'U O ITS.'70'~os _ __ 8 8~ 75 7S~1 ~~~~ \~ /,, ""'~i"~ ~''.' —"~-.'~.~"~"s-~..~'j"'' " ~ ~ ~ ~ ~ ~ ii' I~ ~~~'"',-'~~ ~ —~~ ~ ~iggst..... " -A —:., ~~~~~~.............~.:."' i -- _.~ ~,,'I,?~v......~._ X.~..!".....7' Il'",,l..../-'"yN:' I..,~...... ~~~~~%~5~~:<..../... two pairs of sides, a distance equal to their B., length, forming two new t-iangles, as shown by the dotted lines, and measure the sides B'D', and A'D". The three sides of each of these A. D triangles will thus be known, and also the three sides of the triangle BAD, since AD A= A'D'", and BD =- B'D'. Fig. 56. The method of this article may be employed "7 for a figure of six sides as shown in Fig. 56, \ - (in which the dotted lines within the wooded \? / I/ field have their lengths determined by the tri- -.:,' V// angles formed outside of it,) but not for figures' of a greater number of sides. CHAPTER III. SURVEYING BY PERPENDICULAILS: OR By the Second Method. (I03) THE method of Surveying by Perpendiculars is'founded on the Second Mietod of determining the position of a point, explained in Art. (6). It is applied in two ways, either to making a complete Survey by' _Diagonals and Perpendiculars," or to measuring a crooked boundary by " Off-sets." Each will be considered in turn. CHAP. III.] Surveying by Perpendiculars. 69 The best methods of getting perpendiculars on the ground must, however, be first explained. TO SET OUT PERPENDICULARS. (1i ) Surveyoers Cross, The simplest instrument Fig. 57. for this purpose is the Surveyor's Cross, or Cross-Staff, shown in the figure. It consists of a blocl of wood, of any shape, having in it two saw-cuts, made very precisely at right angles to each other, about half an inch deep, and with centre-bit holes made at the bottom of the cuts 1 l to assist in finding the objects. This block is fixed on a i pointed staff, on which it can turn freely, and which 6 should be precisely 8 links (63~ inches) long, for the convenience of short measurements. To use the Cross-staff to erect a perpendicular, set it 4 at the point of the line at which a perpendicular is want- 3 ed. Turn its head till, on looking through one saw-cut, you see the ends of the line. Then will the other sawcut point out the direction of the perpendicular, and thus guide the measurement desired. To find where a perpendicular to the line, from some object, as a corner of a field, a tree, &c., would meet the line, set up the cross-staff at a point of the line which seems to the eye to be about the spot. Note about how far from the object the perpendicular at this point strikes, and move the cross-staff that distance; and repeat the operation till the correct spot is found. (105) To test the accuracy of the in- Fig. 5.8 B C strument, sight through one slit to some point A, and place a stake B in the line of sight of the other slit. Then turn its A ------------—.....- "4 head a ciuarter of the way around, so that the second slit looked through, points to A. Then see if the other slit covers B again, as it will if correct. If it does not do so, but sights to some other point, as B', the apparent error is double the real one, for it now points as far to the right of the trie point, C as it did before to its left. 70 CHAIN SURYEYI G. LPART Ii. This is the first example we have had of the invaluable prin. ciple of-Reversion, which is used in almost every test of the accuracy of Surveying and Astronomical instruments, its peculiar merit being that it doubles the real error, and thus makes it twice as easy to perceive and correct it. (106) The instrument, in its most finished form, is made of a hollow brass cylinder, which has two pairs of slits exactly opposite to each other, one of each pair being narrow and the other wide, with a horse-hair stretched from the top to the bottom of the latter. It is also, sometimes, made with eight faces, and two more pairs of slits added, so as to set off half a right angle. Fig. Q. Another form is a hollow brass sphere, as in the L " figure. This enables the surveyor to set off perpenpendiculars on very steep slopes. Another form of the surveyor's cross consists of two pairs Fig. 60 of plain " Sights," each shaped as in the figure, placed at the ends of two bars at right angles to each other. The slit, and the opening with a hair stretched from its top to its bottom, are respectively at the top of one sight and at the bottom of the opposite sight.* This is used in the same L manner as the preceding form, but is less portable and more liable to get out of order. A temporary substitute for these instruments may be Fig. 61. made by sticking four pins into the corners of a square piece of board; and sighting across them, in the direction of the line and at right angles to it. (107) Optical Square, The most convenient and accurate instrument is, however, the Optical Square. The figures give a perspective view of it, and also a plan with the lid removed. It is a small circular box, containing a strip of looldng-glass, from the upper half of which the silvering is removed. This glass is placed * The French call the narrow opening ailleton, and the wide one croisee, CHAP. III.] To set out Perpendiculars 71 so as to make precisely half a right Fi. 62. angle with the line of sight, which passes through a slit on one side A of the box, and a vertical hair stretched across the opening on the other side, or a mark on the glass. The box is held in the hand over the spot wihere the perpendicular is desired, (a plumb line in the hand will give perfect accuracy) and _ the observer applies his eye to the slit A, looking through the upper or unsilvered part of the glass, and turns the box till he sees the other D end of the line B, through the opening C. The assistant, with a rod, moves along in the direction where the perpendictlar is desired, being seen in the silvered parts of the glass, by reflection through the opening D, till his rod, at E, is seen to coincide with, or to be exactly under, the object B3. Then is the line DE at right angles to the line AB, by the optical principle of the equality of the angles of incidence and reflection. To find where a perpendicular from a distant object would strike the line, walk along the line, with the instrument to the eye, till the image of the object is seen, in the silvered part of the glass, to coincide with the direction of the line seen through the unsilvered part. The instrument may be tested by sighting along the perpendicular, and fixing a point in the original line; on the principle of " Reversion." The surveyor can make it for himself, fastening the glass in the box by four ang alar pieces of cork, and adjusting it by cutting away the cork on one side, and introducing wedges on the other side. The box should be blackened inside. Another form of the optical square contains two glasses, fixed at an angle of 45~' and giving a right angle on the principle of the Sextant. 72 ]IAIN SURVEYINGL [PART IT (108) Chain Perpenduiculars, Perpendiculars may be set out with the chain alone, by a variety of methods. These methods generally consist in performing on the ground, the operations executed on paper in practical geometry, the chain being used, in the ilace of the compasses, to describe the necessary arcs. As these operations, however, are less often used for the method of surveying now to be explained, than for overcoming obstacles to measurement it will be more convenient ) consider them in that connection, in Chapter V. DIAGONALS AND PERPENDICULARS. (109,) In Chapter I, of this Part, we have seen that plats of surveys made with the chain alone, have their contents most easily determined by measuring, on the plat, the perpendiculars of each of the triangles, into which the diagonals measured on the ground have divided the field. In the Mfethod of Surveying by Diagonals and Perpendiculars, now to be explained, the perpendiculars are measured on the ground. The content of the field can, therefore, be found at once, (by adding together the half products of each perpendicular by the diagonal on which it is let fall,) without the necessity of previously making a plat, or of measuring the sides of the field. This is, therefore, the most rapid and easy method of surveying when the content alone is required, and is particularly applicable to the measurement of the ground occupied by crops, for the purpose of determining the number of bushels grown to the acre, the amount to be paid for mowing by the acre, &c. (110) A three-sided feld., Measure the Fig. 63. longest side, as AB, and the perpendicular, AX CD, let fall on it from the opposite angle C. \ j Then the content is equal to half the product of the side by the perpendicular. If obsta3- E 3' cles prevent this, find the point, where a perpendicular let fall from an angle, as A, to the opposite side produced, as BC, would meet it, as at E in the figure. Then half the product of AE by CB is the content of the triangle. CHAP., II.] Diagonals and Perpendiculare 73 (111) A four-sided ield, Fig. 64. Measure the diagonal AC. Leave marks at the points on this diago- nal at which perpendiculars from B A - ---- --- 4 and from D would meet it; find- ing these points by trial, as previously directed in Arts. (104) and (107). The best marks at these B' False Stations," have been described in Art. (90). Return to these false stations and measure the perpendiculars. When these perpendiculars are measured before finishing the measurement of the diagonal, great care is necessary to avoid making mistakes in the length of the diagonal, when the chainmen return to continue its measurement. One check is to leave at the mark as many pins as have been taken up by the hind-chainman in coming to that point from the beginning of the line. Example 9. Required the content of the field of Fig. 64. Ans. OA. 2R. 29P. The field may be platted from these measurements, if desired, but with more liability to inaccuracy than in the first method, in which the sides are measured. The plat of the figure is 3 chains to 1 inch. The field-notes may be taken by writing the measurements on a sketch, as in the figure; or in more complicated cases, by the column method, as below. A new symbol may be employed, this mark,!, or, to show the False Station, from which a perpendicular is to be measured. Example 10. Calculation. mF;^ 1110 toB sq. lcs. Fromn 200 on 480 F. S. *q ABC -- x 480 x 110 = 26400 ADC- 1 x 480 x 175 42000 175 [to ____ rom 280 on 480 F.. sq. chains 6.8400 480 to C e - d i~2 O 480 lto C MAres m0.68 280 It is still easier to take the two -1 200 triangles together; multiplying From A (0 the diagonal by the sum of the perpendiculars and dividing by two. 74d CHIN SUR$VEYI TG, [PARTT II (112) A many-sided field. Fig. 65, and the accompanying field-notes represent the field which was surveyed by the Firsi Method and platted in Fig. 51. Fig. 65. elI From 160 on 7.75 F. S. ABC = 114252457 Examplem 55 on 11.2. Saleula.ono -j 5.07 or, A..R. 22P. 1... -i —o l The content of the triangles may 1.60 have been taken together, as hi 5.07 on 7.37 F.8.___ be expressed thus: _ From C 0 r the previous field. 2.53 to1D sq.7' k. F^rom 1.60 on 7.75 F. S. ABC = x 1142 x 267=152457 -~- ~~ 93- to AEC = X1142x493=281503 g 5ro.m.45 on 11.42 F t..Co t c a= x l75x253 f 8037 From 4.95 on 11.42 F.._ chains 58.8746 r o. 0 have slieen talken together, ais ib From C A the previous held. 11.42 to A Content calculated from thp 1 495 F perpendiculars will generally var F'rom A ry slightly from that obtained by measuring on the plat. CHAP. ii.] Offsets, 75 (113) A small field which has many sides, may sometimes be conveniently surveyed by taking one diagonal and measuring the perpendiculars let fall on it from each angle of the field, and thus dividing the whole area into triangles and trapezoids; as i Fig. 36, page 48. The line on which the perpendiculars are to be let fall, may also be outside of the field, as in Fig. 37, page 48. Such a survey can be platted very readily, but the length of the perpendiculars renders the plat less accurate. This procedure supplies a transition to the method of " Offsets," which is explained in the next article. OFFSETS. (114) Offsets are short perpendiculars, measured from a straight line, to the angles of a crooked or zigzag line, near which the straight line runs. Thus, in the figure, Fi. g66 let ACDB be a crooked fence, D bounding one side of a fields Chain A. — l —-- - along the straight line AB, which runs from one end of the fence to the other, and, when opposite each corner, note the distance from the beginning, or the point A, and also measure and note the perpendicular distance of each corner C and D from the line. These corners will then be " determined" by the Second Mlethod, Art. (6). The Field-notes, corresponding to Fig. 66, are as in the margin. The measure-__ ments along the line are written in the 0 3001toB. column, as before, counting from the be- ginning of the line, and the offsets are Di 25 250 written beside it, on the right or left, oppo- C1 30 100 site the distance at which they are taken. A sketch of the crooked line is also usually From A 0 0 made in the Field-notes, though not absolately necessary in so simple a case as this. The letters C and D would not be used in practice, but are here inserted to show the connection between the Field-notes and the plat. 76 CHAIN URVEYING, [PART HI In taking the Field-Notes, the wicths of the offsets should not be drawn proportionally to the distances between them, but the breadths should be greatly exaggerated in proportion to the lengths. (115) A more extended example, with a little different notation, is given below. In the figure, which is on a scale of 8 chains to one inch for the distances along the line, the breadths of the offsets are exaggerated to four tines their true proportional dimensions. Fig. 67. B 1500 0 1250 20 0 1000 0 30 750 50 500 40 250 0 A (116) The plat and Field-notes of the position of two houses, determined by offsets, are given below on a scale of 2 chains to 1 inch. Fig. 68. 250 to B 80 185 30 201 150 90 1OnL 3050 From A. A (i17) Double offsets are sometimes convenient; and sometimes triple and quadruple ones. Below are given the notes and the plat, 1 chain to 1 inch, of a road of varying width, both sides of which are determined by double offsets. It will be seen that the line AB crosses one side of the road at 160 links from A, and the other side of it at 220. CHAP. II.] Offsets. 77 Two methods of keeping the Field-notes are given. In the first form, the offsets to each side of the road are given separately and connected by the sign +. In the second form, the total distance of the second offset is given, and the two measurements connected by the word I to." This is easier both for measuring and platting. Fie. 69 i. \;: 260 30+60 260 30 to 90 240 10+70 240 10 to 80 0 220 50 0 220 50 20 200 30 20 200 30 40 180 10 40 180 10 45 1O0 0 45160 0 50+ 0 140 5to 0 140 55+ 5 120 60to 5 120 50+20 100 70 to20 100 45+15 80 60 to 1 80 50+10 60 60 to10 60 50 +20 40 70 to 20 40 55+20 20 75 to 20 20 60+ 0 A 60 to0 A (118) These offsets may generally be taken with sufficient accuracy by measuring them as nearly at right angles to the base line as the eye can estimate. The surveyor should stand by the chain, facing the fence, at the place which he thinks opposite to the corner to which he wishes to take an offset, and measure " square" to it by the eye, which a little practice will enable him to do with much correctness. 78 CGAIN SURVEYING, [PART II. The offsets may be measured, if short, with an Offset-staff a light stick, 10 or 15 links in length, and divided accordingly; or if they are long, with a tape. They are generally but a few links in length. A chain's length should be the extreme limit, as laid down by the English " Tithe Commissioners," and that should be employed only in exceptional cases. When the " Cross-staff" is in use, its divided length of 8 links, renders the offset-staff needless. When offsets are to be taken, the method of chaining to the end of a line, described in Art. (23), page 21, is somewhat modified. After the leader arrives at the end of the line, he should draw on the chain till the follower, with the back end of the chain, reaches the last pin set. This facilitates the counting of the links to the places at which the offsets are taken. The offsets are to be taken to every angle of the fence or other crooked line; that is, to every point where it changes its direction. These angles or prominent bends can be best found by one of the party walking along the crooked fence and directing another at the chain what points to measure opposite to. If the line which is to be thus determined is curved, the offsets should be taken to points so near each other, that the portions of the curved line lying between them may, without much error, be regarded as straight. It will be most convenient, for the subsequent calculations, to take the offsets at equal distances apart along the straight line from which they are measured. In the case of a crooked brook, such as is shown in the figure given below, offsets should be taken to the most prominent angles, such as are marked a a a in the figure, and the intermediate bends may be merely sketched by eye. Fig. 70. a When offsets from lines measured around a field are taken inside of these bounding lines, they are sometimes distinguished as Insets. CHAP. iiI.] Offsets, 79 (119) Platting. The most rapid method of platting the offsets, is by the use of a Platting Scale (described in Art. 49) and an Offset Scale, which is a short scale divided on its edges like a platting scale, but having its zero in the middle, as in the figure. Fig. 71 CIA The platting scale is placed parallel to the line, with its zero point opposite to the beginning of the line. The offset scale is slid along the platting scale, till its edge comes to a distance on the latter at which an offset had been taken, the length of which is marked off with a needle point from the offset scale. This is then slid on to the next distance, and the operation is repeated. If one person reads off the field-notes, and another plats, the operation will be greatly facilitated. The points thus obtained are joined by straight lines, and a miniature copy of the curved line is thus obtained; all the operations of the platting being merely repetitions of the measurements made on the ground. If no offset scale is at hand, make one of a strip of thick drawing paper, or pasteboard; or use the platting scale itself, turned crossways, having previously marked off from it the points from which the offsets had been taken. In plats made on a small scale, the shorter offsets are best estimated by eye. On the Ordnance Survey of Ireland, the platting of offsets is facilitated by the use of a combination of the offset scale and the platting scale, the former being made to slide in a groove in the latter, at right angles to it. (120) Calcilating Clotehnt When the crooked line determined by offsets is the boundary of a field, the content, enclosed 80 sCHAIN SURVEYING, [PART II oetween it and the straight line surveyed, must be determined, that it may be added to, or subtracted from, the content of the field bounded by the straight lines. There are various methods of effecting this. The area enclosed between the straight and the crooked lines is divided up by the offsets into triangles and trapezoids, the content of which may be calculated separately by Arts. (65) and (67), and then added together. The content of the plat on page 759 will, therefore, be 1500 + 4125 + 625 = 6250 square links = 0.625 square chain. The content of the plat on page 76, will in like manner be found to be, on the left of the straight line 30,000 square links, and on its right 5,000 square links. (121) When the offsets have been taken at equal distances, the content may be more easily obtained by adding together half of the first and of the last offset, and all the intermediate ones, and multiplying the sum by one of the equal distances between the offsets. This rule is merely an abbreviation of the preceding one. Thus, in the plat of page 76, the distances being equal, the content of the offsets on the left of the straight line will be 120 x 250 - 30,000 square links, and on the right 20 x 250 = 5,000 square links; the same results as before. When the line determined by the offsets is a curved line, " Simpson's rule" gives the content more accurately. To employ it, an even number of equal distances must have been measured in the part to be calculated. Then add together the first and last offset, four times the sum of the even offsets, (i. e. the 2d, 4th, 6th, &c.,) and twice the sum of the odd offsets, (i. e. the 3d, 5th, 7th, &c.,) not including the first and the last. Multiply the sum by one of the equal distances between the offsets, and divide by 3. The quotient will be the area. Example 12. The offsets from a straight line to a curved fence, were 8, 9, 11, 15, 16, 14, 9, links, at equal distances of 5 links. What was the content included between the curved fence and the straight line? Ans. 371.666 CHAP III.] Offsets. 81 (122) Many erroneous rules have been given on this part of the subject. One rule directs the surveyor to divide the sum of all the offsets by one less than their number, and multiply the quotient by the whole length of the straight line; or, what is the same thing, to multiply the sum of all the offsets by the common distance between them. This will be correct only when the offsets at each end of the line are nothing, i. e, when the curved line starts from the straight line and returns to it at the beginning and end of one of the equal distances. In all other cases it will give too much. A second rule directs the surveyor to divide the sum of all the offsets by their number, and then to multiply the quotient by the whole straight line. This may give too much, or too little, according to circumstances. Suppose offsets of 10, 30, 20, 80, 50, 30, links, to have been taken at equal distances of a chain. The correct content of the enclosed space is 200 x 100 = 2 square chains. The first of the above rules would give 2.2 square chains, and the second would give 1.8333 chains. (123) Reducing to one triangle the many-sided figure which is formed by the offsets, is the method of calculation sometimes adopted. This has been fully explained in Part I, Art. (78), &c. The method of Art. (83) is best adapted for this purpose. (124) Equalizing, or giving acld takling, is an approximate mode of calculation much used by practical surveyors. A crooked line, determined by offsets, having been platted, a straight line is drawn on the plat, across the crooked line, leaving as much space outside of the straight line as inside of it, as nearly as can be estimated by the eye, " Equalizing" it, or' Giving and taking" equal Fig. 72. portions. The straight line is best determined by laying across the irregular outline the straight edge of a piece of transparent horn, or tracing paper, or glass, or a fine thread or horse-hair 6 82 GICAIN SURVEYING, [PART IL stretched straight by a light bow of whalebone. In practical hands, this method is sufficiently accurate in most cases. The stu. dent will do well to try it on figures, the content of which he has previously ascertained by perfectly accurate methods. Sometimes this method may be advantageously combined with the preceding; short lengths of the croooked boundary being " Equalized," and the fewer resulting zigzags reduced to one line by the method of Art. (78), &c. CHAPTER IV. SURVEYITNG BY TIEE PRECEDING iIETI iODS (4 BINBEDB 125) All the methods which have been explained in the three preceding chapters-Surveying by Diagonals, by Tie-lines, and by Perpezdiculars, particularly in the form of offsets-are frequently required in the same survey. The method by.Diagonals should be the leading one; in some parts of the survey, obstacles to the measurement of diagonals may require the use of Tie-lines; and if the fences are crooked, straight lines are to be measured near them, and their crooks determined by Offsets. (1g2) Offsets are necessary additions to almost every other method of surveying. In the smallest field, surveyed by diagonals, unless all the fences are perfectly straight lines, their bends must be determined by offsets. The plat (scale of 1 chain to 1 inch), and field-notes, of such a case are given below. A sufficient num CHAP. iv.] Diagonals, Tie-lines and Offsets, 83 ber of the sides, diagonals, and proof-lines, to prove the work, should be platted before platting the offsets. Fig. 73. B 3. 