 684 
 
 746 
 
 20 
 
 22 
 
 22 
 
 82 
 
 142 
 
 202 
 
 262 
 
 322 
 
 383 
 
 443 
 
 504 
 
 565 
 
 625 
 
 687 
 
 748 
 
 22 
 
 24 
 
 24 
 
 84 144 
 
 204 
 
 264 
 
 324 
 
 385 
 
 445 
 
 506 
 
 567 
 
 627 
 
 689 
 
 750 
 
 24 
 
 26 
 
 26 
 
 86 
 
 146 
 
 206 
 
 266 
 
 326 
 
 387 
 
 447 
 
 508 
 
 569 
 
 629 
 
 691 
 
 752 
 
 26 
 
 28 
 
 28 
 
 88 
 
 148 
 
 208 
 
 268 
 
 328 
 
 389 
 
 449 
 
 510 
 
 571 
 
 632 
 
 693 
 
 754 
 
 28 
 
 30 
 
 30 
 
 90 
 
 150 
 
 210 
 
 270 
 
 331 
 
 391 
 
 451 
 
 512 
 
 573 
 
 634 
 
 695 
 
 756 
 
 30 
 
 32 
 
 32 
 
 92 
 
 152 
 
 212 
 
 272 
 
 333 
 
 393 
 
 4.^3 
 
 514 
 
 575 
 
 636 
 
 697 
 
 758 
 
 32 
 
 34 
 
 34 
 
 94 
 
 154 
 
 214 
 
 274 
 
 335 
 
 395 
 
 455 
 
 516 
 
 577 
 
 638 
 
 699 
 
 760 
 
 34 
 
 36 
 
 36 
 
 96 
 
 156 
 
 216 
 
 276 
 
 3a7 
 
 397 
 
 457 
 
 518 
 
 579 
 
 640 
 
 701 
 
 762 
 
 36 
 
 38 
 
 38 
 
 98 
 
 158 
 
 218 
 
 278 
 
 339 
 
 399 
 
 459 
 
 520 
 
 581 
 
 642 
 
 703 
 
 764 
 
 38 
 
 40 
 
 40 
 
 100 
 
 160 
 
 220 
 
 280 
 
 341 
 
 401 
 
 461 
 
 522 
 
 583 
 
 644 
 
 705 
 
 766 
 
 40 
 
 42 
 
 42 
 
 102 
 
 162 
 
 222 
 
 2*^2 
 
 343 
 
 403 
 
 4r.3 
 
 524 
 
 685 
 
 646 
 
 707 
 
 768 
 
 42 
 
 44 
 
 44 
 
 104 
 
 164 
 
 224 
 
 284 
 
 345 
 
 405 
 
 465 
 
 526 
 
 587 
 
 648 
 
 709 
 
 770 
 
 44 
 
 46 
 
 46 
 
 106 
 
 166 
 
 226 
 
 286 
 
 347 
 
 407 
 
 467 
 
 528 
 
 589 
 
 650 
 
 711 
 
 772 
 
 46 
 
 48 
 
 48 
 
 108 
 
 168 
 
 228 
 
 288 
 
 349 
 
 409 
 
 469 
 
 530 
 
 591 
 
 652 
 
 713 
 
 774 
 
 48 
 
 50 
 
 50 
 
 no 
 
 170 
 
 230 
 
 290 
 
 351 
 
 411 
 
 471 
 
 532 
 
 593 
 
 654 
 
 715 
 
 777 
 
 50 
 
 52 
 
 52 
 
 112 
 
 172 
 
 232 
 
 292 
 
 353 
 
 413 
 
 473 
 
 534 
 
 595 
 
 656 
 
 717 
 
 779 
 
 62 
 
 54 
 
 54 
 
 114 
 
 174 
 
 234 
 
 294 
 
 355 
 
 415 
 
 476 
 
 536 
 
 597 
 
 658 
 
 719 
 
 781 
 
 64 
 
 56 
 
 56 
 
 116 
 
 176 
 
 236 
 
 296 
 
 357 
 
 417 
 
 478 
 
 538 
 
 599 
 
 660 
 
 721 
 
 783 
 
 66 
 
 68 
 
 58 
 
 118 
 
 178 
 
 238 
 
 298 
 
 359 
 
 419 
 
 480 
 
 640 
 
 601 
 
 662 
 
 723 
 
 785 
 
 68 
 
 M. 
 
 0^ 
 
 1^ 
 
 2° 
 
 3° 
 
 4^ 
 
 5° 
 
 6° 
 
 70 
 
 9P 
 
 9° 
 
 10° 
 
 IP 
 
 12° 
 
 M. 
 
 146 
 
TABLE IX.-Continued. 
 
 M. 
 
 13° 
 
 14° 
 
 15° 
 
 16° 
 
 17° 
 
 18° 
 
 19° 
 
 20° 
 
 21° 
 
 22° 
 
 M. 
 
 
 