60. C -- "..... 2 - 4/ D C B 0 360 8z 340 6 315 D = 10 275 si 5 215 - C 0 150 0 31o 115 10 A F 80 5 A 65 8 01 248 B or A 11 180 _ 0 105 0 B 65 5 0 125 D Or Li 11 90 23 62 0 13 12 22 15 110 0 A Li 13 90 _- -~_ __ __ _ ( 0 50 0 -Example B3. Required the con- 30 9 tent of the above field. Ans. C 0 r (127) Field-books, The difficulty and the importance of keep ing the Field-notes clearly and distinctly, increase with each new combination of methods. For this reason, three different methods of keeping the Field-notes of the same survey will now be given, (from Bourns' Surveying), and a careful comparison by the student of the corresponding portions of each will be very profitable to him. 84 CRAIN SURVEYINT [PART IT FIELD-BOOK No. 1. )ioo,,.,5/ /o 0{ tot., k No.} I i.7)e'teStc mehd. ao.,20 in 6o o A 9.. \ ".... -/ -% / V ildBook No. 1 (Fig. 74) shews the Sketch methoI explai id mAro (91) X^ ^^c~,;; \ ^\ \ /"^ \f \\1^ -~ \9 \^ y~~~~~i /~~~~c \ -^ ^~~~'-\^ ^)dtq \ 1 0 MecroQ ^ Fg 4 hesteSec ehd xli ed in Art, (94).~~~~~~~~~~~~~~: CHAP. I. Diagonals, Tielines and Offsets. 8 FIELD-B00K No. 2.,4570 A 4-080 390 &00 18300 It~o / 4000) ~ 1d lt 1250 3 e65 1200 2 480 / 1020 200'3060 / O 680 190 2300 c V-4 10 / X 1390 1 -160 - 1121 )'{ 3o100 1 - 60 ~ p100 i400' 620 zo 0'60 20 580 f~roA 0 fo X x A Field-Book No. 2 (Fig. 75) shews the Coolum metfod, ex~lain. ed in Art. (95). a6 CHAINl SURVEYINg. LPaA FIELD-BOOK No, 30 1120 700 130 -:7 i "'/ ~\1 Qa',00 I 6i0! 0'9,,' 20 G15 X OTh s 0fi0 20 580 A 00fr Fld Bok No I ( )'sI cI I t ^\^\~^/ 102/ /50 4i H \/v thirs figure, at A. CHAP. IV.] Diagonals, Tie-lines and Offsets, 87 (128) It will easily appear from the sketch of Field-book No. 1, how much time and labor may be saved, or lost, by the manner of doing the work. Thus, beginning at A, and measuring 750 links, a pole should be left there, and the line to the right measured tc 17 chains, or C, leaving a pole at 12.30 as a new starting point by and by. Then from C measure 1.9 chains to A again; then mea, sure from A to B, and from B back to the pole left at 7.50 on the main line. (129) The example which will now be given shows part of the Field-notes, the plat, (on a scale of 6 inches to 1 mile [1:10,560]), and a partial calculation of the "Filling up" of a large triangle, the angular points of which are supposed to have been determined by the methods of Geodesic Surveying. They should be well studied.* Fig. 77. D 7\j III UJ H__WTTT\\^ ^.M \__\S C apt. FROME, in his "Trigonometrical Survey," fiom which this example has been condensed, remarks, " It may, perhaps, be thought that too much stress is laid on forms; but method is a most essential part of an undertaking of magnitule: and without excellent preliminary arrangements to ensure uniformity in all the most trifling details, the work never could go on creditably." 88 CHAIN SURVEYING1 [PART IX _____________ G~C 2564 180 D A A F 2452 ^ z a 1;oo00 4050 Q 1700 0 62 8890 N 2324 1530 84 42 3730 iq 23~ ~ 1420 40 0 3540 0 0 1340 0 3420 30 H 1264 0 FronmA A to D A _'72 2484 S 1240 52 A - 1140 86 A 40 2332 950 100 6 2206 772 60 0 2056 0 0 684 0 1805 40 34 502 329 0 1550 50 50 450 ~ I 70 342 3275 54 X 1442 0 82 220 3120 62 FromC A to A 2940 85 FromD A to C.; C 2572 60 D In the above specimen of a field-book, (which resembles that on page 85), all offsets, except those having relation to the bomndary lines, are purposely omitted, to prevent confusion, the example being given solely to illustrate the method of calculating these larger divisions. Rough diagrams are drawn in the field-book not to any scale, but merely bearing some sort of resemblance to the lines measured on the ground, for the purpose of showing, at any period of the work, their directions and how they are to be connected; and also of eventually assisting in laying down the diagram and content plat. On these rough diagrams are written the distinctive letters by which each line is marked in the field-book, and also its length, and the distances between points marked upon it, from which other measurements branch off to connect the interior portions of the district surveyed. (130) Calculations. The calculation of one of the figures, U, is given below in detail. It is composed of the triangle DPQ, with offsets along the sides PQ; and of the triangle DWX, with offsets CHAP. IV.] Dlagonals, Tie-lies and Offsets. 89 along the sides PW. and WX. From the content thus obtained must be subtracted the offsets on PQ, belonging to the' figure 7, and those on WX belonging to the figure JK. When the offsets are triangles, (right angled, of course), the base and perpendicular are put down as two sides; when they are trapezoids, the two parallel sides and the distance between them occupy the columns of" sides." TRIANGIE IST 2D CONTENT DIVISION. OR SIDESIDESIDE IN TRAPEZOID.. CHAINS. DPQ 1168011698 1078 86.2650 ( 52 250.6500 ) PQ 52 30 80.3280 ( 30 216.3240) ~ I~ I ~~l I~ 1.3020 Additives. DWX 1370 1442 770 51.8339 PW 30 310.4650 56 114.3192 WX 56 36 104.4784 ( 36 90.1620).9596 Total Additives, 140.8255 ~ -50 174.4350 PQ 50 30 292 1.1'680 ( 30 66.0990) 1.7020 I ( --- 52 142.3692 Subtractives. WX 52 64 232 1.3456 ( 64 - 88.2816) 1.9964 Total Subtractives, 3.6984 Total Additives, 140.8255 Difference, 137.1271 90 (CHI SURVEYING. [PART II. The other figures, comprised within the large triangle, are record. ed and calculated in a similar manner. An abridged register of the results is given below. DIFFERENCE IN DIVISION. ADDITIVES. SUBTRACTIVE. RECHIN SQUARE CHAINS. DNS DNS DWX 5 / and offsets. oNUV 140.4893 and offsets. [ 1 0DNO DPQ 1001882 and offsets. and offsets. ANO NRM 1 ____ and offsets. and offsets. 10 HTN I ] NRM Offsets. 81.6307 and offsets. a CNS HTN CNS H}sets, ~ 109.5064 and offsets. and offsets. } DPQ 11 ^ DWX Offsets. 137.1271 and offsets. Total,. 672.9195 The accuracy of the preceding calculations of the separate figures must now be tested by comparing the sum of their areas with that of the large triangle ACD, which comprises them all. Their a'rea must previously be increased by the offsets on the lines CS and CH, which had been deducted from a, and which amount respectively to 3.5270 and 2.8690. The total areas will then equal 679.3155 square chains. That of the triangle ACD is 679.5032; a difference of less than a fifth of a square chain, or a fiftieth of an acre; or about one-fortieth of one per cent. on the total area. (13I) The six lines. In most cases, great or small, szx fundamental lines will need to be measured; viz. four approximate boundary lines, forming a quadrilateral, and its two diagonals. Small triangles, to determine prominent points, can be formed within and without these main lines by the FIRST METHOD, Art. (5), and the lesser irregularities can be determined by offsets. CUP. iv.] Diagonals, Tie-lines and Offsets. 91 Fig. 78. A TET 7 B... 7 / ^^^i - \ ^ 1^^^^r I D /. I Thus in he above figure, two ght lines A and CD are Thus, in the above figure, two straight lines AB and CD are measured through the entire length and breadth of the farm, or township, which is to be surveyed. The connecting lines AC, OB, BD and DA are -also measured, uniting the extremities of the first two lines. The last four lines thus form a quadrilateral, which is divided into two triangles by one of the first measured lines, while the second serves as a proof-line. The distance from the intersection of the two diagonals to the extremities of each, being measured on the ground and on the plat, affords an additional test. Other points of the district surveyed (as E, G, K., &c., in the figure,) are determined by measuring the distances from them to known points (as M, N, P, R, &c., in the figure) situated on some of the six fundamental lines, thus forming the triangles T, T. The intersection 0 of the main diagonals, and also the intersections of the various minor lines with the main lines and with each other, should all be carefully noted, as additional checks when the work comes to be platted. 9~ tCHlIN (SlRVgYING. [PART It The larger figures are determined first, and the smaller ones based upon them, in accordance with this important principle in all surveying operations, always to work from the whole to the parts, and from greater to less. The unavoidable inaccuracies are thus subdivided and diminished. The opposite course would accumulate and magnify them. These additional lines, which form secondary triangles, should be so chosen and ranged as to pass through and near as many obh jects as possible, in order to require as few and as short offsets as the position of the lines will permit; the smaller irreguarities being determined by offsets as usual. It is better to measure too many lines than too few, and to establish unnecessary " false stations," rather than not to have enough. (132) Exceptional cases. The preceding arrangement of lines, though in most cases the best, may sometimes be varied with advantage. Unless the farm surveyed be of a shape nearly as broad as long, the two diagonals will cross each other obliquely, instead of nearly at right angles, as is desirable. Fig. 79 When the'farm is much T3 longer than it is wide, two systems, of six lines each, ~./ may be used with much advantage, as in Fig. 79. E)'"t J/'i j," Several such may be com- \ j) bined when necessary. lg. 80. In a case like that in Fig.'. -- 80, five lines will be better than six, and will tie one an- \ other together, their points of intersection being carefully \ noted, cn xv~o ] Diagonals, Tie-lines and Offsets. 93 Fig. 81 In the farm represented in (, _. Fig. 81, the system of lines', \ there shown is the best, and \.__~.'. they will also tie one another. i =B1 ^ (133) luch difficulty will often be found in ranging and mea. suring the long lines required by this method in extensive surveys. Various contrivances for overcoming the obstacles which maybe met with, will be explained in the following chapter. It will often be convenient to measure the minor lines along roads, lanes, paths, &c., although they may not lie in the most desirable directions. Steeples, chimneys, remarkable trees, and other objects of that character, may often be sighted to, and the line measured towards them, with much saving of time and labor. The point where the measured lines cross one another should always be noted, and they will thus form a very complete series of tie-lines.* A view of the district to be surveyed, taken from some elevated position, will be of much assistance in planning the general direction of the lines to be measured. (134) lnaccesslte Areas. Fig. 82. A combination of offsets and,A\ --- tie-lines supplies an easy me-. —-. thod of surveying an inaccessible area, such as a pond, swamp, forest, block of houses, 1 &c., as appears from the figure; in which external bound-'' - ing lines are taken at will and -- * To find the exact point of intersection of these lines, which are only' visual lines," (explained in Art. (19),) three persons are necessary: one stands at some point of one of the lines and sights to some other point on it; a second does the same on the second line; by signs they direct, to right or left, the movements of a third person, who holds a rod, till he is placed in both of the lines and thus at their intersection, on the principle of Art. (1I). 94 CHAIN SURVEYING, [PART I. measured, and tied by " tie-lines" measured between these lines, prolonged when necessary, as in Art. (101), while offsets from them determine the irregularities of the actual boundaries of the pond, &c. These offsets are insets, and their content is, of course, to be subtracted from the content of the principal figure. Even a circular field might thus be approximately measured from the outside. If the shape of the field admits of Fig. 83. it, it will be preferable to measure /- -! four lines about the field in such directions as to enclose it in a rect -- it angle, and to measure offsets from the - / sides of this to the angles of the L -"I'; field. (135) When one of the lines with which Fig. 84 an inaccessible field is surrounded, as in] the last two figures, cuts a corner of the -- field, as in Fig. 84, the triangle ABC is c/ to be deducted from the content of the enclosing figure, and the triangle CDE m added to it. The triangle DEF is also D — 4 —to be added, and the triangle FGH deducted. To do this directly, it would be necessary to find the points of intersection,. AP C and F. But this may be difficult, and can be dispensed with by obtaining the difference of each pair of triangles. The ~p; difference of ABC and CDE will be obtained at once by multiplying the differ-....-iT. ence of the offsets AB and DE by half of BE; and the difidrence of DEF and FGH by multiplying the difference of DE and Giby half of EG.* * For, making the triangle Dmn == ABC, then mnEC = En X A (man -+ CE) (DE - AB) X A EB; and so with the other pair of triangles. CAP. Iv.] Diagonals, Tie-lines and Perpendiculars. 95 (136) Roadso A winding Road may also be surveyed thus, as is shown in Fig. 85; straight lines being measured in the road, Fig. 85.'i' -' \\ @, their changes in direction determined by tie-lines, tying one line to the preceding one prolonged, as explained in Chapter II, of this Part, and points in the road-fences, on each side of these straight lines, being determined by offsets. A River may also be supposed to be represented by the above winding lines; and the lower set of lines, tied to one another as before, and with offsets from them to the water's edge, will be sufficient for making an accurate survey of one side of the river. (137) Towns. A town could be surveyed and mapped in the same manner, by measuring straight lines through all the streets, determining their angles by tie-lines, and taking offsets from them to the blocks of houses. 96 CHAIN SUlRtVEYING. [PART I1 CHAPTER V. OBSTACLES TO MEASURElMENT IN CiAIN SURVEYISNG (138) In the practice of the various methods of surveying which have been explained, the hills and valleys which are to be crossed, the sheets of water which are to be passed over, the woods and houses which are to be gone through-all these form obstacles to the measurement of the necessary lines which are to join certain points, or to be prolonged in the same direction. Many special precautions and contrivances are, therefore, rendered necessary; and the best methods to be employed, when the chain alone is to be used, will be given in the present chapter. (139) The methods now to be given for overcoming the various obstacles met with in practice, constitute a LAND-GEOMETRY. Its problems are performed on the ground instead of on paper: its compasses are a chain fixed at one end and free to swing around with the other; its scale is the chain itself; and its ruler is the same chain stretched tight. Its advantages are that its single instrument, (or a substitute for it, such as a tape, a rope, &c.) can be found anywhere; and its only auxiliaries are equally easy to obtain, being a few straight and slender rods, and a plumb-line, for which a pebble suspended by a thread is a suficient substitute. Many of these problems require the employment of perpendicular and parallel lines. For this reason we will commence with this class of Problems. The Demonstrations of these problems will be placed in an Appendix to this volume, which will be the most convenient arrangement for the two great classes of students of surveying; those who wish merely the practice without the principles, and those who wish to secure both. Tlhe elegant " Theory of Transversals' will be an important element in some of these demonstrations. All of them will constitute excellent exercises for students. CHAP. V.] Obstacles to Measuremento 97 PROBLEM S ON PERPENDICULARS* Problem 1. To erect a perpendicular at any point of a?ine. (140) First Method. Let A be the Fig. 86. point at which a perpendicular to the line is ^ to be set out. Measure off equal distances AB, AC, on each aide of the point. Take _ a portion of the chain not quite 1 times as kiag as AB or AC, fix one end of this at 1B, and describe an arc with the other end.' Do the same from C. The intersection of these arcs will fix a point D. AD will be the perpendicular required. Repeat the operation on the other side of the line. If that is impossible, repeat it on the side with a different length of chain. (141) Second dlethod. Mleasure off as be- Fi. 87 fore, equal distances AB, AC, but each about only one-third of the chain. Fasten the ends of the chain with two pins at B and C. Stretch A3 it out on one side of the line and put a pin at the middle of it, D. Do the same on the other N side of the line, and set a pin at E. Then is DE a perpendicular to BC. If it is impossible to perform the operation on both sides of the line, repeat it on the'same side with a different length of chain, as shown by the lines BF and CF in the figure, so as to get a second point. (142) Other Methods. All the methods to be given for the next problem may be applied to this. Many of these methods would seldom be required in practice, but cases some. times occur, as every surveyor of much experience in Field-work has found to his serious inconvenience, in which some peculiarity of the local circumstances forbids any of the usual methods being applied. In such cases the collection here given will be found of great value. In all the figures, the given and measured lines are drawn with fine full lines, the visual lines, or lines of sight, with broken lines, and the lines of the result with heavy full lines. The points which are centres around which the chain is swung, are enclosed in circles. The alphabetical order of the letters attached to the points shows in what order they are taken. 7 98 CHAIN SURVEYING, [PART II Prioblem 2. To erect a perpendicular to a line at a given point, when the point is at or near the end of the line. (g14) First Miethod. Measure Fig. 88. 50 40 links along the line. Let one assistant hold one end of the chain at that point; let a second hold the 20 link mark which is nearest the other end, at the given point A, and let a 40 A- - third take the 50 link mark, and 0 20 tighten the chain, drawing equally on both portions of it. Then will the 50 link mark be in the perpendicular desired. Repeat the operation on the other side of the line so as to test the work. The above numbers are the most easily remembered, but the longer the lines measured the better; and nearly the whole chain may be used, thus: Fix down the 36th link from one end at A, and the 4th link from the same end on the line at B. Fix the other end of the chain also at B. Take the 40th link mark from this last end, and draw the chain tight, and this mark will be in the perpendicular desired. The sides of the triangle formed by the chain will be 24, 32 and 40. (144) Otherwise: using a 50 feet ig89. tape, hold the 16 feet mark at A; hold the 48 feet mark and the ringend of the tape together on the line; take the 28 feet mark of the tape, and 0 / draw it tight; then will the 28 feet B 16 A mark be in the perpendicular desired. o 0 (145) Second Method. Hold one end Fig. 90. of the chain at A and fix the other end at a point B, taken at will. Swing the chain around B as a centre, till it again meets the line at C. Then carry the same end around B (the other end remaining at B) till it comes in the line of CB at D. AD is the perpen- __ dicular required.'. — cKAP. v.] ODstacles to Measurements 99 (146) T71rd Method. Let A be the given Fig. 91. point. Choose any point B3. Measure BA. Set off, on the given line, AC = AB. On CB 2 AC2 3 produced set off from C, a distance C= B This will fix the point D, and AD will be the perpendicular required. — CA (147) Fourth JMethod. From the Fig. 92. given point A set off on the given line li P any distance AB. From B, in any convenient direction, set off BC = AB. T- A Then on the given line, set off AD = AC. On CB prolonged, set off CE = C AD. Join DE; and on DE, from D, set off DF 2 AB. Then will the line AF be perpendicular to the line AD at the point A. Problem 3, To erect a perpencdicular to an inaccessible line, at a given point of it. (148) Tirst Mfethod. Get points in the direction of the inaccessible line prolonged, and from them set out a parallel to the line, by methods which are given in Art. (16t), &c. Find by trial the point in which a perpendicular to this second line (and therefore to the first line) will pass through the required point. (149) Second Method. If the line is not only inaccessible, but cannot have its direction prolonged, the desired perpendicular can be obtained only by a complicated trigonometrical operation. Problem 4, To let fall a perpendicular from a given point t a given line. (150) Pirst iethod. Let P be Fig. 93. the given point, and AB the given line. Measure some dlstance, a chain or less, from C to P, and then fix one end of the chain at P, and swing it.A - around till the same distance meets - 100 CHAIN SURVEYING. [PART I, the line at some point D. The middle point E of the distance CD will be the required point, at which the perpendicular from P would meet the line. (151) Second Jetflod. Stretch a chain, or a portion of it, from the given point P, to some point, as A, of the Fig. 94. given line. Hold the end of the distance at AS <, and swing round the other end of the chain from P, so as to set off the same distance along the given line from A to some point B. Mea- -~ ~ sure BP. Then will the distance BC from B to the foot of the BP2 desired perpendicular =- AB (152) Other Methods. All the methods given in the next problem can be applied to this one. Problem 5, To let fall a perpendicular to a line, from a point nearly opposite to the end of the line. (153) First Method. Stretch a chain from the given point P, to some point, as A, of the given line. Fix to Fig. 95. the ground the middle point B of the chain P AP, and swing around the end which was at B]'\ P, or at A, till it meets the given line in a / point C, which will be the foot of the re- quired perpendicular. A. / (154) Second Method. Take any point, Fig. 96 e as A, on the given line. Measure a dis- tance AB. Let the end of this distance on the chain be held at B, and swing around the end of the chain, till it comes in the B i j line of AP at some point C, thus making BC = AB. Measure AC and AP. Then the distance AD, from A to the foot of the O.,APXAC perpendicular required = 2 AB CHAP. v.] Obstacles to leasurement. 101 (155) Third Method. At any convenient Fig. 97. P point, as A, of the given line, erect a perpen- / dicular, of any convenient length, as AB, and mark a point C on the given line, in the line / of P and B. Measure CA, CB and CP. C - fk D Then the distance from C to the foot of the perpendicular, i. e. CD = CA Problem 6. To let fall a perpendicular to a line, from an inaccessible point. (156) First Method. Let P be the gien Fig. 98. point. At any point A, on the given line, set out a perpendicular AB of any convenient length. Prolong it on the other side of the line the., same distance. Mark on the given line a /' - / point D in the line of PB; and a point E in D'~, ~T7 A,- the line of PC. Mark the point F at the in- \ tersection of DC and BE prolonged. The line c v"' FP is the line required, being perpendicular a to the given line at the point G. (157) Second Method. - Let A and B Fi 99 P be two points of the given line. From A let fall a perpendicular, AC, to the visual line BP; and from B let fall a perpendi- f cular, BD, to the visual line AP. Find the point at which these perpendiculars intersect, as at E (seeArt.(133)), and the " line PE, prolonged to F, will give the perpendicular required. Probleam T To let fall a perpendicular from a given point to an inaccessible line. 102 CHAIN SURVEYING. [PART 1 (158) First MZethod. Let P be Fig. 100. the given point and AB the given A --—, —, a line. By the preceding problem, let fall perpendiculars from A to BP, at C; and from B to AP, at D; the i I line PE, passing from the given point P to the intersection of these perpendiculars, is the desired perpendicular to the inaccessible line AB. This method will apply when only two points of the line are visible. (159) Second Method. Through the given point, set out, by the methods of Art. (1.5), &c., a line parallel to the inaccessible line. At the given point erect a perpendicular to the parallel line, and it will be the required perpendicular to the inaccessible line. PROBLEMS ON PARALLELS. Problem 1, To run a line, from a gzven point, parallel to a given line. (160) Pirst Miethod. Let fall a perpendicular from the point to the line. At another point of the line, as far off as possible, erect a perpendicular, equal in length to the one just let fall. The line joining the end of this line to the given point will be the parab lel required. (1H1) Second Dethod. LetAB be Fg. 101. D C the given line, and P the given point. A - Take any point, as C, on the given line, and from it set off equal distances, as long as possible, CD on the given line, and CE, on the line CP. Measure P DE. From P set off PF =- CE; and from F, with a distance - DE, and from P, with a distance = CD, describe arcs intersecting in G. PG will be the parallel required. If it is more convenient, PC may be prolonged, and the equal triangle, CDE, be formed on the other side of the line AB. cap. v.] Obstacles to Measurement. 103 (I62) Third Mlethod. Measure from Fa. 102. P to any point, as C, of the given line, and An ~ - - put a mark at the middle point, D, of that line. From any point, as E, of the given D line, measure a line to the point D, and continue it till DF -= DE. Then will the line PF be parallel to AB. (163) Fourth Method. Meeasure from Fig. 103. P to any point C, of the given line, and D continue the measurement till CD = OP. From D measure to any point E of the given line, and continue the measurement A ~till EF = ED. Then will the line PF be parallel to AB. If more convenient, CD may be made one-half, or any other fraction, of CP,. and EF be then made twice, &c., DE. (164) Fifth 1M7ethod. From any Fig. 14. point, as C, of the line, set off equal A — B.; —.~-~ distances along the line, to D and E., Take a point F, in the line of PD. p >H Stake out the lines FC and FE, and,/ also the line EP, crossing the line CF' in the point G. Lastly, prolong the line DG, till it meets the line EF in the point H. PH is the parallel required. Problem 2. To run a line from a given point parallel to an inaccessible line. (16t) First Jlethod. Let AB Fig. 105. be the given line, and P the given B point. Set a stake at C, in the line n of PA, and another at any conven- SQ:- B ient point, D. Through P, set out, \ f ",. -. by the preceding problem, a parallel \, to DA, and set a stake at the point, F as E, where this parallel intersects DC prolonged. Through E 104 CHAIN SURVEING. [PART II set out a parallel to BD, and set a stake at the point F, where this parallel intersects BC prolonged. PF is the parallel required (66) eSecond JMethod. Set a stake Fi. 106. at any point, C, in the line of AP, and A another at any convenient place, as at D. Through P set out a parallel to AD,.. ntersecting CD in E. Through E set \.'?\ D out a parallel to DB, intersecting CB in "' — E F. The line PF will be the parallel re- quired. (X67) Alnement and Measurement, We are now prepared, having secured a variety of methods for setting out Perpendi:ulars and Parallels in every probable case, to take up the general subject of overcoming Obstacles to Measurement. Before a line can be measured, its direction must be determined. This operation is called Ranging the line; or Alinig it; or Boning it.* The word Alinementt will be found very convenient for expressing the direction of a line on the ground, whether between two points, or in their direction prolonged. This branch of our subject naturally divides itself into two parts, &he first of which is preliminary to the second; viz: I. Of Obstacles to Allaneent; or how to establish the direction of a line in any situation. I~. Of Obstacles to lIeasirement; or how to find thle length of a line which cannot be actually measured. I. OBSTACLES TO ALINEMENT. (168) All the cases which can occur under this head, may be reduced to two; viz: A. To find points in a line beyond the given points, i. e. to prolong the line. B. To find points in a line between two given points of it, i. e. o interp2olate points in the line. * This word, like many others used in Engineering, is derived from a French word, Borner, to mark out, or limit; indicating that the Normans introduced the lrt of Surveying into E ngland. t Slightly modified from the French Alignement. CHAP, V.1 Obstacles to Measurement, 105 A. TO PROLONG A LINE (169) By ranging with rods, When two points ii a line are given, and it is desired to Fig. 107. prolong the line by ranging h it out with rods, three per- t sons are required, each furnished with a straight slender rod, and with a plumb-line, or other means of keeping their rods vertical. One holds his rod at one of the given points, A, in the figure, and another at B. A third, C, goes forward as far as he can without losing sight of the first two rods, and thei, looking back, puts himself " in line" with A and B, i. e. so that when his eye is placed at C, the rod at B hides or covers the rod at A. This he can do most accurately by holding a plumb-line before his eye, so that it shall cover the first two rods. The lower end of the plumb-bob will then indicate the point where the third rod should be placed; and so with the rest. The first man, at A, is then signalled, and comes forward, passes both the others, and puts himself at D, "' in line" with C and B. The man at B, then goes on to E, and " lines" himself with D and C: and so they proceed, in this " hand over hand" operation, as far as is desired. Stakes are driven at each point in the line, as soon as it is determined. (t70) The rods should be perfectly straight, either cylindrical or polygonal, and as slender as they can be without bending. They should be painted in alternate bands of red and white, each a foot, or link, inlength. Their lower ends should be pointed with iron, and a projecting bolt of iron will enable them to be pressed down by the foot into the earth, so that they can stand alone. When this is done, one man can range out a line. A rod can be set perfectly vertical, by holding a plumb-line before the eye at some distance from the rod, and adjusting the rod so that the plumb-line covers it from top to bottom; and then repeating the operation in a direction at right angles to the former. A stone dropped from top to bottom of the rods will approximately attain the same end. When the lines to be ranged are long, and great accuracy is re quired, the rods may have attached to them plates of tin with opex? 306 CHAIN SURVEYING, [PART I. ings cut out of them, and black horse-hairs stretched from Fig. 108 top to bottom of the openings. A small telescope must then be used for ranging these hairs mrn line. In a hasty survey, straight twigs, with their tops split to receive a paper folded as in the figure, may be used. (171) By perpelediulars. Fi. 109. Tbe straight line, AB in the - - - -- figure, is supposed to be stop- C D E F ped by a tree, a house, or other obstacle, and it is desired to prolong the line beyond this obstacle. From any two points, as A and B, of the line, set off (by some of the methods which have been given) equal perpendiculars, AC and BD, long enough to pass the obstacle. Prolong this line beyond the obstacle, and from any two points in it, as E and F, measure the perpendiculars EG and FH, eoual to the first two, but in a contrary direction. Then will G and H be two points in the line AB prolonged, which can be continued by the method of the last article. The points A and B should be taken as far apart as possible, as should also the points E and F. Three or more perpendiculars, on each side of the obstacle, may be set off, in order to increase the accuracy of the operation. The same thing may also be done on the other side of the line, as another confirmation, or test, of the accu. racy of the prolonged line. (172) By equilateral triangles. Fig. 110 The obstacles, noticed in the last arti- A CI Go cle, may also be overcome by means of three equilateral triangles, formed by the chain. Fix one end of the chain, C and also the end of the first link from its other end, at B; fix the end of the 33d link at A; take hold of the 66th D link, and draw the chain tight, pulling equally on each part, and put a pin at the point thus found, C, in the figure. An equilateral triangle will thus be formed, each side being 33 links. Prolong the line AC, past the obstacle, to some point, as D. Make another cHAeP v.] Obstacles to leasurement, 107 equilateral triangle, DEF, as before, and thus fix the point F. Prolong DF, to a length equal to that of AD, and thus fix a point G. At G form a third equilateral triangle GHK, and thus fix a point K. Then will KG give the direction of AB prolonged. (173) By symmetrical triangles. Let AB be the line to be prolonged. Take any conv- ig. 111. nient point, as C. Range A B <"f>" -., out the line AC, to a point -,'\- - A', such that CA'= CA. \ "'',/ Range out CB, so that CB', -,/ = CB. Range backwards /,.- C',' A'B', to some point D, such / that DC prolonged will pass A the obstacle. Find, by ranging, the intersection, at E, of DB and AC. From C, measure, on CA', the distance CE'G CE. Then range out DC and B'E' to their intersection in P, which will be a required point in the direction of AB prolonged. The symmetrical points are marked by corresponding letters. Several other points should be obtained in the same manner. In this, as in all similar operations, very acute intersections should be avoided as far as possible. (174) By transversals, Let AB be Fig. 112. the given line. Take any two points C v and D, such that the line CD will pass the obstacle. Take another point, E, in the intersection of CA. and DEB. Measure AE, AC, CD, BD and BE. Then the distance from D to P, a point in the required prolongation, will be CDXBDxAE BE x AC-D x AE' Other points in the prolongation may be obtained in the same manner, by merely moving the single point C, in the line of EA; in which case the new distances CA and CD w alone require to be measured. 108 CHItN $SURlVEYI ING LPART aL CDxBD If AE be made equal to AC, then is DP -BE —B- ) CDxAE f BE be made equal to BD, then is DP =- C AE' The minus sign in the denonmnators must be understood as only meaning that the difference of the two terms is to be taken, without regard to which is the greater. (175) By harmonic conjugates. i. 13. Let AB be the given line. Set a V 3 B P stake at any point C. Set stakes at > —, points, D, on the line CA, and at \ " -,''',@- E on the line CB; these points,, \ " \,, D and E, being so chosen that the -,'L' / line DE will pass beyond the obsta- ^ cle. Set a fourth stake, F, at the, i intersection of the lines AE and P DB. Set a fifth stake, G, any- c where in the line CF; a sixth stake, H, at the intersection of GB and DG prolonged; and a seventh, K, at the intersection of CA and EG prolonged. Finally, range out the lines DE and IKH and their intersection at P, will be in the line AB prolonged. (176) By the complete quadrilateral,' Let AB be the given line. Take any conven Fig. 114 lent point C; measure F - C C'from it to B, and onward, -llA' in the same line prolonged, an equal distance to D. > Take any other convenient D" point, E, such that CE and 1 DE produced will clear the obstacle. Measure from E to A, and onward, an equal distance, to F. Range out the lines FC and DE to their intersection in G. Range out FD and CE to tntersect in H. Measure GH. Its middle point, P, is the required point in the line of AB prolonged. The unavoidable acute intersections in this construction are objectionable. CHAP v.] Obstacles to Measuremente 109 B, TO INTERPOLATA POINTS IN A LINE. (177) The most distant given point of the line must be made as conspicuous as possible, by any efficient means, such as placing there a staff, bearing a flag; red and white, if seen against woods, or other dark back-ground; and red and green, if seen against the sky. A convenient and portable signal is shown in the fisgure. Fig. 115..ront View Side View. Back View. The figure represents a disc of tin, about six inches in diameter, painted white and hinged in the middle, to make it more portable. It is kept open by the bar, B, being turned into the catch, C. A screw, S, holds the disc in a slit in the top of the pole. Another contrivance is a strip of tin, which has its ends bent horizontally in contrary directions. As the wind will take strongest hold of the side which is concave towards it, the bent strip will continually revolve, and thus be very conspicuous. Its upper half should be painted red and its lower half white. A bright tin cone set on the staff, can be seen at a great distance when the sun is shining. I78) Ranging to a point, thus made conspicuous, is very simple when the ground is level. The surveyor places his eye at the nearest end of the line, or stands a little behind a rod placed on it, and by signs moves an assistant, holding a rod at some point as nearly in the desired line as he can guess, to the right or left, till his rod appears to cover the distant point, 110 CHAIN STURVEYIlNG [PART II. (179) Across a valley. When a valley, or low spot, inter venes between the two ends Fig. 116. of the line A and Z in the? figure, a rod held in the a j low place, as at B, would seldom be high enough to ~ be seen, from A, to cover'-_ i; - ~' the distant rod at Z. In such a case, the surveyor at A should heid up a plumb-line over the point, at arm's length, and place nis eye so that the plumb-line covers the rod at Z. He should then direct the rod held at B to be moved till it too is covered by the plumb-line. The point B is then said to be " in line" between A and Z. In geometrical language, B has now been placed in the vertical plane determined by the vertical plumb-line and the point Z. Any number of intermediate points can thus be " interpolated," or placed in line between A and Z. (180) Over a hill. When a hill rises between two points and prevents one being seen from the other, as in the fiure, (the upper Fig. 117. ~~~A zT I of which shows the hill in " Elevation," and the lower part m Plan'), two observers, B and C, each holding a rod, may place themselves on the ridge, in the line between the two points, as nearly as they can guess, and so that each can at once see the other and the point beyond him. B looks to Z, and by signals puts C CHAP. v.] Obstacles to Measurement. 111 ", in line." C then looks to A, and puts B in line at B'. B repeats his operation from B', putting C at C', and is then himself moved to B", and so they alternately' line" each other, continually approximating to the straight line between A and Z, till they at last find themselves both exactly in it, at B"' and C"'. (18~) A single person may put himself in line between two points, on the same principle, by laying a straight stick on some support, going to each end of it in turn, and making it point successively to each end of the line. The " Surveyor's Cross," Art. (104), is convenient for this purpose, when set up between the two given points, and moved again and again, until, by repeated trials, one of its slits sights to the given points when looked through in either direction. (182) On water, A simple instru- Fig. 118. ment for the same object, is represented a in the figure. AB and CD are two tubes, about 1 inches in diameter, con- M nected by a smaller tube EF. A piece irl~'0 of looking-glass, GH, is placed in the lower part of the tube AB, and another, KL, in the tube CD. The planes of j li/l the two mirrors are at right angles to each other. The eye is placed at A, and the tube AB is directed to any distant 1Ill object, as X, and any other object be- hind the observer, as Z, will be seen, apparently under the first object in the mirror GH, by refection from the mirror KL, when the observer has succeeded in getting in line between the two objects. M, N, are screws by which the mirror KL may be adjusted. The distance between the two tubes will cause a small parallax, which will, however, be insensible except when the two objects are near together. 112 CHLIN SURVEYINg, [PART II. (183) Through a wood. When a wood intervenes between any two given Fig. 119. points, preventing one B. from being zi __ seen from the A B' Cy "''' - other, as in the figure, in which A and Z are the given points, pro" ceed thus. Hold a rod at some point B' as nearly in the desired line from A as can be guessed at, and as far from A as possible. To approximate to the proper direction, an assistant may be sent to the other end of the line, and his shouts will indicate the direction; or a gun may be fired there; or, if very distant, a rocket may be sent up after dark. Then range out the "I random line" AB', by the method given in Art. (169), noting also the distance from A to each point found, till you arrive at a point Z', opposite to the point Z, i. e. at that point of the line from which a perpendicular there erected would strike the point Z. Measure Z'Z. Then move each of the stakes, perpendicularly from the line AZ', a distance proportional to their distances from A. Thus, if AZ' be 1000 links, and Z'Z be 10 links, then a stake B', 200 links from A, should be moved.2 links to a point B, which will be in the desired straight line AZ; if C' be 400 links from A,'it should be moved 4 links to C, and so with the rest. The line should then be cleared, and the accuracy of the position, of these stakes tested by ranging from A to Z. (184) To an invisible intersection. Let AB and CD be two lines, which, if prolong- Fig. 120. ed, would meet in a'~ 7Z -- point Z, invisible from, \ i;' either of them; and let /- ~ l. P be a point, from which /,a line is required to be,/' t' I set out, tending to this y invisible intersection. c Set stakes at the five given points, A, B, C, D, P. Set a sixth stake at E, in the alinements of AD and CP; and a seventh stake CHAP. V.] Obstacles to Measurement, 113 at F, in the alinements of BC and AP. Then set an eighth stake at G, in the alinements of BE and DF. PG will be the required line. Otherwise; Through P range out a parallel to the line BD. Note the points where this parallel meets AB and CD, and call these points Q and R. Then the distance from B, on the line BD, to a point which shall be in the required line running from P to the BDxQP invisible point, will be = qR 11i OBSTACLES TO MEASUREMENT. (185) The cases, in which the direct measurement of a line is prevented by various obstacles, may be reduced to three. A. When both ends of the line are accessible. B. W/hen one end of it is inaccessible. C. When both ends of it are inaccessible. A, WETi Bi TEt ENDS OF TIEI L1IE ARE ACCESSIBLEo (186) By perpendicalars. On Fig 121 reaching the obstacle, as at A in A /i / D the figure, set off a perpendicular, ~ AB; turn a second right angle at B, C and measure past the obstacle; turn a third right angle at C; and measure to the original line at D. Then will the measured distance, BC, be equal to the desired distance, AD. If the direction of the line is also unknown, it will be most easily obtained by the additional perpendiculars shown in Fig. 109, of Art. (171). Fig. 121'. (187) By equilateral triangles, A BK The method given in Art. (172), for /determining the direction of a line through an obstacle, will also give its length; for in Fig. 121' (Fig. 110 re- F peaked) the desired distance AG is equal to the measured distances AD, or DG. D b~~~ 114 CH1t SlRVYPINGe, [PART aI (188) By symmetrical triagles. Fig. 122 Let AB be the distance required. Measure from A obliquely to some A, —-- - point C, past the obstacle. Measure onward, in the same line, till CD is as long as AC. Place stakes at C and D. From ]B measure to C, and from C measure onward, in E ~d the same line, till CE is equal to CB, Measure ED, and it will be, equal to AB, the distance required. If more convenient, make CD and CE equal, respectively, to half of AC and CB; then will AB be equal to twice DE. (189) By transversals. Let Fig. 123. AB be the required distance. Set A a stake, C, in the line prolonged;,set another stake, D, so that C and B can be seen from it; and a third \'^ stake, E, in the line of BD prolonged, and at a distance from D equal to the distance from D to B. E Set a fourth stake, F, at the intersection of EA and CD. MVeasure AC (FE-AF). AC, AF and FE. Then is AB =- (FE-A). Fig. 124. (190) In a Town, Cases may occur, in the streets of a compactly built town, in which it is impossible to measure along any other lines than those of the streets. The figure represents such a case, in which is required the distance, AB, be- tween points situated on two streets which meet at the point C, and between which runs a cross-street, DE. In this case measure AC, CE, CD, DE and CB. Then is the required distance x CHAP. v.] Obstacles to Mleasurement, 115 AB /4(AC - BC)2 + [DE2 (CE-CD)2 A]CDxC "CDxCE As this expression is somewhat complicated, an example will b$ given: Let AC -=100, CE = 40, CD = 30, DE 21, and CB 80; then will AB - 51.7 BE WHEN O1E END OF TLE LENE IS INACCESSIBLE. (11a) By perpendiCulars. This principle Fig. 125. may be applied in a variety of ways. In Fig. _ 125, let AB be the required distance. At the - point A, set off AC, perpendicular to AB, and of any convenient length. At C, set off a perpen- C 4 dicular to CB, and continue it to a point, D, in the line of A and B. Measure DA. Then is AC2 D (192) Otherwise: At the point A, in Fig. Fig. 126 126, set off a perpendicular, AC. At C set off another perpendicular, CD. Find a point, g - E, in the line of AC, and BD. iMeasure AE AE x CD and EC, Then is AB 3- E X CD -_ If EC be made equal to AE, and ID be set in the line of BE, and also in the perpendicular D from C, then will CD be equal to AB. If EC = 1 AE, then CD - AB. Fig. 127. (193) Otherwise: At A, in Fig. 127, mea _ sure a perpendicular, AC, to the line AB; and at any point, as D, in this line, set off a perpendicular to DB, and continue it to a point E, in the line of CB. Measure DE and also DA. / Then is AB AC X AD DE - ACE D 116 CHAIN SURVEYING. [PART n Fig. 128. (191) By parallels. From A measure B AC, in any convenient direction. From a point D, in the line of BC, measure a line parallel to CA, to a point E, in the line of AB. Measure also AE. Then is AB =A x A BDE — AC /I (195) By a parallelogram. Set a stake, C, Fig. 29 in the line of A and B, and set another stake, D, ^ wherever convenient. With a distance equal to ____ CD, describe from A, an arc on the ground; and, with a distance equal to AC, describe another arc from D, intersecting the first arc in E. Or, i,' -K take AC and CD, so that together they make / one chain; fix the ends of the chain at A and D; D C — take hold of the chain at such a link, that one part of it equals AC, and the other CD, and draw it tight to fix the point E. Set a stake at F, in the intersection of AE and DB. Measure AF and AC x AF AC X CD EF. Then is AB -; or, CB- EF EF (196) By symmetrical triangles. Fig. 130. Let AB be the required distance. From A A measure a line, in any convenient di- -erection, as AC, and measure onward, in the same direction, till CD = AC. Take " any point E in the line of A and B. Measure from E to C, and onward in the [ / same line, till CF = CE. Then find by trial a point G, which shall be at the same time in the line of C and B, and in G the line of D and F. Measure the distance from G to D, and it will be equal to the required distance from A to B. If more convenient, make CD =- AC, and CF= CE, as shown by the finely dotted lines in the figure. Then will DG —= AB. CHAP, V.] Obstacles to Measurement. 