 2 
 4 
 6 
 
 8 
 
 10 
 12 
 14 
 
 16 
 
 18 
 
 20 
 22 
 24 
 26 
 
 28 
 
 30 
 32 
 34 
 
 36 
 
 38 
 
 40 
 42 
 44 
 46 
 48 
 
 50 
 52 
 
 54 
 56 
 
 58 
 
 787 
 789 
 791 
 793 
 795 
 
 797 
 799 
 801 
 803 
 805 
 
 807 
 809 
 811 
 813 
 816 
 
 818 
 820 
 822 
 824 
 826 
 
 828 
 830 
 832 
 834 
 836 
 
 838 
 840 
 842 
 844 
 846 
 
 848 
 851 
 853 
 855 
 857 
 
 859 
 861 
 863 
 865 
 867 
 
 869 
 871 
 873 
 875 
 877 
 
 879 
 882 
 884 
 886 
 888 
 
 890 
 892 
 894 
 896 
 898 
 
 900 
 902 
 904 
 906 
 908 
 
 910 
 913 
 915 
 917 
 919 
 
 921 
 923 
 925 
 927 
 929 
 
 931 
 933 
 935 
 937 
 939 
 
 942 
 944 
 946 
 948 
 950 
 
 952 
 954 
 956 
 
 958 
 960 
 
 962 
 964 
 966 
 969 
 971 
 
 973 
 975 
 
 977 
 979 
 981 
 
 983 
 985 
 987 
 989 
 991 
 
 994 
 
 996 
 
 998 
 
 1000 
 
 1002 
 
 1004 
 1006 
 1008 
 1010 
 1012 
 
 1014 
 1016 
 1019 
 1021 
 1023 
 
 1025 
 1027 
 1029 
 1031 
 1033 
 
 1035 
 1037 
 1039 
 1042 
 1044 
 
 1046 
 1048 
 1050 
 1052 
 1054 
 
 1056 
 1058 
 1060 
 1063 
 1065 
 
 1067 
 1069 
 1071 
 1073 
 1075 
 
 1077 
 1079 
 1081 
 
 1084 
 1086 
 
 1088 
 1090 
 1092 
 1094 
 1096 
 
 1098 
 1100 
 1102 
 1105 
 1107 
 
 1109 
 
 nil 
 
 1113 
 1115 
 1117 
 
 1119 
 1121 
 1123 
 1126 
 1128 
 
 1130 
 1132 
 1134 
 1136 
 1138 
 
 1140 
 1142 
 1145 
 1147 
 1149 
 
 1151 
 1153 
 1155 
 1157 
 1159 
 
 1161 
 1161 
 1166 
 1168 
 1170 
 
 1172 
 1174 
 1176 
 1178 
 1181 
 
 1183 
 1185 
 1187 
 1189 
 1191 
 
 1193 
 1195 
 1198 
 1200 
 1202 
 
 1204 
 1206 
 1208 
 1210 
 1212 
 
 1215 
 1217 
 1219 
 1221 
 1223 
 
 1225 
 
 1227 
 1229 
 1232 
 1234 
 
 1236 
 1238 
 1240 
 1242 
 1245 
 
 1247 
 1249 
 1251 
 1253 
 1255 
 
 1257 
 1259 
 1261 
 1264 
 1266 
 
 1268 
 1270 
 1272 
 1274 
 1276 
 
 1278 
 1281 
 1283 
 
 1285 
 1287 
 
 1289 
 1291 
 1293 
 1296 
 1298 
 
 1300 
 1302 
 1304 
 1306 
 130S 
 
 1311 
 1313 
 1315 
 1317 
 1319 
 
 1321 
 1324 
 1326 
 1328 
 1330 
 
 1332 
 1334 
 1336 
 1339 
 1341 
 
 1343 
 1345 
 1347 
 1349 
 1352 
 
 1354 
 1356 
 1358 
 1360 
 1362 
 
 1364 
 1367 
 1369 
 1371 
 1373 
 
 1375 
 1377 
 1380 
 1382 
 1384 
 
 1386 
 1388 
 1390 
 1393 
 1395 
 
 1397 
 1399 
 1401 
 1403 
 1406 
 
 1408 
 1410 
 1412 
 1414 
 1416 
 
 
 2 
 4 
 6 
 
 8 
 
 10 
 12 
 14 
 16 
 18 
 
 20 
 
 22 
 24 
 26 
 
 28 
 
 30 
 32 
 34 
 36 
 38 
 
 40 
 42 
 44 
 46 
 48 
 
 50 
 52 
 
 54 
 56 
 58 
 
 M. 
 
 13° 
 
 14° 
 
 15° 
 
 16° 
 
 17° 
 
 18° 
 
 19° 
 
 20° 
 
 21° 
 
 22° 
 
 M. 
 
 147 
 
TABLE IX.-Continued. 
 
 M. 
 
 23° 
 
 21° 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 30° 
 
 31° 
 
 32° 
 
 M. 
 
 
 2 
 4 
 
 6 
 8 
 
 10 
 12 
 14 
 16 
 
 18 
 
 20 
 22 
 24 
 26 
 28 
 
 30 
 32 
 34 
 
 36 
 38 
 
 40 
 42 
 44 
 46 
 48 
 
 50 
 52 
 54 
 56 
 58 
 
 1419 
 1421 
 1423 
 1425 
 1427 
 
 1430 
 1432 
 1434 
 1436 
 1438 
 
 1440 
 1443 
 1445 
 1447 
 1449 
 
 1451 
 1453 
 1456 
 1458 
 1460 
 
 1462 
 1464 
 1467 
 1469 
 1471 
 
 1473 
 1475 
 1477 
 1480 
 1482 
 
 1484 
 1486 
 1488 
 1491 
 1493 
 
 1495 
 1497 
 1499 
 1502 
 1504 
 
 1506 
 1508 
 L510 
 1513 
 1515 
 
 1517 
 1519 
 1521 
 1524 
 1526 
 
 1528 
 1530 
 1532 
 1535 
 1537 
 
 1539 
 1541 
 1543 
 1546 
 1548 
 
 1550 
 1552 
 1554 
 1557 
 1559 
 
 1561 
 1563 
 1565 
 1568 
 1570 
 
 1572 
 1574 
 1577 
 1579 
 1581 
 
 1583 
 lo85 
 1588 
 1590 
 1592 
 
 1594 
 1596 
 1599 
 1601 
 1603 
 
 1605 
 1608 
 1610 
 1612 
 1614 
 
 1616 
 1619 
 1621 
 1623 
 1625 
 
 1628 
 1630 
 1632 
 1634 
 1637 
 
 1639 
 1641 
 1643 
 1645 
 1648 
 
 1650 
 1652 
 1654 
 1657 
 1659 
 
 1661 
 1663 
 1666 
 1668 
 1670 
 
 1672 
 1675 
 1677 
 1679 
 1681 
 
 1684 
 16M6 
 1688 
 1690 
 1693 
 
 1695 
 1697 
 1699 
 1701 
 1704 
 
 1706 
 1708 
 1711 
 1713 
 1715 
 
 1717 
 1720 
 1722 
 1724 
 1726 
 
 1729 
 1731 
 1733 
 1735 
 1738 
 
 1740 
 1742 
 1744 
 1747 
 1749 
 
 1751 
 1753 
 1756 
 1758 
 1760 
 
 1762 
 1765 
 1767 
 1769 
 1772 
 
 1774 
 1776 
 
 1778 
 1781 
 1783 
 
 1785 
 1787 
 1790 
 1792 
 1794 
 
 1797 
 1799 
 1801 
 1803 
 1806 
 
 1808 
 1810 
 1813 
 1815 
 1817 
 
 1819 
 1822 
 1824 
 1826 
 1829 
 
 1831 
 1833 
 1835 
 1838 
 1840 
 
 1842 
 
 1845 
 1847 
 1849 
 1852 
 
 1854 
 1856 
 1858 
 1861 
 1863 
 
 1865 
 1868 
 1870 
 1872 
 1875 
 
 1877 
 1879 
 1881 
 1884 
 1886 
 
 1888 
 1891 
 1893 
 1895 
 1898 
 
 1900 
 1902 
 1905 
 1907 
 1909 
 
 1912 
 1914 
 1916 
 1918 
 1921 
 
 1923 
 1925 
 
 1928 
 1930 
 1932 
 
 1935 
 1937 
 1939 
 1942 
 1944 
 
 1946 
 1949 
 1951 
 1953 
 1956 
 
 1958 
 1960 
 1963 
 1965 
 1967 
 
 1970 
 1972 
 1974 
 1977 
 1979 
 
 19^1 
 1984 
 1986 
 1988 
 1991 
 
 1993 
 1995 
 1998 
 2000 
 2C02 
 
 2005 
 2007 
 2010 
 2012 
 2014 
 
 2017 
 2019 
 2021 
 2024 
 2026 
 
 2028 
 2031 
 2033 
 2035 
 2038 
 
 2040 
 2043 
 2045 
 2047 
 2050 
 
 2052 
 2054 
 2057 
 2059 
 2061 
 
 2064 
 2066 
 2069 
 2071 
 2073 
 
 2076 
 2078 
 2080 
 2083 
 2085 
 
 2088 
 2090 
 2092 
 2095 
 2097 
 
 
 2 
 
 4 
 6 
 
 8 
 
 10 
 12 
 14 
 16 
 18 
 
 20 
 22 
 24 
 26 
 28 
 
 30 
 32 
 34 
 36 
 38 
 
 40 
 42 
 44 
 46 
 48 
 
 50 
 52 
 
 54 
 56 
 58 
 
 M. 
 
 23° 
 
 24° 
 
 25° 
 
 26° 
 
 27° 23° 
 
 29° 
 
 30° 
 
 31° 
 
 32° 
 
 M. 
 
 148 
 

 
 
 TABLE 
 
 IX- 
 
 -Continued. 
 
 
 
 
 M. 
 
 33" 
 
 34° 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 40° 
 
 41° 
 
 42° 
 
 M. 
 