117 (197) Otherwise: Prolong BA to Fig. 131. some point C. Range out any con- venient line CA' and measure CA' =_ =: CA. The triangle CA'B, is now to be reproduced in a symmetrical triangle;,. -— " //\ situated on the accessible ground. B ~ — / D/ For this object, take, on AC, some point A: s: D, and measure CD' = CD. Find the C point E, at the intersection of AD' and A'D. mind the point F, at the intersection of A'B and CE. Lastly, find the point B', at the intersection of AF and CA'. Then will A'B' = AB. The symmetrical points have corresponding letters affixed to them. (198) By tramversals. Set a stake, C, Fig. 130. in the alinement of BA; a second, D, at any convenient point; a third, E, in the line CD; and a fourth, F, at the intersection of the.. alinements of DA and EB. Measure AC,;. CE, ED, DF and FA. Then is AC x xAFx DE' \ B CE x DF- AF x DE' If the point E be taken in the middle of CD, (as it is in the AC x AF figure) then AB -A x A AC < DE If the point E be taken in the middle of AD, then AB - C DE CE — DE" The minus signs must be interpreted as in Art. (171). (199) By harmonic division. Set Fig. 133. stakes, C and D, on each side of A, and b_ __ so that the three are in the same straight line. Set a third stake at any point, E, - of the line AB. Set a fourth, F, at the - -- intersection of CB and DE; and a fifth, / "> \ G, at the intersection of DB and CE. / / Set a sixth stake, H, at the intersection C "'- T AE x AHE of AB and FG. Measure AE and EH. Then is AB -AE AE -~~ EIP 118 CUAI SURVEYING, [PART 11 (200) To aninaccessible line. The Fig. 134. shortest distance, CD, from a given point,. D' C, to an inaccessible straight line AB, is required. From C let fall a perpendicular to A, by the method of Art. (158). Then set a stake at any point, E, on the I line AC; set a second, F, at the inter-\!., section of EB and CD a third, G, at the intersection of AF and CB; and a fourth, H, at the interseb tion of EG and CD. Measure CH and HF. Then is DCHxClF CH+HF' CHxCF CD - CHHF; or, CD-H CH-F or, CD = 2CHCIFI —3E-I]E" CH —— HoF- 2CE —CF Otherwise; When the inaccessible line is determined by the method of Art. (205) or (206), the distance from any point to it, can be at once measured to its symmetrical representative. (201) To an inaccessi$ e interseetion,. When two lines (as A], CD, in the figure) meet in a Fig. 135. river, a building, or any other inaccessible point, the distance G_____ from any point of either to their A-Cintersection, DE, for example, may be found thus. From any point B, on one line, measure 1 F BD, and continue it, till DF =- DB. From any other point, G, of the former line, measure GD, and continue the line till DH = GD. Continue -HF to meet DC in some point K. MTeasure IKD. KD will be equal to the desired distance DE. BE can be found by measuring FK, which is equal to it. If DF and DH, be made respectively equal to one-half, or onethird, &c., of DB and DG, then will KD and KF be respectively equal to one-half or one-third, &c., of DE and BE. CHAP. v.] Obstacles to Measurement, 119 C. WI E BSTH ENDS OF TIE LINE ARE INACCESiBLE. (202) By siilar triabgles. Let AB Fir. 136. be the inaccessible distance. Set a stake at any convenient point C, and find the distan- ces CA and CB, by any of the methods just given. Set a second stake at any point, D, on the line CA. Measure a distance, equal CB x CD e to a CA D from C, on the line CB, to some point E. Measure CA. AC x D)E DE. Then is AB -A xD CD Fig. 137. If more convenient, measure CD in the A -. /, B contrary direction from the river, as in Fig. 137, instead of towards it, and in other respects proceed as before. E D (23) By parallels. Let AB be the in- Fi. 138s. accessible distance. From any point, as C, A A range out a parallel to AB, as in Art. (1 65), &c. Find the distance CA, by Art. (191), - &c. Set a stake at the point E, the inter- section of CA and DB, and measure CE. D /Then is AB CD x (AC - CE) CE (204) By a parallelogram. Set Fig. 139. a stake at any convenient point C. Set stakes D and E, anywhere in the alinements CA and CB. With' <'-i D as a centre, and a length of the chain equal to CE, describe an arc; and with E as a centre, and a length, of the chain equal to CD, describe another arc, intersecting tho former one at F. A parallelogram, CDEF, will thus be formed. Set stakes at G and H, where the alinements DB and EA intersect the sides of this parallelogram. Measure CD, DF, GF, FH, 120 CHAIN SURVEYING, [PAR II CD x DF x GiH and HG. The inaccessible distance AB -CD x DF F6 x FH If CD - CE then AB - CD2 x G FG- X FH1 (205) By symmetrical triangles, Take any convenient point, as C. Set stakes at two other Fig. 140. points, D and D', in the same A - -- - line, and at equal distances u="_=-Y from C. Take a point E, in 2 the line of AD; measure from \/ / it to C, and onward till CE' p E = E. Take a point F in',Ek — J the line of BD; measure from / it to C, and onward till CF'- / /, \ CF. Range out the lines AC/ / " and E'D', and set a stake at //'Q their intersection, A'. Range n "..'" out the lines BC and F'D', and set a stake at their intersection, B'. Measure A'B'. It will be equal to the desired distance AB. (2@0) Otherwise: Take Fig. 141. any convenient point, as C, A\ - and set off equal distances 7___ =_ —_ _ on each side of it, in the 0_ line of CA, to D andD'. Set \ off the same distances from \ C, in the line of CB, to E and F. / " l E'. Through C, set out a \q \ parallel to DE, or D'E', and, /_ ~ -d set stakes at the points F \ \ and F' where this parallel, intersects AE' and BD'. B3' ARange out the lines AD' and EF', and set a stake at their inters section A'. Range out the lines BE' and DF, and set a stake at their intersection B'. Measure A'B' and it will be equal to the desired distance AB. CHAP. v.] Obrsacles to Measurement. 121 The easiest method of setting out the parallel in the above case, is to fix the middle of the chain at the point C, and its ends on the lines CD, CE'; then carry the middle of the chain from C towards F, and mark the point to which it reaches; and repeat this on the other side of C, as shown by the finely dotted lines in the figure. INACCESSIBLE AREAS. (207) Triangles. In the case of a triangular fieldl in which one side cannot be measured, or determined by any of the methods just given, the two accessible sides may be prolonged to their full length, and an equal symmetrical triangle formed, all of whose sides can be measured. Thus in Fig. 102, page 103, if CDE be the original triangle, of which the side EC is inaccessible, DFP will be equal to it. But if this also be impossible, per- Fig.142. tions of the sides may be measured, asAD, AE, B 1 -... Cs in the figure in the margin, and also DE, and the area of this triangle found by aly of the D — / E methods which have been given. Then is the desired area of the triangle ABC = area of AB x AC A AD x AE' (208) ~Quadrilaterals. In the case Fig. 143. of a four-sided field, whose sides cannot be measured, or prolonged, but whose diagonals can be measured, the area iK may be obtained thus. Measure the diagonals AC and ED; and also the portions AE, EC, into which one of them is divided by the other. Calcu- late the area of the triangle BCE, by the preceding method, or any of those heretofore given. Then the area of the quadrilateral ABCOD - area of BCE x BE x CE (209) Polygons, Methods for obtaining the areas of inaccessible fields of more than four sides, have been given in Arts. (1013,) &c. PART ITII COMPASS SURVEYINGX OR By the Third Mfethod. CHAPTER I. A.LST lR SURTEYiNG IN GENERAfL. (210) Angular Surveying determines the relative positions of points, and therefore of lines, on the THIRD PRINCIPLE, as explained in Art. (7), which should now be referred to. (211) When the two lines which form an angle lie in the same horizontal or level plane, the angle is called a' horizontal angle.* When these lines lie in a plane perpendicular to the former, the angle is called a vertical angle. When one of the lines is horizontal and the other line from the eye of the observer passes above the former, and in the same vertical plane, the angle is called an angle of elevation. When the latter line passes below the horizontal line, and in the same vertical plane, the angle is called an angle of depression. When the two lines which form an angle, lie in other planes which make oblique angles with each of the former planes, the angle is called an oblique angle. Horizontal angles are the only angles employed in common land surveying. * A plane is said to be horizontal, or level, when it is parallel to the surface of standing water, or perpendicular to a plumb-line. A line is horizontal when it lies in a horizontal plane, [eCHAP.. nga lar Surveying in general. 123 (212) The angles between the directions of two lines, which it is necessary to measure, may be obtained by a great variety of instruments. All of them are in substance mere modifications of the very simple one which will now be described, and which any one can make for himself. (213) Provide a circular piece of Fi. 144. wood, and divide its circumference \ (by any of the methods of Geometri- cal Drafting) into three hundred and, sixty equal parts, or "' Degrees," and number them as in the figure. The divisions will be like those of a watch face, but six times as many. These divisions are termed yraduation \s. The figure shows only every fifteenth one. In the centre of the circle, fix a needle, or sharp-pointed wire, and upon this fix a straight stick, or thin ruler placed edge-wise, (called an alidade), so that it may turn freely on this point and nearly touch the graduations of the circle. Fasten the circle on a staff, pointed at the other end, and long enough to bring the alidade to the height of the eyes. The instrument is now complete. It may be called a (oniometer, or Angle-measurer. (214) Now let it be required to measure Fig. 145. the angle between the lines AB and AC. Fix the staff in the ground, so that its centre shall be exactly over the intersection of the two / o lines. Turn the alidade, so that it points, (as ~7X — -- determined by sighting along it) to a rod, or 17 other mark at B, a point on one of the lines, and note what degree it covers, i. e. I The Reading." Then, without disturbing the circle, turn the alidade till it points to C, a point on the other line. Note the new reading. The difference of these readings, (in the figure, 45 degrees), is the difference in the directions of the two lines, or is the angle which one makes with the other. If the dis, 124 COMPASS SURVEYINT [CPART in, tance from A to C be now measured, the point C is "' determined," with respect to the points A and B, on the Third Principle. Any number of points may be thus determined. (215) Instead of the very simple and rude alidade, which has been supposed to be used, needles may be fixed on each end of the alidade; or sights may be added, such as those described in Art. (106); or a small straight tube may be used, one end being covered with a piece of pasteboard in which a very small eye hole is pierced, and threads, called " cross-hairs," being stretch- Fit 146. ed across the other end of it, as in the figure; so that (~) ( their intersection may give a more precise line for determining the direction of any point. (216) When a telescope is substituted for this tube, and supported in such a way that it can turn over, so as to look both backwards and forwards, the instrument (with various other additions, which however do not affect the principle), is called the IEngineer's Transit. With the addition of a level, and a vertical circle, for measuring vertical angles, the instrument becomes a Theodolite; in which, however, the telescope does not usually admit of being turned over. (2I7) The Compass differs from the instruments which have been described, in the following respect. They all measure the angle which one line makes with another. The compass measures the angle which each of these lines makes with a third line, viz: that shown by the magnetic needle, which always points (approxi. mately) in the same direction, i. e. North and South, Fig. 147. in the iJfagnetic Meridian. Thus, in the figure, the N? line AB makes an angle of 30 degrees with the line AN, and the line AC makes an angle of 75 de- gTees with AN. The difference of these angles, A,-c or 45 degrees, is the angle which AC makes with AB, agreeing with the result obtained in Art. (214). S [CHAP.. Angular Surveying in general. 125 (218) Surveying with the compass is, therefore, a less direct operation than surveying with the Transit or Theodolite. But as the use of the compass is much more rapid and easy (only one sight and reading at each station being necessary, instead of two, as in the former case), for this reason, as well as for its smaller cost, it is the instrument most commonly employed in land surveying in this country, in spite of its imperfections and inaccuracies. As many may wish to learn "; Surveying with the Compass," without being obliged to previously learn " Surveying with the Transit," (which properly, being more simple in principle, though less so in practice, should precede it, but which will be considered in Part IV), we will first take up COMPASS SURVEYING. (219) Angular Surveying in general, and therefore Compass Surveying, may employ either of the 3d, 4th and 5th methods of determining the position of a point, given in Part I; that is, any instrument which measures angles may be employed for Polar, Triacngular or Triinear Surveying. The first of these, Polar Surveying, is the one most commonly adopted for the compass, and is therefore the one which will be specially explained in this part. The same method, as employed with the Transit and Theodolite, will be explained in the following part. The 4th and 5th methods will be explained in the next two parts. (220) The method of Polar Surveying embraces two minor methods. The most usual one consists in going around the field with the instrument, setting it at each corner and measuring there the angle which each side makes with its neighbor, as well as the length of each side. This method is called by the French the method of Cherminement. It has no special name in English, but may be called (from the American verb, To progress), the Method of Progression. The other system, the Method of Radiatzon, consists in setting the instrument at one point, and thence measuring the direction and distance of each corner of the field, or other object. The corresponding name of what we have called Triangular Surveying is the Mfethod of Intersections; since it determines points by the intersections of straight lines. 126 CgOIPASS SURVEYING [PART 11I. I____A~~~~~___ _ 1___ _ a (0/"U~~ ~ ^^'~ -— ^R~~^~ ^ ~__bi________ 1) ^ ^ ^ - T T, " ~ ^ -^ T " CHAPTER II. THE COMPASS. (221) The Needle, The most essential part of the compass is the magnetic needle, It is a slender bar of steel, usually five or six inches long, strongly magnetized, and balanced on a pivot, so that it may turn freely, and thus be enabled to continue pointing in the same direction (that of the'" Magnetc Merician," approimately North and South) however much the r Compass Box," to which the pivot is attached, may be turned around. As it is important that the needle should move with the least possible friction, the pivot should be of the hardest steel ground to a very sharp point; and in the centre of the needle, which is to rest on the pivot, should be inserted a cap of agate, or other hard material. Iridium for the pivot, and ruby for the cap, are still better. If the needle be balanced on its pivot before being magnetized, one end will sink, or " Dip," after the needle is magnetized. To bring it to a level, several coils of wire are wound around the needle so that they can be slid along it, to adjust the weight of its two ends and balance it more perfectly. The North end of the needle is usually cut into a more ornap mental form than the South end, for the sake of distinction. The principal requisites of a compass needle are, intensity of directive force and susceptibility. " Shear steel" was found by Capt. Kater to be the kind capable of receiving the greatest magnetic force. The best form is that of a rhomboid, Fig. 149. or lozenge, cut out in the middle, so as to dimi-,s nish the extent of surface in proportion to the mass, as it is the latter on which the directive force depends. Beyond a certain limit, say five inches, no additional power is gained by increasing the length of the needle. On the contrary, longer ones are apt to have their strength diminished by several consecutive poles being formed. Short needles, made very hard, are therefore to be preferred. 128 COMPASS SURVEYlING. [PART III The needle should not come to rest very quickly. If it does, it indicates either that it is weakly magnetized, or that the friction on the pivot is great. Its sensitiveness is indicated by the number of vibrations which it makes in a small space before coming to rest. A screw, with a milled head, on the under side of the plate which supports the pivot, is used to raise the needle off this pivot, when the instrument is carried about, to prevent the point being dulled by unnecessary friction. (222) The SightSe Next after the needle, which gives the di rection of the fixed line, whose angles with the lines to be surveyed are to be measured, should be noticed the Sights, which show the directions of these last lines. At each end of a line passing through the pivot is placed a " Sight," consisting of an upright bar of brass, with openings in it of various forms; usually either slits, with a circular aperture at their top and bottom*; or of the form described in Art. (10 ); all these arrangements being intended to enable the line of sight to be directed to any desired object, with precision. (223) A Telescope which can move up and down in a vertical plane, i. e. a plunging telescope, or one which can turn completely over, is sometimes substituted for the sights. It has the great advantage of giving more distinct vision at long distances, and of admitting of sights up and down very steep slopes. Its accuracy of vision is however rendered nugatory by the want of precision in the readings of the needle. If a telescope be applied to the comu pass, a graduated circle with vernier should be added, thus converting the compass into a "Transit." The Telescope will be found minutely described in Part IV, " Transit Surveying." (22i) The divided circ1e, We now have the means of indicating the directions of the two lines whose angle is to be measur. ed. The number of degrees contained in it is to be read from a circle, divided into degrees, in the centre of which is fixed the * An inside and an outside view, or " Elevation," of such sights, are given on each side of the figure of the Compass, on page 126. It is itself drawn in " Military Perspective." OHAP. ii.] The Compasse 129 pivot bearing the needle. The graduations are usually made to half a degree, and a quarter of a degree or less can then be' estimated." The pivot and needle are sunk in a circular box, so that its top may be on a level with the needle. The graduations are usually made on the top of the surrounding rim of the box, but should also be continued down its inside circumference so that it may be easier to see with what division the ends of the needle coincide. The degrees are not numbered consecutively from 0~ around tX 8603; but run froml 0 to 90~, both ways from the two diametrically opposite points at which a line, passing through the slits in the middle of the sights, would meet the divided circle. The lettering of the Surveyor's Compass has one important dif ference from that of the Mariner's Compass. When we stand facing the North, the East is on our right hand, and the West on our left. The graduated card of the Mariner's Compass which is fastened to the needle, and turns with it, is marked accordingly. But, in the Surveyor's compass, one of the 0 points being marked N, or North, (or indicated by a fleur-delis,) and the opposite one S, or South, the 90-degrees-point on the right of this line, as you stand at the S end and look towards the N, is marked W, or West; and the left hand 90-degrees-point is marked E, or LEast. The reason of this will be seen when the method of using the compass comes to be explained in the following chapter. (225) The Points. In or- Fig. 150. dinary land surveying, only four points of the compass have / %' -\ names, viz: North, South, East and West; the direction of a! line being described by the an- " qA!1;a gle which it makes with a North I and South line, to its East or to /s% -t its West. But for nautical pur- k? /t \ poses, the circle of the compass i/' }" / is divided into 32 points, the names of which are shown in 9 180 COMPASS SURVEYIING [PART III. the figure. Two rules embrace all the cases. 