 
 
 2100 
 
 2171 
 
 2244 
 
 2318 
 
 2393 
 
 2468 
 
 2545 
 
 2623 
 
 2702 
 
 2782 
 
 
 
 2 
 
 2102 
 
 2174 
 
 2247 
 
 2320 
 
 2895 
 
 2471 
 
 2548 
 
 2625 
 
 2704 
 
 2784 
 
 2 
 
 4 
 
 2104 
 
 2176 
 
 2249 
 
 2323 
 
 2398 
 
 2473 
 
 2550 
 
 2628 
 
 2707 
 
 2787 
 
 4 
 
 6 
 
 2107 
 
 2179 
 
 2252 
 
 2325 
 
 2400 
 
 2476 
 
 2553 
 
 2631 
 
 2710 
 
 2790 
 
 6 
 
 8 
 
 2109 
 
 2181 
 
 2254 
 
 2328 
 
 2403 
 
 2478 
 
 2555 
 
 2633 
 
 2712 
 
 2792 
 
 8 
 
 10 
 
 2111 
 
 2184 
 
 2257 
 
 2330 
 
 2405 
 
 2481 
 
 2558 
 
 2636 
 
 2715 
 
 2795 
 
 10 
 
 12 
 
 2114 
 
 2186 
 
 2259 
 
 2333 
 
 2408 
 
 2484 
 
 2560 
 
 2638 
 
 2718 
 
 2798 
 
 12 
 
 14 
 
 2116 
 
 2188 
 
 2261 
 
 2335 
 
 2410 
 
 2486 
 
 2563 
 
 2641 
 
 2720 
 
 2h01 
 
 14 
 
 16 
 
 2119 
 
 2191 
 
 2264 
 
 2338 
 
 2413 
 
 2489 
 
 2566 
 
 2644 
 
 2723 
 
 2803 
 
 16 
 
 18 
 
 2121 
 
 2193 
 
 2266 
 
 2340 
 
 2415 
 
 2491 
 
 2568 
 
 2646 
 
 2726 
 
 2806 
 
 18 
 
 20 
 
 2123 
 
 2196 
 
 2269 
 
 2343 
 
 2418 
 
 2494 
 
 2571 
 
 2649 
 
 2728 
 
 2809 
 
 20 
 
 22 
 
 2126 
 
 2198 
 
 2271 
 
 2345 
 
 2420 
 
 2496 
 
 2573 
 
 2651 
 
 2731 
 
 2811 
 
 22 
 
 24 
 
 2128 
 
 2200 
 
 2274 
 
 2348 
 
 2423 
 
 2499 
 
 2576 
 
 2654 
 
 2-33 
 
 2814 
 
 24 
 
 26 
 
 2131 
 
 2203 
 
 2276 
 
 2350 
 
 2425 
 
 2501 
 
 2578 
 
 2657 
 
 2736 
 
 2817 
 
 26 
 
 28 
 
 2133 
 
 2205 
 
 2279 
 
 2353 
 
 2428 
 
 2504 
 
 2581 
 
 2659 
 
 2739 
 
 2820 
 
 28 
 
 30 
 
 2135 
 
 2208 
 
 2281 
 
 2355 
 
 2430 
 
 2506 
 
 2584 
 
 2662 
 
 2742 
 
 2822 
 
 30 
 
 32 
 
 2138 
 
 2210 
 
 2283 
 
 2358 
 
 2433 
 
 2509 
 
 2586 
 
 2665 
 
 2744 
 
 2825 
 
 32 
 
 34 
 
 2140 
 
 2213 
 
 2286 
 
 2360 
 
 2435 
 
 2512 
 
 25S9 
 
 2667 
 
 2747 
 
 2828 
 
 34 
 
 36 
 
 2143 
 
 2215 
 
 2288 
 
 2363 
 
 2438 
 
 2514 
 
 2591 
 
 2670 
 
 2750 
 
 2830 
 
 36 
 
 38 
 
 2145 
 
 2217 
 
 2291 
 
 2365 
 
 2440 
 
 2517 
 
 2594 
 
 2673 
 
 2752 
 
 2833 
 
 38 
 
 40 
 
 2147 
 
 2220 
 
 2293 
 
 2368 
 
 2443 
 
 2519 
 
 2597 
 
 2675 
 
 2755 
 
 2«36 
 
 40 
 
 42 
 
 2150 
 
 2222 
 
 2296 
 
 2370 
 
 2445 
 
 2522 
 
 2599 
 
 2678 
 
 2758 
 
 2839 
 
 42 
 
 44 
 
 2152 
 
 2225 
 
 2298 
 
 2373 
 
 2448 
 
 2524 
 
 2602 
 
 2680 
 
 2760 
 
 2841 
 
 44 
 
 46 
 
 2155 
 
 2227 
 
 2301 
 
 2375 
 
 2451 
 
 2527 
 
 2604 
 
 2683 
 
 2763 
 
 2844 
 
 46 
 
 48 
 
 2157 
 
 2230 
 
 2303 
 
 2378 
 
 2453 
 
 2530 
 
 2607 
 
 2686 
 
 2766 
 
 2847 
 
 48 
 
 50 
 
 2159 
 
 2232 
 
 2306 
 
 2380 
 
 2456 
 
 2532 
 
 2610 
 
 2688 
 
 2768 
 
 2849 
 
 50 
 
 52 
 
 2162 
 
 2235 
 
 2308 
 
 2383 
 
 2458 
 
 2535 
 
 2612 
 
 2691 
 
 2771 
 
 2652 
 
 52 
 
 54 
 
 2164 
 
 2237 
 
 2311 
 
 2385 
 
 2461 
 
 2537 
 
 2615 
 
 2694 
 
 2774 
 
 2855 
 
 54 
 
 56 
 
 2167 
 
 2239 
 
 2313 
 
 2388 
 
 2463 
 
 2540 
 
 2617 
 
 2i'96 
 
 2776 
 
 2858 
 
 56 
 
 58 
 
 2169 
 
 2242 
 
 2316 
 
 2390 
 
 2466 
 
 2542 
 
 2620 
 
 2699 
 
 2779 
 
 2860 
 
 58 
 
 M. 
 
 33° 
 
 34° 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 40° 
 
 41° 
 
 42° 
 
 M. 
 
 149 
 
TABLE iX.-Continued. 
 
 M. 
 
 43° 
 
 44° 
 
 45° 
 
 46° 
 
 47° 
 
 48° 
 
 49° 
 
 50° 
 
 51° 
 
 52° 
 
 M. 
 
 
 2 
 4 
 6 
 
 8 
 
 10 
 12 
 14 
 16 
 
 18 
 
 20 
 22 
 24 
 
 26 
 
 28 
 
 30 
 32 
 34 
 36 
 38 
 
 40 
 42 
 
 44 
 
 46 
 48 
 
 50 
 52 
 54 
 66 
 58 
 
 2863 
 2866 
 2869 
 
 2871 
 2874 
 
 2877 
 2880 
 2882 
 2885 
 2888 
 
 2891 
 2893 
 2896 
 2899 
 2902 
 
 2904 
 2907 
 29 
 2Q13 
 2915 
 
 2918 
 2921 
 
 2^24 
 2926 
 2929 
 
 29^2 
 2"35 
 2937 
 2^^0 
 2943 
 
 2946 
 2949 
 2951 
 2954 
 2957 
 
 2960 
 2963 
 2965 
 2968 
 2971 
 
 2974 
 2976 
 2979 
 2982 
 2985 
 
 2988 
 2991 
 2993 
 2996 
 2999 
 
 3002 
 3005 
 3007 
 3010 
 3013 
 
 3016 
 3019 
 3021 
 
 3024 
 3027 
 
 3030 
 3033 
 3036 
 3038 
 3041 
 
 3044 
 3047 
 3050 
 3053 
 3055 
 
 305'8 
 3061 
 3064 
 3067 
 3070 
 
 3073 
 3075 
 30''8 
 3081 
 3084 
 
 3087 
 3090 
 3093 
 3095 
 30i.8 
 
 3101 
 3104 
 3107 
 3110 
 3113 
 
 3116 
 3118 
 3i21 
 3124 
 3127 
 
 3130 
 3133 
 3136 
 3139 
 3142 
 
 3144 
 3147 
 3150 
 3153 
 3156 
 
 3159 
 3162 
 3165 
 3168 
 3171 
 
 3173 
 3176 
 3179 
 
 3182 
 3185 
 
 3188 
 3191 
 3194 
 3197 
 3200 
 
 3203 
 3206 
 3209 
 3212 
 3214 
 
 3217 
 3220 
 3223 
 3226 
 3229 
 
 3232 
 3235 
 3238 
 3241 
 3244 
 
 3247 
 
 3250 
 3253 
 3256 
 3259 
 
 3262 
 3265 
 3268 
 3271 
 3274 
 
 3277 
 3280 
 3283 
 
 3286 
 3239 
 
 3292 
 3295 
 3298 
 3301 
 3303 
 
 3306 
 3309 
 3312 
 3316 
 3319 
 
 3322 
 3325 
 3328 
 3331 
 3334 
 
 3337 
 3340 
 3343 
 3346 
 3349 
 
 3352 
 3355 
 3358 
 3361 
 3364 
 
 3367 
 3370 
 3373 
 3376 
 3379 
 
 3382 
 3385 
 3388 
 3391 
 3394 
 
 3397 
 3400 
 3403 
 3407 
 3410 
 
 3413 
 3416 
 3419 
 3422 
 3425 
 
 3428 
 3431 
 3434 
 3*37 
 3440 
 
 3443 
 3447 
 3450 
 3453 
 3*56 
 
 3459 
 3*62 
 3*65 
 3*68 
 3471 
 
 3474 
 3478 
 3481 
 3484 
 3487 
 
 3490 
 3493 
 3496 
 3499 
 3503 
 
 3506 
 3509 
 3512 
 3515 
 3518 
 
 3521 
 3525 
 3528 
 3531 
 3534 
 
 3537 
 3540 
 3543 
 3547 
 3550 
 
 3553 
 3556 
 3559 
 3^62 
 3566 
 
 3569 
 3572 
 3575 
 3578 
 3582 
 
 3585 
 3588 
 3591 
 3594 
 3598 
 
 3601 
 3604 
 3607 
 3610 
 3614 
 
 3617 
 3620 
 3623 
 3626 
 3630 
 
 3633 
 3636 
 3639 
 3643 
 3646 
 
 3649 
 3652 
 3655 
 3659 
 3662 
 
 3665 
 3668 
 3672 
 3675 
 3678 
 
 3681 
 3685 
 3688 
 3691 
 3695 
 
 3698 
 3701 
 3''04 
 3708 
 3711 
 
 3714 
 3717 
 3721 
 3724 
 3727 
 
 3731 
 3734 
 3737 
 37*1 
 3744 
 
 3747 
 3750 
 3754 
 3757 
 3760 
 
 
 2 
 4 
 6 
 8 
 
 10 
 12 
 14 
 16 
 18 
 
 20 
 22 
 24 
 26 
 
 28 
 
 30 
 32 
 34 
 36 
 38 
 
 40 
 42 
 44 
 46 
 48 
 
 50 
 52 
 
 54 
 56 
 58 
 
 M. 
 
 43° 
 
 44° 
 
 45° 
 
 46° 
 
 47° 
 
 48° 
 
 49° 
 
 50° 
 
 51° 
 
 52° 
 
 M. 
 
 150 
 

 
 
 TABLE 
 
 IX- 
 
 -Continued. 
 
 
 
 M. 
 
 53" 
 
 54° 
 
 55° 
 
 56° 
 
 57° 
 
 58° 
 
 59° 
 
 60° 
 
 61° 
 
 62° 
 
 M. 
 
 
 
 3764 
 
 3865 
 
 3968 
 
 4074 
 
 4183 
 
 4294 
 
 4409 
 
 4527 
 
 4649 
 
 4775 
 
 
 