10 When the letters indicating two points are joined together, the point half way between the two is meant; thus, N. E. is half way between North and East; and N. N. E. is half way between North and North East. 20 When the letters of two points are joined together with the intermediate word by, it indicates the point which comes next after the first, in going towards the second; thus, N. by E, is the point which follows North in going towards the East; S. E. by S. is the next point from South East, going towards the South. (22() Eccentricity. The centre-pin, or pivot of the needle, ought to be exactly in the centre of the graduated circle; the needle ought to be straight; and the line of the sights ought to pass exactly through this centre and through the 0 points of the circle. If this is not the case, there will be an error in every observation. This is called the error of eccentricity. When the maker of a compass is about to fix the pivot in place, he is in doubt of two things; whether the needle is perfectly straight, and whether the pivot is exactly in the cen- Fig. 151. tre. In figures 151 and 152, both of these are represented as being excessively in error. Firstly, to examine if the needle be_____ straight. Fix the pivot temporarily, so that the ends of the needle may cut oppo- site degrees, i. e. degrees differing by 1800. The condition of things at this Fia stage of progress, will be represented by Fig. 151. Then turn the compass-box half way around. The error will now be doubled, as is shown by Fig. 152, in which the former position of the needle is indi- /\ catel by a dotted line.* Now bend the needle, as in Fig. 153, till it cuts divisions midway between those xut by it in * This is another example of the fruitful pr;nciole of Reverswon, first noticed in Art. (105). CHAP, Ii.] The Compass. 181 its present and in its former position Fig 153. This makes it certain that the needle is straight, or that its two ends and its cen- \ tre lie in the same straight line. Secondly, to put the pivot in the cen- tre. Move it till the straightened needle cuts opposite divisions. It is then certain that the direction of the needle passes through the centre. Turn the compass box one-quarter around, and if the needle does not then cut opposite divisions, move the pivot till it does. Repeat the operation in various positions of the box. It will be a sufficient test if it cuts the opposite divisions of 0~, 45~ and 900. To fix the sights precisely in line, draw a hair through their slits and move them till the hair passes over the 0 points on the circle. The surveyor can also examine for himself, by the principle of Reversion, whether the line of the sights passes th h through the centre or not. Sight to any very near object. Read off the number of degrees indicated by one end of the needle. Then turn the compass half around, and sight to the same object. If the two readings do not agree, there is an error of eccentricity, and the arithmetical mean, or half sum of the two readings is the correct one. Fig. 154. Fig. 155. In Fig. 154, the line of sight AB is represented as passing to one side of the centre, and the needle as pointing to 460. In Fig. 155, the compass is supposed to have been turned half around and the other end of the sights to be directed to the same object. Suppose that the needle would have pointed to 450, if the line of 132 COMPAS SURVEYING. [PART III, sight had passed through the centre. The needle will now point to 440, the error being doubled by the reversion, and the true reading being the mean. This does not, however, make it certain that the line of the sights passes through the 0 points, which can only be tested by the hair, as mentioned above.'(2'2) Levels. On the compass plate are two small spirit levels. They consist of glass tubes, slightly curved upwards, and nearly filled with alcohol, leaving a bubble of air within them. They are so adjusted that when the bubbles are in the centres of the tubes, the plate of the compass shall be level. One of them lies in the direction of the sights, and the cther at right angles to this direction. (228) Tangent Scale, This is a convenient, though not essential, addition to the compass, for the purpose of measuring the slopes of ground, so that the proper allowance in chaining may be made. In the figure of the compass, page 126, may be seen, on the edge of the left hand sight, a small projection of brass with a hole through it. On the edge of the other sight are engraved lines numbered from 00 to 200, the 0~ being of the same height above the compass plate that the eye-hole is. To use this, set the compass at the bottom of a slope, and at the top set a signal of exactly the height of the eye-hole from the ground. Level the compass very carefully, particularly by the level which lies lengthwise, and, with the eye at the eye-hole, look to the signal and note the number of the division on the farther sight which is cut by the visual ray. That will be the angle of the slope; the distances of the engraved lines from the 00 line being tangents (for the radius equal to the distance between the sights) of the angles corresponding to the numbers of the lines. (229) Vernier. The compass box is connected with the plate, which carries it and the sights, so that it can turn around on this plate. This motion is given to it by a screw, (called a slow-motion, or Tangent screw), the head of which is the nearest one in CHAP II.I The Compass. 13i the figure on page 126. If two marks be made opposite to each other, one on the projecting part of the compass box, and the other on the plate to which the sights are fastened, these marks will separate when the slow-motion screw is turned. Their distance apart (in angular measurement, i. e. fractions of a circle), in any position, is measured by a contrivance called a Vernier, which is the minutely divided arc of a circle seen between the left hand sight and the compass box. It will be better to defer explaining the mode of reading the vernier for the present, since it is rarely used with the compass, and an entire chapter will be given to it in Part IV. Its principle is similar to that of the Vernier Scale, described in Art. (;0). Its applications in "Field-work" will be noticed under that head. (280) Tripodo The compass, like most surveying instruments, is usually supported on a Tripod, consisting of three legs, shod with iron, and so connected at top as to be movable in any direction. There are many forms of these. Lightness Fig. 156. Fig. 157. of these. Lightness and stiffness are the qualities desired. The most usual form is shewn in the figures: of the Transit and the Theodolite; at the be- ginning of Part IV. Of the two represented in Figs. 156 and 157, the first has the advantage of being very easily and cheaply made; and the second that of being light and yet capable of very firmly resisting horizontal torsion. The joints, by which the instrument is connected with the tripod, are also various. Fig. 158 is the " Ball-and-socket joint," most usual in this country. It takes its name from the ball, in which 134 COMPASS SURVEYING. ['ART III Fig. 158. Fig. 159. Fig. 160 under the compass plate, and the socket in which this ball turns. It admits of motion in any direction, and can be tightened or loosened by turning the upper half of the hollow piece enclosing it, wich is screwed on the lower half. Fig. 159 is called the " Shelljoint." In it the two shell-shaped pieces enclosing the ball are tightened by a thumb-screw. Fig. 160, is " Cugnot's joint." It consists of two cylinders, placed at right angles to each other, and through the axes of which pass bolts, which turn freely in the cylinder and can be tightened or loosened by thumb-screws at their ends. The combination of the two motions which this joint permits, enables the instrument which it carries, to be placed in any desired position. This joint is mhch the most stable of the three. (23t ) Jacob's Staff. A single leg, called a C Jacob's Staff," has some advantages, as it is lighter to carry in the field, and can be made of any wood on the spot where it is to be used, thus say ing the expense of a tripod and the trouble of its transportation Its upper end is fitted into the lower end of a brass head which has a ball and socket joint, and axis above. Its lower end should be shod with iron, and a spike running through it is useful for pressing it into the ground with the foot. Of course it cannot be conveniently used on frozen ground, or on pavements. It may, however, be set before or behind the spot at which the angle is to be mea CHAP. I.] The Compass, 135 sured, provided that it is placed very precisely in the line of direc. tion from that station to the one to which a sight is to be taken. (232) The Prismatic Compass. The peculiarity of this instrument (often called Schmalcalder's) is that a glass triangular prism is substituted for one of the sights. Such a prism has this peculiar property that at the same time, it can be seen through, so that a sight can be taken through it, and that its upper surface reflects like a mirror, so that the numbers of the degrees immediately under it, can be read off at the same time that a sight to any object is taken. Another peculiarity, necessary for profiting by the last one, is, that the divided circle is not fixed, but is a card fastened to the needle and moving around with it, as in the Mariner's Compass. The minute description, which follows, is condensed from Simms. In the figure, A repre- Fi. 161. sents the compass box, and B the card, which, being attached to the magnetic E needle, moves as it moves, around the agate centre, a, on which it is suspend- c ed. The circumference of the card is usually di- vided to' or' of a iegree. C is a prism, which the observer looks through. The perpendicular thread A e of the sight-vane, E, and the divisions on the card, appear together on looking through the prism, and the division with which the thread coincides, when the needle is at rest, is the " Bearing" of whatever object the thread may bisect, i. e. is the angle which the line of sight makes with the direction of the needle. The prism is mounted with a hinge joint, ). The sight-vane has a fine thread stretched along its opening, in the direction of its length, which is brought to bisect any object, by turning the box around horizontally. F is a mirror, made to 136 C0MPASS SURVEYING. [PART 113 slide on or off the sight-vane, E; and it may be reversed at plea? sure, that is, turned face downwards; it can also be inclined at any angle, by means of its joint, d; and it will remain stationary on any part of the vane, by the friction of its slides. Its use is to reflect the image of an object to the eye of an observer when the object is much above or below the horizontal plane. The colored glasses represented at G, are intended for observing the sun. At se is shown a spring, which being pressed by the finger at the time of observation, and then released, checks the vibrations of the card, and brings it more speedily to rest. A stop is likewise fixed to the other side of the box, by- which the needle may be thrown off its centre. The method of using this instrument is very simple. First raise the prism in its socket, b, until you obtain a distinct view of the divisions on the card. Then, standing over the point where the angles are to be taken, hold the instrument to the eye, and, looking through the slit, C, turn around till the thread in the sight-vane bisects one of the objects whose bearing is required; then by touching the spring, e, bring the reedle to rest, and the division on the card which coincides with the thread on the vane, will be the bearing of the object from the north or south points of the magnetic meridian. Then turn to any other object, and repeat the operation; the difference between the bearing of this object and that of the former, will be the angular distance of the objects in question. Thus, suppose the former bearing to be 40 380', and the latter 100 15', both east, or both west, Fig. 16. from the north or south, the angle will be 300 16'. The divisions are generally numbered 50, 100, 150, &c. around the circle to 8600. // \\\ The figures on the compass card _______ are ieversed, or written upside down, as in the figure (in which / only every fifteenth degree is mark- a ed), because they are again reversed by the prism. CHAP. ii.] The Compass. 137 (233) The prismatic compass is generally held in the hand, the bearing being caught, as it were, in passing; but more accurate readings would of course be obtained if it rested on a support, such as a stake cut flat on its top. In the former mode, the needle never comes completely to rest, particularly in the wind. In such cases, observe the extreme divisions between which the needle vibrates, and take their arithmetical mean. (234) Defcts of compass. The compass is deficient m both precision and correctness.* The former defect arises from the indefiniteness of its mode of indicating the part of the circle to which it points. The point of the needle has considerable thickness; it cannot quite touch the divided circle; and these divisions are made only to whole or half degrees, though a fraction of a division may be estimated, or guessed at. The Vernier does not much better this, as we shall see when explaining its use. Now an inaccuracy of one quarter of a degree in an angle, i. e. in the difference of the directions of two lines, causes them to separate from each other 5{ inches at the end of 100 feet; at the end of 1000 feet nearly 41 feet; and at the end of a mile, 23 feet. A difference of only one-tenth of a degree, or six minutes, would produce a difference of 13 feet at the end of 1000 feet; and 9- feet at the distance of a mile. Such are the differences which may result from the want of precision in the indications of the compass. But a more serious defect is the want of correctness in the compass. Its not pointing exactly to the true north does not indeed affect the correctness of the angles measured by it. But it does not point in the same or in a parallel direction, during even the same day, but changes its direction between sunrise and noon nearly a quarter of a degree, as will be fully explained in Chapter YII. The effect of such a difference we have just seen. This direction " The student nmust not confound these two qualities. To say that tLe sun allp pears to rise in thle eastern quarter of the heavens and to set in the western, is ecrrcct, but not v rcise. A watch with a second hand indicates the time of day precisely, but not always correctly. The statement that two and two make five, is precise, but is not usually regarded as correct. 138 COMIPASS SURVEYIfNG. [PART III. may also be greatly altered in a moment, without the knowledge of the surveyor, by a piece of iron being brought near to the corm pass, or by some other local attraction, as will be noticed hereafter. This is the weak point in the compass. Notwithstanding these defects, the compass is a very valuable instrument, from its simplicity, rapidity and convenience in use; and though never precise, and seldom correct, it is generally not very wrong. CHAPTER III. THE FIELD WORK. (235) Taking Bearings. The " Bearing" of a line is the an. gle which it makes with the direction of the needle. Thus, in Fig. 147, page 124, the angle NAB is the Bearing of the line AB, and NAC is the Bearing of AC. The Bearing and length of a line are named collectively the Course. To take the Bearing of any line, set the compass exactly over any point of it by a plumb-line suspended from beneath the centre of the compass, or, approximately, by dropping a stone. Level the compass by bringing the air bubbles to the middle of the level tubes. Direct the sights to a rod held truly vertical, or "; plumb," at another point of the line, the more distant the better. The two ends are usually taken. Sight to the lowest visible point of the rod. When the needle comes to rest, note what division on the circle it points to; taking the one indicated by the North end of the needle, if the North point on the circle is farthest from you, and vice versa. In reading the division to which one end of the needle points, the eye should be placed over the other end, to avoid the error which might result from the " parallax," or apparent change of place, of the end read from, when looked at obliquely. CSaP. iii.] The Field Work. 139 The bearing is read and recorded by noting between what letters the end of the needle comes, and to what number; naming, or writing down, firstly, that letter, N or S, which is at the 03 point nearest to that end of the needle from which you are reading; secondly, the number of degrees to which it points, and thirdly, the letter, E or W, of the 90~ point which is nearest to the same end of the needle. Thus ir, the figure~ if when the sights wero directed along a line, (the North Fig. 163. point of the compass being most distant from the observer), the North end of the needle was at the / point A, the bearing of the line sighted on, would be North 45~0 B 1 p East; if the end of the needle was at B, the bearing would be East; if at C, S. 30~ E; if at D, South; if at E, S. 600 W; if at F, West; if at G, N. 60~ W; if at I, North. D (236) We can now understand why W is en the right hand of the compass-box, and E on the left. Let the direction from the centre of the compass to the point Fig. 164. B in the figure, be required, and suppose the sights in the first place to be pointing in the direction of the needle, S N, and the North sight to be ahead. When the sights (and H i / W ---- the circle to which they are fasten- ed) have been turned so as to point in the direction of B, the point of ~ A the circle marked E, will have come round to the North end of the needle, (since the needle remains immovable,) and the reading will therefore be East," as it should be. The effect on the reading is the same as if the needle had moved to the left the same quantity which the sights have moved to the right, and the left side is therefore properly marked " East," and vice versa. So, too, if the bearing of the line to C be desired, half-way between North and 140 COMPASS SRVEYING. [PART Ii East, i. e. N. 450 E.; when the sights and the circle have turned 45 degrees to the right, the needle, really standing still, has apparently arrived at the point half-way between N. and E., i. e. N. 450 E. Some surveyors' compasses are marked the reverse of this, the E on the right and the W cn the left. These letters must then be reversed in the mind before the bearing is noted down. (237) Readlng with Vernier When the needle does not point precisely to one of the division marks on the circle, the fractional part of the smallest space is usually estimated by the eye, as has been explained. But this fractional part may be measured by the Vernier, described in Art. (229), as follows. Suppose the needle to point between N. 31~ E. and N. 311-0 E. Turn the tangent screw, which moves the compass-box, till the smaller division (in this case 310) has come round to the needle. The Vernier will then indicate through what space the compass-box has moved, and therefore how much must be added to the reading of the needle. Suppose it indicates 10 minutes of a degree. Then the bearing is N. 310 10' E. It is, however, so difficult to move the Vernier without disturbing the whole instrument, that this is seldom resorted to in practice. The chief use of the Vernier is to set the instrument for running lines and making an allowance for the variation of the needle, as will be explained in the proper place. A VernierA Vernier arc is sometimes attached to one end of the needle and carried around by it. (238) Practical HBnts Mark every station, or spot, at which the compass is set, by driving a stake, or digging up a sod, or piling up stones, or otherwise, so that it can be found if any error, or other cause, makes it necessary to repeat the survey. Very often when the line of which the bearing is recuired, is a fence, &c., the compass cannot be set upon it. In such cases, set the compass so that its centre is a foot or two from the line, and set the flag-staff at precisely the same distance from the line at the other end of it. The bearing of the flag-staff from the compass will be the same as that of the fence, the two lines being parallel CHAP. III.] The Field Work. 141 The distances should be measured on the real line. If more convenient the compass may be set at some point on the line prolonged, or at some intermediate point of the line, " in line" between its extremities. In setting the compass level, it is more important to have it level crossways of the sights than in their direction; since if it be not so, on looking up or down hill through the upper part of one sight and the lower part of the other, the line of sight will not be parallel to the N and S, or zero line, on the compass, and an incorrect bearing will therefore be obtained. The compass should not be levelled by the needle, as some books recommend, i. e. so levelled that the ends of the needle shall be at equal distances below the glass. The needle should be brought so originally by the maker, but if so adjusted in the morning, it will not be so at noon, owing to the daily variation in the dip. If then the compass be levelled by it, the lines of sight will generally be more or less oblique, and therefore erroneous. If the needle touches the glass, when the compass is levelled, balance it by slihd ing the coil of wire along it. The same end of the compass should always go ahead. The North end is preferable. The South end will then be nearest to the observer. Attention to this and to the caution in the next paragraph,. will prevent any confusion in the bearings. Always take the readings from the same end of the needle; from the North end, if the North end of the compass goes ahead; and vice versa. This is necessary, because tne two ends will not always cut opposite degrees. With this precaution, however, the angle of two meeting lines can be obtained correctly from either end, provided the same one is used in taking the bearings of both the lines. Guard against a very frequent source Fig. 165. of error with beginners, in reading from the wrong number of the two between 1 j which the needle points, such as reading 340 for 260, in a case like that in the figure. 142 COMPASS SURVEYING. [PART III Check the vibrations of the needle by gently raising it off the pivot so as to touch the glass, and letting it down again, by the screw on the under side of the box. The compass should be smartly tapped after the needle has settled, to destroy the effect of any adhesion to the pivot, or friction of dust upon it. All iron, such as the chain, &c., must be kept at a distance from the compass, or it will attract the needle, and cause it to deviate from its proper direction. The surveyor is sometimes troubled by the needle refusing to traverse and adhering to the glass of the compass, after he has briskly wiped this off with a silk handkerchief, or it has been carried so as to rub against his clothes. The cause is the electricity excited by the friction. It is at once discharged by applying a wet finger to the glass. A compass should be carried with its face resting against the side of the surveyor, and one of the sights hooked over his arm. In distant surveys an extra centre pin should be carried, (as it is very liable to injury, and its perfection is most essential), and, also, an extra needle. When two such are carried, they should be placed so that the north pole of one rests against the south pole of the other. (239) When the magnetism of the needle is lessened or destroyed by time, it may be renewed as follows. Obtain two bar magnets. Provide a board with a hole to admit of the axis, so that its collar may fit fairly, and that the needle may rest flat on it, without bearing at the centre. Place the board before you, with the north end of the needle to your right. Take a magnet in each hand, the left holding the North end of the bar, or that which has the mark across, downwards; and the right holding the same mark upwards. Bring the bars over the axis, about a foot above it, without approaching each other within two inches:-bring them down vertically on the needle, (the marks as directed) about an inch on each side of its axis; slide them outwards to its ends with slight pressure; raise them up; bring them to their former position, and repeat this a number of times. CHAP. IIi.] The Field Work, 143 (240) Back Sights. To test the accuracy of the bearing of a line, taken at one end of it, set up the compass at the other end, or point sighted to, and look back to a rod held at the first station, or point where the compass had been placed originally. The reading of the needle should now be the same as before. If the position of the sights had been reversed, the reading would be the Reverse Bearing; a former bearing of N. 300 E. would then be S. 800 W. and so on. (241) Local attraction If the Back-sight does not agree with the first or forward sight, this latter must be taken over again. If the same difference is again found, this shows that there is local attraction at one of the stations; i. e. some influence, such as a mass of iron ore, ferruginous rocks, &c., under the surface, which attracts the needle, and makes it deviate from its usual direction. Any high object, such as a house, a tree, &c., has recently been found to produce a similar effect. To discover at which station the attraction exists, set the compass at several intermediate points in the line which joins the two stations, and at points in the line prolonged, and take the bearing of the line at each of these points. The agreement of several of these bearings, taken at distant points, will prove their correctness. Otherwise, set the compass at a third station; sight to each of the two doubtful ones, and then from them back to this third station. This will show which is correct. When the difference occurs in a series of lines, such as around a field, or along a road, proceed Fig. 16(. thus. Let C be the station at c which the back-sight to B dif- A fers from the foresight from B to C. Since the back-sight from B to A is supposed to have agreed with the foresight from A to B, the local attraction must be at C, and the forward bearing must be corrected by the difference just found between the fore and back sights, adding or subtracting it, according to circumstances. An easy method is to draw a 144 COMPISS SURVIEYING, [PART Ml. figure for the case, as in Fig. 167. In Fg. 167. it, suppose the true bearing of BC, as, given by a fore-sight from B to C, to be'I // N. 400 E., but that there is local at" / traction at C, so that the needle is drawn aside 10~, and points in the direction 8'N', instead of AN. The back-sight from C to B will then give a bearing / of N. 500 E.; a difference, or correc- < tion for the next fore-sight, of 100. If the next fore-sight, from C to D, be N. 700 E, this 10o must be subtracted from it, making the true fore-sight N. 600 E. A general rule may also be given. When the back-sight is greater than thefore-sight, as in this case, subtract the difference from the next fore-sight, if that course and the preceding one have both their letters the same (as in this case, both being N. and E.), or both their letters different; or add the difference if either the first or last letters of the two courses are different. When the back-sight is less than the fore-sight, add the difference in the case in which it has just been directed to subtract it, and subtract it where it was before directed to add it. (242) Alngls of deflectin. When the compass indicates much local attraction, the difference between the directions of two meeting lines, (or the " angle of deflection' of one from the other), can still be correctly measured, by taking the difference of the bearings of the two lines, as observed at the same point. For, the error caused by the local attraction, whatever it may be, affects both bearings equally, inasmuch as a "Bearing" is the angle which a line makes with the direction of the needle, and that here remains fixed in some one direction, no matter what, during the taking of the two bearings. Thus, in Fig. 167, let the true bearing of BC, i. e. the angle which it makes with the line SN, be, as before, N. 40l E., and that of CD N. 600 E. The true " angle of deflection" of these lines, or the angle B'CD, is therefore 200. Now, if local attraction at C causes the needle to point in the direcS'N', 10~ to the left of its proper direction, BC will bear N. 50s CHAP. III.] The Field Work. 145 E., and CD N. 70~ E., and the difference of these bearings, i. e. the angle of deflection, will be the same as before. (243) Angles between Courses. To determine the angle of deflection of two courses meeting at any point, the following simple rules, the reasons of which will appear from the accompanying figures, are sufficient. Fig. 168. Case 1. When the first letters of the bearing are alike, (i. e. both N. or both j / S.), and the last letters also alike, (i. e. both E. or both W.), take the difference / of the bearings. Example., IfAB bears \W-~ — N. 300 E. and BC bears N. 100 E., the angle of deflection CBB' is 200. JFig. 169. Case 2. When the first letters are alike and the last letters different; take \ the sum of the bearings. Ex. If AB bears N. 400 E. and BC bears N. 20~ W —---- -< W.; the angle CBB' is 600. 9.o...o Fig. 170. N / Case 3. When the first letters are / different and the last letters alike, sub- tract the sum of the bearings from 180~., Ex. If AB bears N. 300 E. and BCO...t... bears S. 40c0 E.; the angle CBB'is 1100. 10 146 COMPSS SURVEYING. [PART III Fig. 170. Case 4. When both the first and 30 last letters are different, subtract the " difference of the bearings from 1800o Ex. If AB bears S. 300 W. and B wC —.. --- bears N. 700 E.; the angle CBB' is / I 140~/ / I If tle angles included between the courses are desired, they will be at once found by reversing one bearing, and then applying the above rules; or by subtracting the results obtained as above from 1800; or an analogous set of rules could be formed for them. (244) To change Bearings. It is convenient in certain calculations to suppose one of the lines of a survey to change its direction so as to become due North and South; that is, to become a new Meridian line. It is then necessary to determine what the bearings of the other lines will be, supposing them to change with it. The subject may be made plain by supposing the survey to be platted in the usual way, with the North uppermost, and the plat to be then turned around, till the line to be changed is in the desired direction. The effect of this on the other lines will be readily seen. A (general Rule can also be formed. Take the difference between the original bearing of the side which becomes a Meridian and each of those bearings which have both their letters the same as it, or both different from it. The changed bearings of these lines retain the same letters as before, if they were originally greater than the original bearing of the new Meridian line; but, if they were less, they are thrown on the other side of the N. and S. line, and their last letters are changed; E. being put for W. and W for E. Take the sum of the original bearing of the new Meridian line, and each of those bearings which have one letter the same as one letter of the former bearing, and one different. If this sum exceeds CLAP, IIi.] The Field Ilork. 14i 900, this shews that the line is thrown on the other side of the East or West point, and the difference between this sum and 1800 will be the new bearing and the first letter will be changed, N. being put for S. and S. for N. Example. Let the Bearings of the sides of a field be as follows N. 320 E.; N. 80~ E.; S. 480 E.; S. 180 W.; N. 783o W.; North. Suppose the first side to become due North; the changed bearings will then be as follows: North; N. 480 E.; S. 800 E.; S. 140 E.; S. 7410 W.; N. 320 W. To apply the rule to the "' North" course, as above, it must be called N. 00 W.; and then by the Rule, 323 must be added to it. The true bearings can of course be obtained from the changed bearings, by reversing the operation, taking the sum instead of the difference, and vice versa. (245) Line SiTrveying. This name may be given to surveys of lines, such as the windings of a brook, the curves of a road, &c., by way of distinction from JFarm Surveying, in which the lines surveyed enclose a space. To survey a brook, or any similar line, set the compass at, or near, one end of it, and take the bearing of an imaginary or visual line, running in the general average direction of the brook, Fig. 172. such as AB in the figure. Measure this line, taking offsets to the various bends of the brook, as to the fence explained in Art. ( 1i). Then set the compass at B, and take a back-sight to A, and if they agree, take a fore-sight to C, and proceed as before, noting particularly the points where the line crosses the brook. To survey a road, take the bearings and lengths of the lines Fig. 173. I = 148 COMPASS SURVEYING. [PART II, which can be most conveniently measured in the road, and mea, sure offsets on each side, to the outside of the road. When the line of a new road is surveyed, the bearings and lengths of the various portions of its intended centre line should be measured, and the distance which it runs through each man's land should be noted. Stones should be set in the ground at recorded distances from each angle of the line, or in each line prolonged a known distance, so as not to be disturbed in making the road. In surveying a wide river, one bank may be surveyed by the method just given, and points on the opposite banks, as trees, &c., may be fixed by the method of intersections, founded on the Fourth Method of determining the position of a point; and fully explained in Part IV. (246) Checks by intersecting bearings. At each station at which the compass is set, take bearings to some remarkable object, such as a church steeple, a distant house, a high tree, &c. At least three bearings should be taken to each object to make it of any use: since two are necessary to determine it, (by our Fourth Method), and, till thus determined, it can be no check. When the line is platted, by the methods to be explained in the next chapter, plat also the lines given by these bearings. If those taken to the same object from three different stations, intersect in the same point, this proves that there has been no mistake in the sur. vey or platting of those stations. If any bearing does not intersect a point fixed by previous bear ings, it shows that there has been an error, either between the last station and one of those which fixed the point, or in the last bearing to the point. To discover which it was, plat the following line. of the survey, and, at its extremity, set off the bearing from it to the point;'and if the line thus platted passes through the point, it proves that there was no error in the line, but only in the bearing to the point. If otherwise, the error was somewhere in the line between the stations from which the bearings to that point were taken. chAP. III.] The Field Work, 149 (247) Keeping the Field-notes. The simplest and easiest method for a beginner is to make a rough sketch of the survey by eye, and write down on the lines their bearings and lengths. An improvement on this is to actually lay down the precise bear ings and lengths of the lines in the field-book in the manner to be explained in the chapter on Platting, Art. (269). (248) A second method is to draw a straight line up the page of the field-book, and to write on it the bearings and lengths of the lines. The only advantage of this method is that the line will not run off the side of the page, as it is apt to do in the preceding method. (219) A third method is to represent the line surveyed, by a double column, as in Part II, Chapter I, Art. (95), which should be now referred to. The bearings are written obliquely up the columns. At the end of each course, its length is written in the column, and a line drawn across it. Dotted lines are drawn across the column at any intermediate measurement. Offsets are noted as explained in Art. (114). The intersection-bearings, described in Art. (246), should be entered in the field-book before the bearings of the line, in order to avoid mistakes of platting, in setting off the measured distances on the wrong line. (250) A bourth method is to write the Stations, Bearings, and Distances in three columns. This is compact, and has the advantage, when applied to farm surveying, of presenting a form suitable for the subsequent calculations of Content, but does not give facilities for noting offsets. Examples of these four methods are given in Art. (4 5); which contains the field-notes of the lines bounding a field. (251) New-York Canal Iaps. The following is a description of the original maps of the survey of the line of the New-York Erie Qanal, as published by the Canal Commissioners. The figure represents a portion of such a map; but, necessarily, with all its lines black; rei and blue lines being used on the real map. 150 COMPASS SURVEYING, rPART BI Fig. 174. W. -}a L c S E 3-:2 $ 440 E links. The offsets at each station are represente~ by red lines d by te o se l, po The IRED LINE described along the inner edge of the towing pa thes the linbas nd the io h all the measu rem ents i the directhe magnetic meridian at the time of the survey. The lengths of the several portions are inserted at the end of each, in chains and links. The offsets at each station are represented by red lines drawn across the canal in such a direction as to bisect the angles formed by the two contiguous portions of the red or base line, upon the towingl path. The intermediate ofsets are set off at right angles to the base line; and the distances onll both are given from it irn links. The intermediate offsets are represented by red dotted lines, and the distances to them upon the base line are reckoned, in each case, from the last preceding station. The same is likewise done with the other distances upon the base line; those to the Bridges being taken to the lines joining the nearest angles, or corner posts of their abutments; those to the Locks extending to the lines passing through the centres of the two nearest quoin posts; and those to the Aqueducts, to the faces of their abutments. The space enclosed by the BLUE LINES represents the portion embraced within the limits of the survey as belonging to the state; and the names of the adjoining proprietors are given as they stood at the time of executing the survey. The distances are projected upon a scale of two chains to the inch."' (252) Farm Surveying. A farm, or field, or other space mcluded within known lines, is usually surveyed by the compass thus. Begin by walking around the boundary lines, and setting stakes at all the corners, which the flag-man should specially note, CHAP. III.] The Field Work. 151 so that he may readily find them again. Then set the compass at any corner, and send the flag-man to the next corner. Take tha bearing of the bounding line running from corner to corner, which is usually a fence. Measure its length, taking offsets if necessary. Note where any other fence, or road, or other line, crosses or meets it, and take their bearings. Take the compass to the end of this first bounding line; sight back, and if the back-sight agrees, take the bearing and distance of the next bounding line; and so proceed till you have got back to the point of starting. (253) Where speed is more important than accuracy in a survey, whether of a line or a farm, the compass need be set only at every other station, taking a forward sight, from,he 1st station to the 2d; then setting the compass at the 3d station, taking a backsight to the 2d station (but with the north point of the compass always ahead), and a fore-sight to the 4th; then going to the 5th, and so on. This is, however, not to be recommended. (254) Field-ntes, The Field-notes of a Farm survey may be kept by any of the methods which have been described with reference to a Line survey. Below are given the Field-notes of the same field recorded by each of the methods. First Method. Fig. 175. qN 8310 E Q'~- / ^K ^ ^ ^~k /^ 152 COMPASS SURVEYING, [PART III Second Third Fourth Method. Mdethod. MAethod.* 0 (1) -(1)3.23 STATIONS.] BEARINGS. DISTANCES. Cit^ | q 1I N. 35~ E. 2.70'3 l:} om 2 SN. 83 10 E. 1.29 1 8 3 S. 570 E. 2.22 o (5) 4 S. 34W s. 84. 3.55.o -(5) S I N.56 eW. 8.23 o to 3.54 Fig. 176. Q f 2.77 oCz) 1-~) 0() ^ -0.90 — A I 2.22 -0.25 —6 o(3) u/ ((D1.29 (2) o 1.34 Z^~~ ^ -0.70 —8 osets my e kept(2) in seprte 2.70 & 0(1) - / co.umn will contain the stations from which the measurements are z ~ —1- 0 -(1)(255) The Field-notes of a field, in which offsets occur, may be most easily recorded by the Third Method; as in Fig. 176. When the Field-notes are recorded by the Fourth Method, the offsets may be kept in a separate Table; in which the 1st co.umn will contain the stations from which the measurements are made, the 2d column the distances at which they occur, the 3d In the " Third Method," the bearings should be written obliauely upward, as directed in Art, (249) but arf not so printed here, from typographical diffi culties, CHAP, II.] The Field Work. 153 column the lengths of the offsets, and the 4th column the side of the line, " Right," or " Left," on which they lie. For calculation, four more columns may be added to the table, containing the intervals between the offsets; the sums of the adjoining pairs; and the products of the numbers in the two preceding columns, separated into Right and Left, one being additive to the field, and the other subtractive. (256) Tests of accuracy. 1st. The check of intersections de scribed in Art. (216), may be employed to great advantage, when some conspicuous object near the centre of the farm can be seen from most of its corners. 2nd. When the survey is platted, if the last course meets the starting point, it proves the work, and the survey is then said to " close." 3d. Diagonal lines, running from corner to corner of the farm, like the " Proof-lines" in Chain Surveying, may be measured and their bearings taken. When these are laid down on the plat, their meeting the points to which they had been measured, proves the work. 4th. The only certain and precise test is, however, that by "Latitudes and Departures." This is fully explained in Chapter V, of this Part. (257) A very fallacious test is recommended by several writers on this subject. It is a well-known proposition of Geometry, that in any figure bounded by straight lines, the sum of all the interior angles is equal to twice as many right angles, as the figure has sides less two; since the figure can be divided into that number of triangles. Hence this common rule. "Calculate [by the last paragraph of Art. (243)] the interior angles of the field or farm surveyed; add them together, and if their sum equals twice as many right angles as the figure has sides less two, the angles have been correctly measured." This rule is not applicable to a compass survey; for, in Fig. 167, page 144, the interior angle BCD will conrain the same number of degrees (in that case 160~) whether the bearings of the sides have been noted correctly, as being the 154 COMPASS StRVEYINIG [PAR III. argles which they make with NS-or incorrectly, as being the angles which they make with N'S'. This rule would therefore prove the work in either case. (258) Method of Radiatti0n A field may be surveyed from one station, either within it or without it, by taking the bearings and the distances from that point to each of the corners of the field. These corners are then " determined," by the 3d method Art. (7). This modification of that method, we named, in Art. (220), the Method of Radiation. All our preceding surveys with the compass have been by the Method of Progression. The compass may be set at one corner of the field, or at a point in one of its sides, and the same method of Radiation employed. This method is seldom used however, since, unlike the method of Progression, its operations are not checks upon each other. (259) IIethod of Intersection. A field may also be surveyed by measuring a base line, either within it or without it, setting the compass at each end of the base line, and taking, from each end, the bearings of each corner of the field; which will then be fixed and determined, by the 4th method, Art. (8). This mode of sur veying is the Method of Intersections, noticed in Art. (220). It will be fully treated of in Part V, under the title of Triangular Surveying, (260) Running out old lines, The original surveys of lands in the older States of the American Union, were exceedingly deficient in precision. This arose from two principal causes; the small value of land at the period of these surveys, and the want of skill in the surveyors. The effect at the present day is frequent dissat. isfaction and litigation. Lots sometimes contain more acres than they were sold for, and sometimes less. Lines which are straight in the deed, and on the map, are found to be crooked on the ground. The recorded surveys of two adjoining farms often make one overlap the other, or leave a gore between them. The most difficult and delicate duty of the land-surveyor, is to run out these old boundary lines. In such cases, his first business is to find CHAP. IIi.] The Field Work. 155 monuments, stones, marked trees, stumps, or any other old' corners, or landmarks. These are his starting points. The owners whose lands join at these corners should agree on them. Old fences must generally be accepted by right of possession; though such questions belong rather to the lawyer than to the surveyor.* His business is to mark out on the ground the lines given in the deed. When the bounds are given by compass-bearings, the surveyor must be reminded that these bearings are very far from being the same now as' originally, having been changing every year. The method of determining this important change, and of making the proper allowance, will be found in Chapter VIII, of this Part. (21l) Town Surveying. Begin at the meeting of two or more of the principal streets, through which you can have the longest prospects. Having fixed the instrument at that point, and taken the bearings of all the streets issuing from it, measure all these lines with the chain, taling offsets to all the corners of streets, lanes, bendings, or windings; and to all remarkable objects, as churches, markets, public buildings, &c. Then remove the instrument to the next street, take its bearings, and measure along the street as before, taking offsets as you go along, with the offset-staff. Proceed in this manner from street to street, measuring the distances and offsets as you proceed. Fig. 177..l ID t II // — I / " In the description of land conveyed, the rule is, that known and fixed mon~ aments control courses and distances. So, the certainty of metes and bounds will include and pass all the lands within them, though they vary from the given quantity expressed in the deed. In New-York, to remove, deface or alter land marks maliciously, is an indictable offence."