 2 
 
 3767 
 
 3«68 
 
 3971 
 
 4077 
 
 4186 
 
 4298 
 
 4413 
 
 4531 
 
 4653 
 
 4779 
 
 2 
 
 4 
 
 3770 
 
 3«71 
 
 3975 
 
 4081 
 
 4190 
 
 4302 
 
 4417 
 
 4535 
 
 4657 
 
 4784 
 
 4 
 
 6 
 
 3774 
 
 3875 
 
 3978 
 
 4085 
 
 4194 
 
 4306 
 
 4421 
 
 4539 
 
 4662 
 
 4788 
 
 6 
 
 8 
 
 3777 
 
 3878 
 
 3982 
 
 4088 
 
 4197 
 
 4309 
 
 4425 
 
 4543 
 
 4666 
 
 4792 
 
 8 
 
 10 
 
 3780 
 
 3882 
 
 3985 
 
 4092 
 
 4201 
 
 4313 
 
 4429 
 
 4547 
 
 4670 
 
 4796 
 
 10 
 
 12 
 
 3784 
 
 3b85 
 
 3989 
 
 4095 
 
 4205 
 
 4317 
 
 4433 
 
 4551 
 
 4674 
 
 4801 
 
 12 
 
 14 
 
 3787 
 
 3889 
 
 3992 
 
 4f99 
 
 4208 
 
 4321 
 
 4436 
 
 4555 
 
 4678 
 
 4805 
 
 14 
 
 16 
 
 3790 
 
 3892 
 
 3996 
 
 4103 
 
 4212 
 
 4325 
 
 4440 
 
 4559 
 
 4682 
 
 4809 
 
 16 
 
 18 
 
 3794 
 
 3895 
 
 3999 
 
 4106 
 
 4216 
 
 4328 
 
 4444 
 
 4564 
 
 4687 
 
 4814 
 
 18 
 
 20 
 
 3797 
 
 3899 
 
 4003 
 
 4110 
 
 4220 
 
 4332 
 
 4448 
 
 4568 
 
 4691 
 
 4818 
 
 20 
 
 22 
 
 3800 
 
 3902 
 
 4006 
 
 4113 
 
 4223 
 
 4336 
 
 4452 
 
 4572 
 
 4695 
 
 4822 
 
 22 
 
 24 
 
 3804 
 
 3906 
 
 4010 
 
 4117 
 
 4227 
 
 4340 
 
 4456 
 
 4576 
 
 4699 
 
 4826 
 
 24 
 
 26 
 
 3807 
 
 3909 
 
 4014 
 
 4121 
 
 4231 
 
 4344 
 
 4460 
 
 4580 
 
 4703 
 
 4831 
 
 26 
 
 28 
 
 3811 
 
 3913 
 
 4017 
 
 4124 
 
 4234 
 
 4347 
 
 4464 
 
 4584 
 
 4707 
 
 4835 
 
 28 
 
 30 
 
 3814 
 
 3916 
 
 4021 
 
 4128 
 
 4238 
 
 4351 
 
 4468 
 
 4588 
 
 4712 
 
 4839 
 
 30 
 
 32 
 
 3817 
 
 3919 
 
 4024 
 
 4132 
 
 4242 
 
 4355 
 
 4472 
 
 4592 
 
 4716 
 
 4844 
 
 32 
 
 34 
 
 3821 
 
 3923 
 
 4028 
 
 4135 
 
 4246 
 
 4359 
 
 4476 
 
 4596 
 
 4720 
 
 4848 
 
 34 
 
 36 
 
 3824 
 
 3926 
 
 4031 
 
 4139 
 
 4249 
 
 4363 
 
 4480 
 
 4600 
 
 4724 
 
 4852 
 
 36 
 
 38 
 
 3827 
 
 3930 
 
 4035 
 
 4142 
 
 4253 
 
 4367 
 
 4484 
 
 4604 
 
 4728 
 
 4857 
 
 38 
 
 40 
 
 3«31 
 
 3983 
 
 4038 
 
 4146 
 
 4257 
 
 4370 
 
 4488 
 
 4608 
 
 4733 
 
 4861 
 
 40 
 
 42 
 
 3834 
 
 3937 
 
 4042 
 
 4150 
 
 4260 
 
 4374 
 
 4492 
 
 4612 
 
 4737 
 
 4865 
 
 42 
 
 44 
 
 3838 
 
 3940 
 
 4045 
 
 4153 
 
 4>64 
 
 4378 
 
 4499 
 
 4616 
 
 4741 
 
 4870 
 
 44 
 
 46 
 
 3841 
 
 3944 
 
 4049 
 
 4157 
 
 4268 
 
 4382 
 
 4495 
 
 4620 
 
 4745 
 
 4874 
 
 46 
 
 48 
 
 3844 
 
 3947 
 
 4052 
 
 4161 
 
 4272 
 
 4386 
 
 4503 
 
 4625 
 
 4750 
 
 4879 
 
 48 
 
 50 
 
 3848 
 
 3951 
 
 4056 
 
 4164 
 
 4275 
 
 4390 
 
 4507 
 
 4629 
 
 4754 
 
 4883 
 
 50 
 
 52 
 
 3851 
 
 3954 
 
 4060 
 
 4168 
 
 4279 
 
 4894 
 
 4511 
 
 4633 
 
 4758 
 
 4887 
 
 52 
 
 54 
 
 3854 
 
 3958 
 
 4063 
 
 4172 
 
 4283 
 
 4398 
 
 4515 
 
 4637 
 
 4762 
 
 4892 
 
 54 
 
 56 
 
 3858 
 
 3961 
 
 4067 
 
 4175 
 
 4287 
 
 4401 
 
 4519 
 
 4641 
 
 4766 
 
 4896 
 
 56 
 
 58 
 
 3861 
 
 3964 
 
 4070 
 
 4179 
 
 4291 
 
 4405 
 
 4523 
 
 4645 
 
 4771 
 
 4901 
 
 58 
 
 M. 
 
 53° 
 
 54° 
 
 55° 
 
 56° 
 
 57° 
 
 58° 
 
 59° 
 
 60° 
 
 61° 
 
 62° 
 
 M. 
 
 151 
 

 
 
 TABLE 
 
 X- 
 
 -Continued. 
 
 
 
 
 M. 
 
 63° 
 
 64° 
 
 65° 
 
 66° 
 
 67° 
 
 68° 
 
 69° 
 
 70° 
 
 71° 
 
 72° 
 
 M. 
 
 
 
 4905 
 
 5039 
 
 5179 
 
 5324 
 
 5474 
 
 5631 
 
 5795 
 
 5966 
 
 6146 
 
 6335 
 
 
 
 2 
 
 4909 
 
 5044 
 
 5184 
 
 5328 
 
 5479 
 
 5636 
 
 5800 
 
 5972 
 
 6152 
 
 63^1 
 
 2 
 
 4 
 
 4914 
 
 5049 
 
 5188 
 
 5333 
 
 5484 
 
 5642 
 
 5806 
 
 5978 
 
 6158 
 
 6348 
 
 4 
 
 6 
 
 4918 
 
 5053 
 
 5193 
 
 5338 
 
 5489 
 
 5647 
 
 5811 
 
 5984 
 
 6164 
 
 6.'^-4 
 
 6 
 
 8 
 
 4923 
 
 5058 
 
 5198 
 
 5343 
 
 5495 
 
 5652 
 
 5817 
 
 5989 
 
 6170 
 
 6361 
 
 8 
 
 10 
 
 4927 
 
 5062 
 
 5203 
 
 5348 
 
 5500 
 
 5658 
 
 5823 
 
 5995 
 
 6177 
 
 6367 
 
 10 
 
 12 
 
 4931 
 
 5067 
 
 5207 
 
 5353 
 
 5505 
 
 5663 
 
 5828 
 
 6001 
 
 6183 
 
 6374 
 
 12 
 
 14 
 
 4936 
 
 5071 
 
 5212 
 
 5358 
 
 5510 
 
 5668 
 
 5834 
 
 6007 
 
 6189 
 
 6380 
 
 14 
 
 16 
 
 4940 
 
 5076 
 
 5217 
 
 5363 
 
 5515 
 
 5674 
 
 5839 
 
 6013 
 
 6195 
 
 63h7 
 
 16 
 
 18 
 
 4945 
 
 5081 
 
 5222 
 
 5368 
 
 5520 
 
 5679 
 
 5845 
 
 6019 
 
 6201 
 
 6394 
 
 18 
 
 20 
 
 4^49 
 
 5085 
 
 5226 
 
 5373 
 
 5526 
 
 5685 
 
 5851 
 
 6025 
 
 6208 
 
 6400 
 
 20 
 
 22 
 
 4954 
 
 5090 
 
 5231 
 
 5378 
 
 5531 
 
 5690 
 
 5856 
 
 6031 
 
 6214 
 
 6407 
 
 22 
 
 24 
 
 4958 
 
 5095 
 
 5236 
 
 5383 
 
 5536 
 
 5695 
 
 5862 
 
 6037 
 
 6220 
 
 6413 
 
 24 
 
 26 
 
 4963 
 
 5099 
 
 5'241 
 
 5388 
 
 5541 
 
 5701 
 
 5868 
 
 6043 
 
 6226 
 
 6420 
 
 26 
 
 28 
 
 4967 
 
 5104 
 
 5246 
 
 5393 
 
 5546 
 
 5706 
 
 5874 
 
 6049 
 
 6233 
 
 6427 
 
 28 
 
 30 
 
 4972 
 
 5108 
 
 5250 
 
 5398 
 
 5552 
 
 5712 
 
 5879 
 
 6055 
 
 6239 
 
 6433 
 
 30 
 
 32 
 
 4976 
 
 5113 
 
 5>55 
 
 5403 
 
 5557 
 
 5717 
 
 5885 
 
 6061 
 
 6245 
 
 6440 
 
 32 
 
 34 
 
 4981 
 
 5118 
 
 5260 
 
 5408 
 
 5562 
 
 5723 
 
 5891 
 
 6067 
 
 6252 
 
 6447 
 
 34 
 
 36 
 
 4985 
 
 5122 
 
 5265 
 
 5413 
 
 5567 
 
 5728 
 
 5896 
 
 6073 
 
 6258 
 
 6453 
 
 36 
 
 38 
 
 4990 
 
 5127 
 
 5270 
 
 5418 
 
 5573 
 
 5734 
 
 5902 
 
 6079 
 
 6264 
 
 6460 
 
 38 
 
 40 
 
 4994 
 
 5132 
 
 5275 
 
 5423 
 
 5578 
 
 5739 
 
 5908 
 
 6085 
 
 6271 
 
 6467 
 
 40 
 
 42 
 
 4999 
 
 5136 
 
 5280 
 
 5428 
 
 5583 
 
 5745 
 
 5914 
 
 6091 
 
 6277 
 
 6473 
 
 42 
 
 44 
 
 5003 
 
 5141 
 
 5284 
 
 5433 
 
 5588 
 
 5750 
 
 5919 
 
 6097 
 
 6283 
 
 6480 
 
 44 
 
 46 
 
 5008 
 
 5146 
 
 5289 
 
 5438 
 
 5594 
 
 5756 
 
 5925 
 
 6103 
 
 6290 
 
 6487 
 
 46 
 
 48 
 
 5012 
 
 5151 
 
 5294 
 
 5443 
 
 5599 
 
 5761 
 
 5931 
 
 6109 
 
 6296 
 
 6494 
 
 48 
 
 50 
 
 5017 
 
 5155 
 
 5299 
 
 5448 
 
 5604 
 
 5767 
 
 5937 
 
 6115 
 
 6303 
 
 6500 
 
 50 
 
 52 
 
 5021 
 
 5160 
 
 5304 
 
 5454 
 
 5610 
 
 5772 
 
 5943 
 
 6121 
 
 6309 
 
 6507 
 
 52 
 
 54 
 
 5026 
 
 5165 
 
 5309 
 
 5459 
 
 5615 
 
 577« 
 
 5948 
 
 6127 
 
 6315 
 
 6514 
 
 64 
 
 56 
 
 5030 
 
 5169 
 
 5314 
 
 5464 
 
 5620 
 
 5783 
 
 5954 
 
 6133 
 
 6322 
 
 6521 
 
 66 
 
 58 
 
 5035 
 
 5174 
 
 5319 
 
 5469 
 
 5625 
 
 5789 
 
 5960 
 
 6140 
 
 6328 
 
 6528 
 
 58 
 
 M. 
 
 63° 
 
 64° 
 
 65° 
 
 66° 
 
 67° 
 
 68° 
 
 69° 
 
 70° 
 
 71° 
 
 72° 
 
 M. 
 
 152 
 

 
 
 
 TABLE 
 
 X. 
 
 
 
 
 
 Showing it 
 
 e length of 1 
 
 minute of longitude on difFeren 
 
 t latitudes. 1 
 
 Lat. 
 
 Statute 
 
 Naut. 
 
 T "«- 
 
 Statute 
 
 Naut. 
 
 T 
 
 
 Statute 
 
 Naut. 
 
 Miles. 
 
 Miles. 
 
 J-. 
 
 ■ 
 
 Miles. 
 
 Miles. 
 
 J^ctt. 
 
 Miles. 
 
 Miles. 
 
 0^ 
 
 ' 
 
 1.1527 
 
 1.0000 
 
 27° 
 
 , 
 
 1.0277 
 
 .8916 
 
 54° 
 
 _, 
 
 .6790 
 
 .5891 
 
 
 30 
 
 1 . 1526 
 
 .9999 
 
 
 30 
 
 1. 0232 
 
 .8877 
 
 
 30 
 
 .6708 
 
 .5820 
 
 1 
 
 — 
 
 1.1525 
 
 .9998 
 
 28 
 
 — 
 
 1.0185 
 
 .8836 
 
 55 
 
 — 
 
 .6626 
 
 .5749 
 
 
 30 
 
 1.1523 
 
 .9996 
 
 
 30 
 
 1.0138 
 
 .8795 
 
 
 30 
 
 .6544 
 
 .5678 
 
 2 
 
 — 
 
 1.1520 
 
 .9994 
 
 29 
 
 — 
 
 1.0089 
 
 .8753 
 
 56 
 
 — 
 
 .6460 
 
 .5605 
 
 
 30 
 
 1.1516 
 
 .9991 
 
 
 30 
 
 1.0041 
 
 .8711 
 
 
 30 
 
 6377 
 
 .5.533 
 
 3 
 
 — 
 
 1.1511 
 
 .9986 
 
 30 
 
 — 
 
 .9991 
 
 .8667 
 
 57 
 
 — 
 
 .6293 
 
 .54.59 
 
 
 30 
 
 1.1506 
 
 .9981 
 
 
 30 
 
 .9940 
 
 .8624 
 
 
 30 
 
 .6208 
 
 .5416 
 
 4 
 
 — 
 
 1.1499 
 
 .9976 
 
 31 
 
 — 
 
 .9889 
 
 .8579 
 
 58 
 
 — 
 
 .6123 
 
 .5312 
 
 
 30 
 
 1.1491 
 
 ,9969 
 
 
 30 
 
 .9837 
 
 .8534 
 
 
 30 
 
 .6048 
 
 .5238 
 
 5 
 
 — 
 
 1.1483 
 
 .9962 
 
 32 
 
 — 
 
 .9784 
 
 .8488 
 
 59 
 
 — 
 
 .5951 
 
 .5161 
 
 
 30 
 
 1.1475 
 
 .9954 
 
 
 30 
 
 .9731 
 
 .8442 
 
 
 30 
 
 5865 
 
 .5088 
 
 6 
 
 — 
 
 1.1464 
 
 .9945 
 
 33 
 
 — 
 
 .9^76 
 
 .8395 
 
 60 
 
 — 
 
 .5778 
 
 .5013 
 
 
 30 
 
 1.1453 
 
 .9936 
 
 
 30 
 
 .9622 
 
 .8348 
 
 
 30 
 
 .5691 
 
 .4938 
 
 7 
 
 — 
 
 1.1441 
 
 .9926 
 
 34 
 
 — 
 
 .9566 
 
 .8299 
 
 61 
 
 — 
 
 .5602 
 
 .4861 
 
 
 30 
 
 1.1428 
 
 .9915 
 
 
 30 
 
 .9509 
 
 .8250 
 
 
 30 
 
 .5514 
 
 .4785 
 
 8 
 
 — 
 
 1.1415 
 
 .9903 
 
 35 
 
 — 
 
 .9452 
 
 .8200 
 
 62 
 
 — 
 
 .5425 
 
 .4707 
 
 
 30 
 
 1.1401 
 
 .9891 
 
 
 30 
 
 .9394 
 
 .8150 
 
 
 30 
 
 .5337 
 
 .4630 
 
 9 
 
 — 
 
 1.1386 
 
 .9877 
 
 36 
 
 — 
 
 .9336 
 
 .8099 
 
 63 
 
 — 
 
 .5247 
 
 .4552 
 
 
 30 
 
 1.1348 
 
 .9863 
 
 
 30 
 
 .9277 
 
 .8048 
 
 
 30 
 
 .5157 
 
 .4474 
 
 10 
 
 — 
 
 1.1352 
 
 .9849 
 
 37 
 
 — 
 
 .9217 
 
 .7996 
 
 64 
 
 — 
 
 5066 
 
 .4395 
 
 
 30 
 
 1.1335 
 
 .9833 
 
 
 30 
 
 .9157 
 
 .7943 
 
 
 30 
 
 .4976 
 
 .4317 
 
 11 
 
 — 
 
 1.1316 
 
 .9817 
 
 38 
 
 — 
 
 .9095 
 
 .7890 
 
 65 
 
 — 
 
 .4885 
 
 .4237 
 
 
 30 
 
 1.1297 
 
 .9800 
 
 
 30 
 
 ,9033 
 
 .7837 
 
 
 30 
 
 .4794 
 
 .4158 
 
 12 
 
 — 
 
 1.1276 
 
 .9783 
 
 39 
 
 — 
 
 .8970 
 
 .7782 
 
 66 
 
 — 
 
 .4701 
 
 .4079 
 
 
 30 
 
 1.1254 
 
 .9764 
 
 
 30 
 
 .8907 
 
 .7727 
 
 
 30 
 
 .4610 
 
 .3999 
 
 13 
 
 — 
 
 1.1233 
 
 .9745 
 
 40 
 
 
 .8842 
 
 .7671 
 
 67 
 
 — 
 
 .4517 
 
 .3918 
 
 
 30 
 
 1.1211 
 
 .9726 
 
 
 30 
 
 .8778 
 
 .7615 
 
 
 30 
 
 .4424 
 
 .3838 
 
 U 
 
 — 
 
 1.1187 
 
 .9705 
 
 41 
 
 
 