-Keznt's Commentaries, IV, 515 156 COIPASS SRTVEYINT. [PART III Thus, in the figure, fix the instrument at A, and measure lines in the direction of all the streets meeting there, noting their bearings; then measure AB, noting the streets at X, X. At the second station, B, take the bearings of all the streets which meet there; and measure from B to C, noting the places and the bearings of all the cross-streets as you pass them. Proceed in like manner from C to DI and from D to A, " closing" there, as in a farm survey. Having thus surveyed all the principal streets in a particular neighborhood, proceed then to survey the smaller intermediate streets, and last of all, the lanes, alleys, courts, yards, and every other place which it may be thought proper to represent in the plan. The several cross-streets answer as good check lines, to prove the accuracy of the work. In this manner you continue till you take in all the town or city. (26g) Obstacles hi Compass Surveying. The various obstacles which may be met with in Compass Surveying, such as woods, water, houses, &c., can be overcome much more easily than in Chain Surveying. But as some of the best methods for effecting this involve principles which have not yet been fully developed, it will be better to postpone giving any of them, till they can be all treated of together; which will be done in Part VII. CHAPTER IV. PLATTNG THE SURVEEY (263) The platting of a survey made with the compass, consists in drawing on paper the lines and the angles which have been measured on the ground. The lines are drawn " to scale," as has been fully explained in Part I, Chapter III. The manner of plat" ting angles was referred to in Art. (41), but its explanation has been reserved for this place. (2i64) With a Protractor. Prortractor is an instrumenm made for this object, and is usually a semicircle of brass, as in the figure, with its semi-circumference divided into 180 equal parts, or Fig. 178, degrees, and numbered in both directions. It is, in fact, a minias ture of the instrument, (or of half of it), with which the angles have been measured. To lay off any angle at any point of a straight line, place the Protractor so that its straight side, the diameter of the semi-circle, is on the given line, and the middle of this diameter, which is marked by a notch, is at the given point. With a needle, or sharp pencil, make a mark on the paper at the required number of degrees, and draw a line from the mark to the given point. 158 COMPASS SIRVEYING. [PART III Sometimes the protractor has an arm turning on its centre, and extending beyond its circumference, so that a line can be at once drawn by it when it is set to the desired angle. A Vernier scale is sometimes added to it to increase its precision. A Rectangular Protractor is sometimes used, the divisions of degrees being engraved along three edges of a plane scale. The semi-circular one is preferable. The objection to the rectangular protractor is that the division corresponding to a degree is very Fig. 179., \ \, \ _\_\ \ \ \I/'?/ //// / 7/ /! o'o iio isoiiin Oo loo - o unequal on different parts of the scale, being usually two or three times as great at its ends as at its middle. A Protractor embracing an entire circle, with arms carrying verniers, is also sometimes employed, for the sake of greater accuracy. (265) Platting Bearings. Since' Bearings" taken with the Compass are the angles which the various lines make with the Magnetic MAeridian, or the direction of the compass-needle, which, as we have seen, remains always (approximately) parallel to itself, it is necessary to draw these meridians through each station, before laying off the angles of the bearings. The T square, shown in Fig. 14, is the most convenient instrument for this purpose. The paper on which the plat is to be made is fastened on the board so that the intended direction of the North and South line may be parallel to one of the sides of the board. The inner side of the stock of the T square being pressed against one of the othf the other sides of the board and slid along the edge of the long blade of the square will always be parallel to itself and to the first named side of the board, and will thus represent the meridian passing through any station. CHAP. iv.] Platting the Survey. 159 If a straight-edged drawing Fig. 180. board or table cannot be procured, nail down on a table of any shape a straight-edged ru- /0 ler, and slide along against it the outside of the stock of a T. square, one side of the stock - being flush with the blade. A parallel ruler may also be used, one part of it being ____ screwed down to the board in the proper position. If none of these means are at hand, approximately parallel meridians may be drawn by the edges of a common ruler, at distances apart equal to its width, and the diameter of the protractor made parallel to them by measuring equal distances between it and them. (266) To plat a survey with these instruments, mark, with a fine point enclosed in a circle, a convenient spot in the paper to represent the first station, 1 in the figure. Its place must be so chosen Fig. 181. 3 a S1-~~ K /y^ ^^0~n 1 K ^~4 160 COMPASS SURVEYING, [PART IiI that the plat may not " run off" the paper. With the T square draw a meridian through it. The top of the paper is usually, though not necessarily, called North. With the protractor lay off the angle of the first bearing, as directed in Art. (264). Set off the length of the first line, to the desired scale, by Art. (42), from 1 to 2. The line 1 —— 2 represents the first course. Through 2, draw another meridian, lay off the angle of the second course, and set off the length of this course, from 2 to 3. Proceed in like manner for each course. When the last course is platted, it should end precisely at the starting point, as the survey did, if it were a closed survey, as of a field. If the plat does not " close," or " come together," it shows some error or inaccuracy either in the original survey, if that have not been 1" tested" by Latitudes and Departures, or in the work of platting. A method of correction is explained in Art. (268). The plat here given is the same as that of Fig. 175, page 151. This manner of laying down the directions of lines, by the angles which they make with a meridian line, has a great advantage, in both accuracy and rapidity, over the method of platting lines by the angles which each makes with the line which comes before it. In the latter method, any error in the direction of one line makes all that follow it also wrong in their directions. In the former, the direction of each line is independent of the preceding line, though its position would be changed by a previous error. Instead of drawing a meridian through each station, sometimes only one is drawn, near the middle of the sheet, and, all the bearings of the survey are laid off from some one point of it, as shown in the figure, and numbered to correspond with the stations from which these bearings were taken. The circular protractor is convenient for this. They are then transferred to the places where they are wanted, by a triangle or other parallel ruler, as explained on page 27. The figure at the top of the next page represents the same field platted by this method. A semi-circular protractor is sometimes attached to the stock end of the T square, so that its blade may be set at any desired angle with the meridian, and any bearing be thus protracted with. out drawing a meridian. It has some inconveniences. 1ar.3 Platting the Survey, 161 Fig. 182. S 5 (267) The Compass itself may be used to plat bearings. For this purpose it must be attached to a square board so that the IN and S line of the compass box may be parallel to two opposite edges of the board. This is placed on the paper, and the box is turned till the needle points as it did when the first bearing was taken. Then a line drawn by one edge of the board will be in a proper direction. Mark off its length, and plat the next and the succeeding bearings in the same manner. (268) When the plat of a survey does not "' close," it may be corrected as follows. Let Fig. 183. ABODE be the boundary I lines platted according to C,O the given bearings and B distances, and suppose that the last course comes to E, Ag~ / / instead of ending at A, as \'~ A / / it should. Suppose also I _ r / that there is no reason to suspect any single great error, and that no one of the lines was measured over very rough 11 1(2 C03A$S SURVEYING. [PART n1 ground, or was specially uncertain in its direction when observed, The inaccuracy must then be distributed among all the lines il proportion to their length. Each point in the figure, B, C, D,E, must be moved in a direction parallel to EA, by a certain distance which is obtained thus. Multiply the distance EA by the distance AB, and divide by the sum of all the courses. The quotient will be the distance BB'. To get CC', multiply EA by AB + BC, and divide the product by the same sum of all the courses. To get DD', multiply EA by AB + BC + CD, and divide as before. So for any course, multiply by the sum of the lengths of that course and of all those preceding it, and divide as before. Join the points thus obtained, and the closed polygon AB'C'D'A will thus be formed, and will be the most probable plat of the given survey.' The method of Latitudes and Departures, to be explained hereafter, is, however, the best for effecting this object. (269) Field Platting. It is sometimes desirable to plat the courses of a survey in the field, as soon as they are taken, as was mentioned in Art. (247), under the head of " Keeping the fieldnotes." One method of doing this is to have the paper of the Field-book ruled with parallel lines, at unequal distances apart, and to use a rectangular pro- Fig. 184. tractor (which may be made -___ of Bristol-board, or other stout - drawing paper,) with lines ruled across it at equal distances, X of some fraction of an inch. A _ bearing having been taken and ________ noted, the protractor is laid on l-: the paper and its centre placed at the station where the bearing is to be laid off. It is then turned till one of its cross-lines coincides with some one of the lines on the paper, which represent East and West lines. The long side of the protractor will then be on a meridian and the proper angle (400 in the figure) can be at once marked off. The length of the course can also be set off by the equal spaces between the cross-lines, letting each space represent any convenient number of links. This was demonstrated by Dr. BOWDITCH, in No. 4, of " The Analyst.' CHAP. Iv.] Platting the Survey, 163 (270) A common rectangular protractor without any cross-lines, or a semi-circular one, can also Fig. 185. be used for the same purpose. The parallel lines on the paper (which, in this method, may be equi-cistant, as in common t ruled writing paper) will now / / represent mercidans. Place the centre of the protractor on the meridian nearest to the station at which the angle is to be laid off, and turn it till the _ _ _ given number of degrees is cut by the meridian. Slide the protractor up or down the meridian (which must continue to pass through the centre and the proper degree) till its edge passes through the station, and then draw by this edge a lino, which will have the bearing required. (271) Paper ruled into squares, (as are sometimes the righthand pages of surveyors' field-books), may be used for platting bearings in the field. The lines running up the page may be called North and South lines, and those running across the page will then be East and West lines. Any course of the survey will be the hypothenuse of a right-angled triangle, and the ratio of its other t.-,~~~~~~~~ ^. - ^Fig. 186. two sides will determine the rigangle. Thus, if the ratio of _ C B the two sides of the right-an- _ gled triangle, of which the line _ __ AB in the figure is the hypoth-. / _ 7 enuse, is 1, that line makes an 1f / angle of 450 with the meridian. - If the ratio of the long to the 7short side of the right-angled t - triangle of which the line AC __ //// I_-I is the hypothenuse, is 4 to 1, the line AC mnakes an angle of 140 with the. meridian. The line AD, the hypothenuse of an 164 COMPASS SURVEYING. [PART II1 equal triangle, which has its long side lying East and West, makes likewise an angle of 14~ with that side, and therefore makes an angle of 760 with the meridian.i To facilitate the use of this method, the following table has baen prepared. TABLE FOR PLATTING BY SQUARES. ~' 0 0 ~ ~~ ] Ratio of Rati o of Rto of 1 (0 Io si ong side to' long side to t4 loner sideto o0 Q i lona side to long side to D o short side. -C short side. short side. S 1~ 57.3 to 1 89o 160 3.49 to 1 74o 310 1.664 to 1 590 30 19.1 to 1 870 180 3.08 to 1 720 330 1.540 to 1 570 4014.13 to 1 860 190 2.90 to 171 3401.483 to 1 56 5011.4 to 1 850 200 2.5 to1700 350 1.428 to 1 550 60 9.5 to 1 840 210 2.61 to 1 690 36C 1.376 to 1 540 70 8.1 to 1 830 220 2.48 to 1 680 370 1.327 to 1 530 80 7.1 to 1 820 2302.36 to 1 670 380 1.280 to 1 520 90 6.3 to 11810 240 2.25 to 1660 3901.235 to 1 510 10 5.7 to 1 800 250 2.14 to 1 650 4001.1921 to 1500 110 5.1 to 1 790 260 2.05 to 1 64 410 1.150 to 1 490 120 4.9 to 11780 270 1.96 to 1 630 420~1.111 to 11480 130 4.3 to 11770 280 1.88 to 1 620 4301.072 to 1 470 140 4.0 to 1 760 290 1.80 to 11610 440 1.036 to 11460 150 3.7 to 1 750 300 1.73 to 1 600 450 1.000 to 1 450 To use this table, find in it the ratio corresponding to the angie which you wish to plat. Then count, on the ruled paper, any number of squares to the right or to the left of the point which represents the station, according as your bearing was East or West; and count upward or downward according as your bearing was North or South, the number of squares given by multiplying the first number by the ratio of the Table. Thus; if the given bearing from A in the figure, was N. 200 E. and two squares were counted to the right, then 2 x 2.75 = 5 squares, should be counted upward, to E, and AE would be the required course. (272) With a paper protractor. Engraved paper protractors may be obtained from the instrumentmakers, and are very convex This and all the following ratios may be obtained directly fiom Trigonome. trical Tables; for the ratio of the long side to the short side, the latter being baken as unity, is the natural cotangent of the angle. CHAP. IV.] Platting the Survey, 165 nient. A circle of large size, divided into degees and cuarters, is engraved on copper, and impressions from it are taken on drawing paper. The divisions are not numbered. Draw a straight line to represent a meridian, through the centre of the circle, in any convenient direction. Number the degrees from 0 to 90~, each way from the ends of this meridian, as on the compass-plate. The protractor is now ready for i. 187. use. Choose a convenient point for the first station.' Suppose the first bearing to/ \ be N. 30~ E. The line pass- ng through the centre of the 2. W E circle and through the oppo- site points N. 30~ E. and S. 30~ W. has the bearing re-/ i/ quired. But'it does not pass S through the station 1. Transfer it thither by draiwing through station I a line parallel to it, which will be the course required, its proper length being set off on it from 1 to 2. Now suppose the bearing from 2 to be S. 60~ E. Draw through 2 a line parallel to the line passing through the centre of the circle and through the opposite points S. 60~ E., and N. 60~ W., and it will be the line desired. On it set off the proper length from 2 to 3, and so proceed. When the plat is completed, the engraved sheet is laid on a clean one, and the stations " pricked through," and the points thus obtained on the clean sheet are connected by straight lines. The pencilled plat is then rubbed off from the engraved sheet, which can be used for a great number of plats. If the central circle be cut out, the plat, if not too large, can be made directly on the paper where it is to remain. The surveyor can make such a paper protractor for himself, with great ease, by means of the Table of Chords at the end of this volume, the use of which is explained in Art. (27;). The engraved ones may have shrunk after being printed. Such a circle is sometimes drawn on the map itself. This will be particularly convenient if the bearings of any lines on the map, 166 COM]PA$S SURVEYING. [PART In. not taken on the ground, are likely to be required. If the map be very long, more than one may be needed. (273) Drawuig-Board Protractor. Such a divided circle, as has just been described, or a circular protractor, may be placed on a drawing board near its cenrie, and so that its 0~ and 900 lines are parallel to the sides of the drawing board. Lines are then to be drawn, through the centre and opposite divisions, by a ruler long enough to reach the edges of the drawing board, on which they are to be cut in, and numbered. The drawing board thus becomes, in fact, a double rectangular protractor. A strip of white paper may have previously been pasted on the edges, or a narrow strip of white wood inlaid. When this is to be used for platting, a sheet of paper is put on the board as usual, and lines are drawn by a ruler laid across the 0~ points and the 90~ points, and the centre of the circle is at once found, and should be marked'. The bearings are then platted as in the last method. (274) With a scale of chords. On the plane scale contained in cases of mathematical drawing instruments will be found a series of divisions numbered from 0 to 90, and marked CH, or O. This is a scale of chords, and gives the lengths of the chords of any arc for a radius equal in length to the chord of 60~ on the scale. To lay off an angle with this scale, as for Fig. 188. example, to draw a line making at A an angle E of 40~ with AB, take, in the dividers, the dis- / tances from 0 to 60 on the scale of chords; with: this for radius and A for centre, describe an in- / definite arc CD. Take the distance from 0 to 40 on the same scale, and set it off on the arc as a chord, from C to some point D. Join AD, and A prolong it. BAE is the angle required. The Sector, represented on page 36, supplies a modification of this method, sometimes more convenient. On each of its legs is a scale marked C, or CH. Open it at pleasure; extend the corn pass from 60 to 60, one on each leg, and with this radius describe an arc. Then extend the compasses from 40 to 40, and the dis. CHAP. IV.] Platting the Survey. 167 tance will be the chord of 40~ to that radius. It can be set off as above. The smallness of the scale renders the method with a scale of chords practically deficient in exactness; but it serves to illustrate the next and best method. (275) Wiith a Table of chords, At the end of this volume will be found a Table of the lengths of the chords of arcs for every degree and minute of the quadrant, calculated for a radius equal to 1. To use it, take in the compasses one inch, Dne foot, or any other convenient distance (the longer the better) divided into tenths and hundredths, by a diagonal scale, or otherwise. With this as radius describe an arc as in the last case. Find in the table of chords the length of the chord of the desired angle. Take it from the scale just used, to the nearest decimal part which the scale will give. Set it off as a chord, as in the last figure, and join the point thus obtained to the starting point. This gives the angle desired. The superiority of this method to that which employs a protractor, is due to the greater precision with which a straight line can be divided than can a circle. A slight modification of this method is to take in the compasses 10 equal parts of any convenient length, inches, half inches, quarter inches, or any other at hand, and with this radius describe an arc as before, and set off a chord 10 times as great as the one found in the Table, i. e. imagine the decimal point moved one place to the right. If the radius be 100 or 1000 equal parts, imagine the decimal paJnt moved two, or three, places to the right. Whatever radius may be taken or given, the product of that radius into a chord of the Table, will give the chord for that radius. This gives an easy and exact method of getting a right angle; by describing an are with a radius of 1, and setting off a chord equal to 1.4142. If the angle to be constructed is more than 900, construct on the other side of the given point, upon the given line prolonged, an angle equal to what the given angle wants of 180~; i. e. its Su?7plement, in the language of Trigonometry. 168 COMPASS SURVEYINGT [PART 11I, This same Table gives the means of measuring any angle. With the angular point for a centre, and 1, or 10, for a radius, describe an arc. Measure the length of the chord of the arc between the legs of the angle, find this length in the Table, and the angle corresponding to it is the one desired.* (276) With a Table of natural sines, In the absence of a Table of chords, heretofore rare, a table of natural sines, which can be found anywhere, may be used as a less convenient substitute. Since the chord of any angle equals twice the sine of half the angle, divide the given angle by two; find in the table the natural sine of this half angle; double it, and the product is the chord of the whole angle. This can then be used precisely as was the chord in the preceding article. An ingenious modification of this method has been much used. Describe an arc from the given point as centre, as in the last two articles, but with a radius of 5 equal parts. Take, from a Table. the length of the natural sine of half the given angle to a radius of 10. Set off this length as a chord on the arc just described, and join the point thus obtained to the given point.t (277) By Latitudes and Departures. When the Latitudes and Departures of a survey have been obtained and corrected, (as explained in Chapter V), either to test its accuracy, or to obtain its content, they afford the easiest and best means of platting it. The description of this method will be given in Art. (285). * This Table will also serve to find the natural sine, or cosine, of any angle. Multiply the given angle by two; find, in the Table, the chord of this double angle; and half of this chord will be the natural sine required'or, the chord of any angle is equal to twice the sine of half the angle. To find the cosine, pro. ceed as above, with the angle which added to the given angle would make 90~. Another use of this Table is to inscribe regular polygons in a circle by setting off the chords of the arcs which their sides subtend. Still another use is to divide an arc or angle into any number of equal parts by setting off the fractional are or angle. Fig. 189. tThe reason of this is apparent from the figure. DE is the sine of half the angle BAC, to a radius of 10 equal parts, and o > BC is the chord directed to be set off, to a / radius of 5 cqual parts. BO'is equal to DE; I for BC = 2.BF, by Trigonometry, and DE -- 2.BF, by similar triangles; hence BO = DE. CHAPTER V. LATITUDES AND DEPARTUREH (278) DefiBitions, The LATITUDE of a point is its distance North or South of some " Parallel of Latitude," or line running East or West. The LONGITUDE of a point is its distance East or West of some "61 eridian," or line running North and South. In Compass-Surveying, the Magnetic Meridian, i. e. the direction in which the Magnetic Needle points, is the line from which the Longitudes of points are measured, or reckoned. The distance which one end of a line is due North or South of the other end, is called the Difference of Latitttde of the two ends of the line; or its Northing or Southing; or simply its Latitude. The distance which one end of the line is due East or West of the other, is here called the Difference of Longitude of the two ends of the line; or its Easting or WVesting; or its PDepartzure. Latitudes and IDepyartures are the most usual terms, and will be generally used hereafter, for the sake of brevity. This subject may be illustrated geographically, by noticing that a traveller in going from New-York to Buffalo in a straight line, would go about 150 miles due north, and 250 miles due west. These distances would be the differences of Latitude and of Longitude between the two places, or his Northing and Westing. Returning from Buffalo to New-York, the same distances would be his Southing and Easting.