 .8712 
 
 .7558 
 
 68 
 
 — 
 
 .4330 
 
 .37.57 
 
 
 30 
 
 1.1163 
 
 .9682 
 
 
 30 
 
 .8646 
 
 .7501 
 
 
 30 
 
 .4238 
 
 .3676 
 
 15 
 
 — 
 
 1.1137 
 
 .9661 
 
 42 
 
 — 
 
 .8579 
 
 .7443 
 
 69 
 
 — 
 
 .4143 
 
 .3594 
 
 
 30 
 
 1.1111 
 
 .9638 
 
 
 30 
 
 .8512 
 
 .7385 
 
 
 30 
 
 .4049 
 
 .3513 
 
 16 
 
 — 
 
 1.1083 
 
 .9615 
 
 43 
 
 
 .8443 
 
 ,7325 
 
 70 
 
 — 
 
 .3954 
 
 .3430 
 
 
 30 
 
 1.1056 
 
 .9571 
 
 
 30 
 
 .8375 
 
 .7266 
 
 
 30 
 
 .3859 
 
 .33.58 
 
 17 
 
 — 
 
 1.1026 
 
 .9566 
 
 44 
 
 
 
 .8305 
 
 ,7205 
 
 71 
 
 — 
 
 .3764 
 
 .3265 
 
 
 30 
 
 1.0997 
 
 .9540 
 
 
 30 
 
 .8235 
 
 .7143 
 
 
 30 
 
 .3668 
 
 ,3183 
 
 18 
 
 — 
 
 1.0966 
 
 .9513 
 
 45 
 
 — 
 
 .8164 
 
 .7080 
 
 72 
 
 — 
 
 .3572 
 
 .3099 
 
 
 30 
 
 1.0935 
 
 .9486 
 
 
 30 
 
 .8093 
 
 ,7020 
 
 
 30 
 
 .3476 
 
 .3016 
 
 19 
 
 — 
 
 1.0903 
 
 .9458 
 
 46 
 
 
 .8021 
 
 ,6960 
 
 73 
 
 — 
 
 .3380 
 
 .2933 
 
 
 30 
 
 1.0870 
 
 .9430 
 
 
 30 
 
 .7949 
 
 ,6891 
 
 
 30 
 
 .3284 
 
 .2850 
 
 20 
 
 — 
 
 1.0836 
 
 .9401 
 
 47 
 
 __ 
 
 .7875 
 
 .6831 
 
 74 
 
 — 
 
 .3187 
 
 .2765 
 
 
 30 
 
 1.0802 
 
 .9372 
 
 
 30 
 
 .'J 802 
 
 .6769 
 
 
 30 
 
 .3090 
 
 .2681 
 
 21 
 
 — 
 
 1.0766 
 
 .9340 
 
 48 
 
 — 
 
 .7727 
 
 .6704 
 
 75 
 
 — 
 
 .2993 
 
 .2596 
 
 
 30 
 
 1 0728 
 
 .9308 
 
 
 30 
 
 .7652 
 
 .6639 
 
 
 30 
 
 .2895 
 
 ,2.512 
 
 22 
 
 — 
 
 1.0692 
 
 •9276 
 
 40 
 
 
 .7576 
 
 .6573 
 
 76 
 
 — 
 
 .2797 
 
 .2427 
 
 
 30 
 
 1.0654 
 
 .9243 
 
 
 30 
 
 .7500 
 
 .6507 
 
 
 30 
 
 .2699 
 
 .2343 
 
 23 
 
 — 
 
 1.0616 
 
 .9209 
 
 50 
 
 — 
 
 .7422 
 
 .6440 
 
 77 
 
 — 
 
 .2601 
 
 .2256 
 
 
 30 
 
 1.0577 
 
 .9175 
 
 
 30 
 
 .7345 
 
 .6347 
 
 
 30 
 
 .2503 
 
 .2171 
 
 24 
 
 — 
 
 1.0536 
 
 .9140 
 
 51 
 
 
 .7268 
 
 ,6306 
 
 78 
 
 — 
 
 .2404 
 
 ,2085 
 
 
 30 
 
 1.0495 
 
 .9105 
 
 
 30 
 
 .7190 
 
 .6238 
 
 
 30 
 
 ,2305 
 
 .2000 
 
 25 
 
 — 
 
 1 0453 
 
 .9069 
 
 52 
 
 — 
 
 .7111 
 
 ,6169 
 
 79 
 
 — 
 
 .2206 
 
 .1914 
 
 
 30 
 
 1.0411 
 
 .9032 
 
 
 30 
 
 .7032 
 
 ,6101 
 
 
 30 
 
 .2107 
 
 .1823 
 
 26 
 
 — 
 
 1.0367 
 
 .8994 
 
 53 
 
 
 .6952 
 
 .6031 
 
 80 
 
 — 
 
 .2008 
 
 ,1742 
 
 
 
 30 
 
 1.0323 
 
 .8056 
 
 
 30 
 
 .6872 
 
 .5962 
 
 
 
 
 
 153 
 
TABLE XI. 
 
 This table contains the correction, in minutes, to be added 
 
 to the middle latitude, to obtain the correct 
 
 middle latitude. 
 
 1^ 
 
 ^2 
 
 DIFFERENCE OF LATITUDE. 
 
 3° 
 
 40 
 
 5° 
 
 50 70 
 
 8° 
 
 9° 
 
 10° 
 
 12° 
 
 14° 
 
 16° 
 
 15^ 
 
 18 
 
 21 
 
 2' 
 1 
 
 1 
 
 3' 
 3 
 
 2 
 
 5' 
 4 
 4 
 
 7' 
 6 
 5 
 
 9' 
 
 8 
 
 7 
 
 12' 
 
 10 
 
 9 
 
 15' 
 13 
 12 
 
 18' 
 16 
 15 
 
 26' 
 23 
 21 
 
 36' 
 32 
 29 
 
 47' 
 41 
 
 37 
 
 24 
 30 
 35 
 
 1 
 
 1 
 1 
 
 2 
 
 2 
 2 
 
 3 
 3 
 3 
 
 5 
 5 
 
 4 
 
 7 
 6 
 6 
 
 9 
 
 8 
 8 
 
 11 
 10 
 10 
 
 14 
 13 
 12 
 
 20 
 18 
 
 18 
 
 27 
 25 
 24 
 
 35 
 32 
 32 
 
 40 
 45 
 50 
 
 1 
 1 
 1 
 
 2 
 2 
 2 
 
 3 
 3 
 4 
 
 5 
 5 
 5 
 
 6 
 6 
 
 7 
 
 8 
 8 
 9 
 
 10 
 10 
 11 
 
 13 
 13 
 14 
 
 18 
 18 
 20 
 
 25 
 
 25 
 28 
 
 32 
 32 
 36 
 
 55 
 
 58 
 60 
 
 1 
 2 
 2 
 
 3 
 3 
 3 
 
 4 
 4 
 4 
 
 6 
 6 
 6 
 
 8 
 8 
 9 
 
 10 
 11 
 11 
 
 13 
 14 
 14 
 
 16 
 17 
 
 18 
 
 22 
 24 
 26 
 
 31 
 33 
 35 
 
 40 
 43 
 46 
 
 62 
 64 
 66 
 
 2 
 2 
 2 
 
 3 
 3 
 4 
 
 5 
 5 
 5 
 
 7 
 7 
 8 
 
 9 
 10 
 11 
 
 12 
 13 
 14 
 
 15 
 16 
 18 
 
 19 
 20 
 22 
 
 27 
 29 
 32 
 
 37 
 40 
 43 
 
 49 
 52 
 57 
 
 This table is to be entered at the top with the 
 Difference of the two latitudes, and at the side with the 
 Middle Latitude; under the former, and opposite the 
 latter, is the correction, in minutes, to be added to the 
 middle latitude, to obtain the corrected middle latitude. 
 
 154 
 
TABLE XII. 
 Distance of Objects by Two Bearings. 
 
 Useful in rounding headlands in the night. 
 
 Ti tie 
 
 P. 
 
 3 
 
 3% 
 
 4 
 
 4^ 
 
 5 
 
 5^ 
 
 6 
 
 6^ 
 
 7 
 
 75^ 
 
 9 
 10 
 
 FIRST BEARINGS FROM HEADING OF 
 VESSEL. — POINTS. 
 
 2 2^ 3 3H 4 4M 5 5M 6 6K 7 7^ 8 8^ 
 
 .5561. 
 
 .6341 
 
 .707 
 
 .773 
 
 .831 
 
 .882 
 
 .924 
 
 .957 
 
 .981 
 
 .995 
 
 .000 
 
 .995 
 
 .981 
 
 .957 
 
 .924 
 
 322.42 
 001.62 
 8J 1.23 
 691.00 
 60 .85 
 
 12.85 
 
 1.91 
 
 1.45 
 
 1.18 
 
 1.00 
 
 .88 
 
 .79 
 
 .72 
 
 .67 
 
 .63 
 
 .60 
 
 .58 
 
 .57 
 
 25 
 
 193.62 
 66 2.44 3 
 
 .35 1,85 3.66 
 141.502.02 
 001.27 1.64 
 901.111.39 
 
 .821.001.22 
 
 .91 1.09 
 .85 1.00 
 .80| .93 
 .77 .88 
 
 I 
 
 26 
 
 2.86 4.52 
 2.173.04 
 1.76 2.30 
 1.501.87 
 1.311.59 
 1.181.39 
 1.08 1.25 
 1.001.14 
 
 4.74 
 
 3.184.91 
 
 2.413.30 5.03 
 
 1.962.503.38 
 
 1.662.032.57 
 
 1.461. r2'2. 08 
 
 1.311.511.77 
 
 5.10 
 
 3.43 5.13 
 
 2.603.44J 
 
 2.11 2. 6113. 435. ( 
 
 I 
 
 5.10 
 
 Rui^E. — The number in column of first bearing, and 
 in line of second bearing, multiplied by the distance seen 
 between the times of taking the bearings, is the distance 
 of object from the vessel at time of taking the second 
 bearing, in units of the distance seen. 
 
 And this distance multiplied by the sine of the 
 second bearing, is the distance of the line of ship's course 
 from the object, at right angles. 
 
 155 
 

 TABLE XIII. 
 
 Table for Reducing Longitude to Time. 
 
 o 
 
 M. 
 
 
 M. S. 
 
 ' 
 
 M. S. 
 
 / 
 
 M. S. 
 
 ' 
 
 M. s. 
 
 ' 
 
 M. S. 
 
 ' 
 
 M. S. 
 
 1 
 
 4 
 
 1 
 
 4 
 
 11 
 
 .44 
 
 21 
 
 1.24 
 
 31 
 
 2.04 
 
 41 
 
 2.44 
 
 51 
 
 3.24 
 
 2 
 
 8 
 
 2 
 
 8 
 
 12 
 
 .48 
 
 22 
 
 1.28 
 
 32 
 
 2.08 
 
 42 
 
 2.48 
 
 52 
 
 3.28 
 
 3 
 
 12 
 
 3 
 
 12 
 
 13 
 
 .52 
 
 23 
 
 1.32 
 
 33 
 
 2.12 
 
 43 
 
 2.52 
 
 53 
 
 3.32 
 
 4 
 
 16 
 
 4 
 
 16 
 
 14 
 
 .56 
 
 24 
 
 1 36 
 
 34 
 
 2.16 
 
 44 
 
 2.56 
 
 54 
 
 3.36 
 
 5 
 
 20 
 
 5 
 
 20 
 
 15 
 
 1.00 
 
 25 
 
 1.40 
 
 35 
 
 2.20 
 
 45 
 
 3.00 
 
 55 
 
 3.40 
 
 6 
 
 24 
 
 6 
 
 24 
 
 16 
 
 1.04 
 
 26 
 
 1.44 
 
 36 
 
 2.24 
 
 46 
 
 3.04 
 
 56 
 
 3.44 
 
 7 
 
 28 
 
 7 
 
 28 
 
 17 
 
 1.08 
 
 27 
 
 1.48 
 
 37 
 
 2.28 
 
 47 
 
 3.08 
 
 57 
 
 3.48 
 
 8 
 
 32 
 
 8 
 
 32 
 
 18 
 
 1.12 
 
 28 
 
 1.52 
 
 33 
 
 2.32 
 
 48 
 
 3.12 
 
 58 
 
 3.52 
 
 9 
 
 36 
 
 9 
 
 36 
 
 19 
 
 1.16 
 
 29 
 
 1.56 
 
 39 
 
 2.36 
 
 49 
 
 3 16 
 
 59 
 
 3.56 
 
 10 
 
 40 
 
 10 
 
 40 
 
 20 
 
 1.20 
 
 30 
 
 2.00 
 
 40 
 
 2.40 
 
 50 
 
 3.20 
 
 60 
 
 4.00 
 
 
 The above table is wanted for finding the error of 
 
 the standard time watch, on local mean time, or on local 
 
 apparent time, when expanding an amplitude into a 
 
 tat 
 
 >le of time azimuths of the sun, for the purpose of 
 
 fin 
 
 ding compass errors. 
 
 156 
 
PLATE I.— PEARSON'S DIAGRAM. 
 
 Var. East 
 
 Vkr.We^t 
 
 158 
 
PEARSON^S DIAGRAIYI.— PLATE I. 
 
 Va.rBa.st Vkr.W^s^ 
 
 159 
 
Se mA(UrtulcLr:an3L ' Qua^ranfal 
 
 NortJ]Q ^ gee pa-ye ^ qSoujJi 
 
 PLATE II. 
 
 160 
 

 ^n 
 
 __..^_... 
 
 ^ -* ^ > 
 
 k^.-Zkt.^ 
 
 161 
 
 PLATE II. 
 
PLATE III. 
 
 162 
 
2EARSON'S 
 — DUMB 
 
 COM R ASS. 
 
 MANUFACTURED BY 
 
 L. BEOKMANN, TOLEDO, O. 
 
Pearson's Dumb Compass 
 
 MANUFACTURED AFTER THE /NVENTORS 
 DIRECTIONS BY 
 
 L. BEGKMANN, TOLEDO, 0. 
 
 This little Instrument, so important to Masters and Pilots of 
 Vessels, consists of a Tripod with Ball and Socket, on which 
 turns a plate with index, which can be set parallel to the line of 
 Keel. On this lower plate turns a circle divided into Degrees 
 and Compass points, on which revolves an Alidade provided 
 with an Index to read off the Degrees and Compass points; with 
 a jDair of sight vanes and a pair of levels to set the whole instru- 
 ment horizontal. 
 
 Mostly made of brass and executed in the best manner and 
 first-class workmanship. 
 
 PRICE $30.00. 
 
 FOR FURTHER PARTICULARS, ADDRESS 
 
 L. BECKMANN, 
 
 319 ADAMS ST., TOLEDO, O. 
 
 Also keep in stock a full line of Aneroid Barometers, Marine 
 Glasses and Compasses, Parallel Kulers, Etc.