* In mathematical language, the operation of finding the Latitude and Longitude of a line from its Bearing and Length, would be called the transformation of Polar Co-ordinates into Rectangular Co-ordinates. It consists in determining, by our Second Principle, the position of a point which had originally been determined by the Third Princilple. Thus, in the figure, (which is the same as * It should be remembered that the following discussions of the Latitudes and Longitudes of the points of a survey will not always be fully applicable to those of distant places, such as the cities just named, in consequence of the surface of the earth not being a plane. 170 COMIPASS SUJEYENGI [PART III that of Art.(9)), the point S is determin- Fi 190. ed by the angle SAC and by the dis- tance AS. It is also determined by the distances AC and CS, measured at right angles to each other; and then, supposing A' c CS to run due North and South, CS will be the Latitude, and AC the Departure of the line AS. (279) Calclation of Latitudes and Departires' Let AB be a given line, of which the length Fig. 191. AB, and the bearing (or'angle, BAC, which it makes with the Magnetic Meridian), are known. It is required C to find the differences of Latitude and I of Longitude between its two extremities A and B: that is, to find AC and / CB; or, what is the same thing, BD B -- -- -— 1 — and DA. It will be at once seen that ABt is the hypothenuse of a right-angled tri- angle, in which the 1" Latitude" and the " Departure' are the sides about the right angle. We therefore know, from the principles of trigonometry, that AC = AB. cos. BAC, BC - AB. sin. BAC. Hence, to find the Latitude of any course, multiply the natural cosine of the bearing by the length of the course; and to find the Departure of any course, multiply the natural sine of the bearing by the length of the course. If the course be Northerly, the Latitude will be North, and will be marked with the algebraic sign +, plus, or additive; if it be Southerly, the Latitude will be South, and will be marked with the algebraic sign -, minus, or subtractive. If the course be Easterly, the Departure will be East, and marked +-, or additive; if the course be Westerly, the Departure will be West, and marked -, or subtractive. CHAP. vo Latitudes and Departuies. 171 (280) Formulase The rules of the preceding article may be expressed thus; Latitude = Distance x cos. Bearing, Departure = Distance x sin. Bearing.* From these formulas may be obtained others, by which, when any two of the above four things are given, the remaining two can be found. IWhen fte Bearing and Latitude are given; Distance = tnt.ee_ = Latitude X sec. Bearing, cos. Bearillng Departure = Latitude x tang. Bearing. When the Bearing and Departure are given; Distance = Dep.al l.'e = Departure x cosec. Bearing, sin. Bearing Latitude = Departure x cotang. Bearing. When the -Distance and Latitude are given; Latit ude Cos. Bearing Latitude DilstanceO Departure = Latitude x tang. Bearing. When the -Distance and.Departure are given; Departrure Sin. Bearing D= - ne Distance Latitude = Departure x cotang. Bearing. When the Latitude and Departure are given; Tang. of Bearing Depirture Distance = Latitude x sec. Bearing. Still more simply, any two of these three-Distance, Latitude and Departure-being given, we have Distance = v (Latitude2 + Departure2) Latitude =,/(Distance2 - Departure2) Departure = v (Distance2 -Latitude2) (281) Traverse Tables. The Latitude and Departure of any distance, for any bearing, could be found by the method given in Art. (279), with the aid of a table of Natural Sines. But to Whenever sines, cosines, tangents, &c., are here named, they mean the natu ral sines &c., of an arc described wiith a radius equal to one, or to t'e unit by which the sines, &c., are measured. 172 COMPASS SURVEYINS. [PART III facilitate these calculations, which are of so frequent occurrence and of so great use, Traverse Tables have been prepared, originally for navigators, (whence the name Traverse), and subsequently for surveyors.' The Traverse Table at the end of this volume gives the Latitude and Departure for any bearing, to each quarter of a degree, and for distances from 1 to 9. To use it, find in it the number of degrees in the bearing, on the left hand side of the page, if it be less than 45~, or on the right hand side if it be more. The numbers on the same line running across the page,t are the Latitudes and Departures for that bearing, and for the respective distances-1, 2, 3, 4, 5, 6, 7, 8, 9,which are at the top and bottom of the page, and which may represent chains, links, rods, feet, or any other unit. Thus, if the bearing be 15~, and the distance 1, the Latitude would be 0.966 and the Departure 0.259. For the same bearing, but a distance of 8, the Latitude would be 7.727, and the Departure 2.071. Any distance, however great, can have its Latitude and Departure readily obtained from this table; since, for the same bearing, they are directly proportional to the distance, because of the similar triangles which they form. Therefore, to find the Latitude or Departure for 60, multiply that for 6 by 10, which merely moves the decimal point one place to the right; for 500, multiply the numbers found in the Table for 5, by 100, i. e. move the decimal point two places to the right, and so on. Merely moving the decimal point to the right, one, two, or more places, will therefore enable this Table to give the Latitude and Departure for any decimal multiple of the numbers in the Table. For compound numbers, such as 873, it is only necessary to find separately the Latitudes and Departures of 800, of 70, and of 3, and add them together. But this may be done, with scarcely any risk of error, by the following simple rule. * The first Traverse Table for Surveyors seems to have been published in 1791, by John Gale. The most extensive table is that of Capt. Boileau, of the British army, being calculated for every minute of bearing, and to five decimal places, for distances from 1 to 10. The Table in this volume was calculated for it, and then compared with the one just mentioned. t In using this or any similar Table, lay a ruler across the page, just above or below the line to be followed out. This is a very valuable mechanica. assistance cHAP. v.] Latitudes and Departures, 173 Write down the Latitude and Departure for the first figure of the given number, as found in the Table, neglecting the decimal point; write under them the Latitude and Departure of the second figure, setting them one place farther to the right; under them write the Latitude and Departure of the third figure, setting them one place farther to the right, and so proceed with all the figures of the given number. Add up these Latitudes and Departures, and cut off the three right hand figures. The remaining figures will be the Latitude and Departure of the given number in links, or chains, or feet, or whatever unit it was given in. For example; let the Latitude and Departure of a course hav ing a distance of 873 links, and a bearing of 20~, be required. In the Table find 20", and then take out the Latitude and Departure for 8, 7 and 3, in turn, placing them as above directed, thus: Distances. Latitudes. Departures. 800 7518 2736 70 6578 2394 3 2819 1026 873 820.399 298.566 Taking the nearest whole numbers and rejecting the decimals, we find the desired Latitude and Departure to be 820 and 299.* When a 0 occurs in the given number, the next figure must be set two places to the right, the reason of which will appear from the following example, in which the 0 is treated like any other number. Given a bearing of 350, and a distance of 3048 links. Distances. Latitudes. Departures. 3000 2457 1721 000 0000 0000 40 3277 2294 8 6553 4589 3048 2496.323 1748.529 Herehethe Latitudes and Departures are 2496 and 1749 links. It is frequently doubtful, in many calculations, when the final decimal is 5, whether to increase the preceding figure by one or not. Thus, 43.5 may be called 43 or 44 with equal correctness. It is better in such cases not to increase the whole number, so as to escape the trouble of changing the original figure, and the increased chance of error. If, however, more than one such a case occurs in the same column to be added up, the larger and smaller number should be taken alternately. 174 COMPASS SUJRVEYING. [PART III When the bearing is over 450, the names of the columns must be read from the bottom of the page, the Latitude of any bearing, as 50~, being the Departure of the complement of this bearing, or 40~0 and the Departure of 40~ being the Latitude of 50~, &c. The reason of this will be at once seen on inspecting the last figure, (page 170), and imagining the East and West line to become a Meridian. For, if AC be the magnetic meridian, as before, and therefore BAG be the bearing of the course AB, then is AC the Lati. tude, and CB the Departure of that course. But if AE be the meridian and BAD (the complement of BAC) be the bearing, then is AD (which is equal to CB) the Latitude, and DB, (which is equal to AC), the Departure. As an example of this, let the bearing be 638~, and the distance 3469 links. Proceeding as before, we have Distances. Latitudes. Departures. 3000 1350 2679 400 1800 3572 60 2701 5358 9 4051 8037 3469. 1561.061 3097.817 The required Latitude and Departure are 1561 and 3098 links. In the few cases occurring in Compass-Surveying, in which the bearing is recorded as somewhere between the fractions of a degree given in the Table, its Latitude and Departure may be found by interpolation. Thus, if the bearing be 103~, take the half sum of the Latitudes and Departures for 10l o and 10h~. If it be 10~ 20', add one-third of the difference between the Lats. and Deps. for 10 and for 10 ~, to those opposite to 10~; and so in any similar case. The uses of this table are very varied. The principal applicae tions of it, which will now be explained, are to Testing the accuracy of surveys; to Supplying omi.ssions in them; to Platting them, and to Calculating their content. * The Traverse Table admits of many other minor uses. Thus, it may be used for solving, approximately, any right-angled triangle by mere inspection, the bearing being taken for one of the acute angles; the Latitude being the side adjacent, the Departure the side opposite, and the Distance the hypothenuse. Any two of these being given, the others are given by the Table. The Table will therefore serve to show the' allowance to be made in chaining on slopes (see Art, CHAP, v.] Latitudes and Departures. 175 (282) Applcation to Testing a Surey. It is self-evident, that when the surveyor has gone completely aroun a field or farm, taking the bearings and distances of each boundary line, till he has got back to the starting point, that he has gone precisely as far South as North, and as far West as East. But the sum of the North Latitudes tells how far North he has gone, and the sum of the South Latitudes how far South he has gone. Hence these two sums will be equal to each other, if the survey has been correctly made. In like manner, the sums of the East and of the West Departures must also be equal to each other. We will apply this principle to testing the accuracy of the survey of which Fig. 175, page 151, is a plat. Prepare seven columns, and head them as below. Find the Latitude and Departure of each course to the nearest link, and write them in their appropriate columns. Add up these columns. Then will the difference between the sums of the North and South Latitudes, and between the sums of the East and West Departures, indicate the degree of accuracy of the survey. -}, -, iLATITUDE. DEPARTURE. STATION. BEARING. DISTANCE. _. S. E. W. 1 N. 350 E. 2.70 2.21' 1.55 2 N. 83b0 E. 1.29.15 1.28 3 S. 57~ E. 2.22 1.21 1.86 4 S. 34 0`W. 3.55 2.93 2.00 5 N. 56 W. 3.23 1.78 2.69 _____ 1 _ 4.14 14.14 4.69 4.69 The entire work of the above example is given below. 350 1638 1147 34{O 2480 1688 57340 40150 4133 2814.~_... ~ 4133 2814 270. 221.140 154.850 - 355. 293.463 199.754 (26)); fr, look ih the column of bearings for the slope of the ground, i. e. the angle it makes witI the horizon, find the given distance, and the Latitude aorre. sponding will be the desired horizontal measurement, and the difference between it and the Distance will be the allowance to be made I76 COMPISS SURTVEYING. [PART IM. 83-0o 113 994 5610 1656 2502 226 1987 1104 1668 1019 8942 1656 2502 129. 14.579 128.212 323. 178.296 269.382 570 1089 1677 50 1089 1677 The nearest link is taken 1089 1677 to be inserted in the Table, ____ and the remaining Decimals 222. 120.879 186.147 are neglected. In the preceding example the respective sums were found to be exactly equal. This, however, will rarely occur in an extensive survey. If the difference be great, it indicates some mistake, and the survey must be repeated with greater care; but if the difference be small it indicates, not absolute errors, but only inaccura, cies, unavoidable in surveys with the compass, and the survey may be accepted. How great a difference in the sums of the columns may be allowed, as not necessitating a new survey, is a dubious point. Some surveyors would admit a difference of 1 link for every 3 chains in the sum of the courses: others only 1 link for every 10 chains. One writer puts the limit at 5 links for each station; another at 25 links in a survey of 100 acres. But every practical surveyor soon learns how near to an equality his instrument and his skill will enable him to come in ordinary cases, and can therefore establish a standard for himself, by which he can judge whether the difference, in any survey of his own, is probably the result of an error, or only of his customary degree of inaccuracy, two things to be very carefully distinguished.' (283) Application to supplying omissions. Any two omis sions in the Field-notes can be supplied by a proper use of the method of Latitudes and Departures; as will be explained in Part VII, which treats of " Obstacles to Measurement," under which head this subject most appropriately belongs. But a knowledge of the fact that any two omissions can be supplied, should not lead * A French writer fixes the allowable difference in chaining at 1-400 of level lines 1-200 of lines on moderate slopes; 1-100 of lines on steep slopes. CHAP..] Latitudes and Departures. 177 the young surveyor to be negligent in making every possible measurement, since an omission renders it necessary to assume all the notes taken to be correct, the means of testing them no longer existing. (284) Balancing a Survey. The subsequent applications of this method require the survey to be previously Balanced. This operation consists in correcting the Latitudes and Departures of the courses, so that their sums shall be equal, and thus' balance." This is usually done by distributing the differences of the sums among the courses in proportion to their length; saying, As the. sum of the lengths of all the courses Is to the whole difference of the Latitudes, So is the length of each course To the correction of its Latitude. A similar proportion corrects the Departures.* It is not often necessary to make the exact proportion, as the correction can usually be made, with sufficient accuracy, by noting how much per chain it should be, and correcting accordingly. In the example given below, the differences have purposely been made considerable. The corrected Latitudes and Departures have been here inserted in four additional columns, but in practice they should be written in red ink over the original Latitudes and Departures, and the latter crossed out with red ink. ATE _ TS CORRECTED CORRECTED LATITUDES. DEPlTURES. STA. BEARING, DIST. _L LATITUDES. DEPARTURES. ____ __ N.+ S.- E.+ W.- N.+ S.- E.+ W.1 N. 2~ E. 10.63 6.54 8.38 6.58 8.34 2 S. 291~ E. 4.10 8.56 2.03 3.55 2.01 3 S. 314 W. 7.69 6.54 4.05 4.058 4 N. 610 W. 7.13 3.46 6.24 3.48 6.21'29.55 10.00 10.10 10.41 10.29 10.06 10.06 10.35 10.35 The corrections are made by the following proportions; the nearest whole numbers being taken: For the Latitudes, For the.Departures. 29.55:10.63:: 10 4 29.55: 10.63: 12: 4 29.55: 4.10: 10: 1 29.55: 4.10::12: 2 29.55:.69:: 10: 3 29.55:.69:: 12: 3 29.55:.13: 10: 2 29.55: T.13:12: 3 10 12 *A demonstration of this principle was given by Dr. Bowditch, in No. 4 of "The Analyst." 12 178 COIMPASS SURVEYING. [PART IIm This rule is not always to be strictly followed. If one line of a survey has been measured over very uneven and rough ground, or if its bearing has been taken with an indistinct sight, while the other lines have been measured over level and clear ground, it is probable that most of the error has occurred on that line, and the correction should be chiefly made on its Latitude and Departure. If a slight change of the bearing of a long course will favor the Balancing, it should be so changed, since the compass is much more subject to error than the chain. So, too, if shortening any doubtful line will favor the Balancing, it should be done, since distances are generally measured too long. (285) Application to Platting. Rule three columns; one for Stations; the next for total Latitudes; and the third for total De. partures. Fill the last two columns by beginning at any convenient station (the extreme East or West is best) and adding up (algebraically) the Latitudes of the following stations, noticing that the South Latitudes are subtractive. Do the same for the Departures, observing thatthe Westerly ones are also subtractive. Taking the example given on page 175, Art. (282), and beginring with Station 1, the following will be the results: TOTAL LATITUDES TOTAL DEPARTURES' FROM STATION 1. FIR1O STATION 1. 1 0.00 0.00 2 +2.21 N. +1.55 E. 3 +2.36 N. +2.83 E. 4 +1.15 N. +4.69 E. 5 -1.78 S. +2.69 E. 1 0.00 0.00 It will be seen that the work proves itself, by the total Latitudes and Departures for Station 1, again coming out equal to zero. To use this table, draw a meridian through the point taken for Station 1, as in the figure on the following page. Set off, upward from this, along the meridian, the Latitude, 221 links, to A, and from A, to the right perpendicularly, set off the Departure, 155 lmks.! This gives the point 2. Join 1....2. From 1 again, set This is most easily done with the aid of a right-angled triangle, sliding one of the sides acdjacent to the right angle along the blade of the square, to which She other side will then be perpendicular. CHAP, v.] Latitudes and Departurese 179 off, upward, 236 Fig. 192. links, to B, and from B, to the right, per- __.__- __. pendicularly, set off 283 links, which will fix thepoit 3. Join 2.... 8; and so pro- c ceed, setting off North Latitudes along the Meridian Lt/ up-wards, and South Latitudes along it downwards; East Departures perpendicularly to the right, D a and West Depar- II tures perpendicularly to the left. The advantages of this method are its rapidity, ease and accu. racy; the impossibility of any error in platting any one course affecting the following points; and the certainty of the plat comig together," if the Latitudes and Departures have been Balanced." CHAPTER VI. CALCTLATING THE CONTENT. (286) lethods, WTYR a field has been platted, by what ever method it may have been surveyed, its content can be obtained from its plat by dividing it up into triangles, and measuring on the plat their bases and perpendiculars; or by any of the other means explained in Part I, Chapter IV. But these are only approximate methods; their degree of accuracy depending on the largeness of scale of the plat, and the skill of the draftsman. The invaluable method of Latitudes and Departures gives another means, perfectly accurate, and not requiring the previous preparation of a plat. It is sometimes called the Rectangular, or the Pennsylvania, or Rittenhouse's, method of calculation.' (287) DefinatiouSe Imagine a Meridian line to pass through the extreme East or West corner of a field. According to the definitions established in Chapter V, Art. (278), (and here recapitulated for convenience of reference), the perpendicular distance of each Station from that Meridian, is the Longitude of that Station; additive, or plus, if East; subtractive, or minus, if West. The distance of the middle of any line, such as a side of the field, from the Meridian, is called the Longitude of that side.t The difference of the Longitudes of the two ends of a line is called the Departure of that line. The difference of the Latitudes of the two ends of a line is called the Latitude of the line. * It is, however, substantially the same as Mr. Thomas Burgh's "Method to determine the areas of right lined figures universally,' published nearly a century ago. + The phrase " Meridian Distance," is generally used for what is here called Longitude"; but the analogy of" Diferences of Longitude" with " Differences of Latitude," usually but anomalously united with the word " Departure," bor. rowed from Navigation, seems to put beyond all question the propriety of the innovation here introduced. CHAP. VI.] Calculating the Content. 181 (288) Longitudes, To give more definteness to the develop mnent of this subject, the figure in the margin will be referred to, and may be considered to represent any space enclosed by straight lines. Let NS be the Meridian passing through the extreme Westerly Station of the field ABODE. From Fi9. 193. the middle and ends of each side ] C draw perpendiculars to the Meridi- l ) / an. These perpendiculars will be --- f- —-- the Longitudes and Departures of \ the respective sides. The Longi- tude, FG, of the first course, AB, is evidently equal to half its Depar- r I ture HB. The Longitude, JK, of the second course, BC, is equal to / A JL + LM + MK, or equal to the v __/ Longitude of the preceding course, Y: / plus half its Departure, plus half \ / the Departure of the course itself. Ti --- ---—. The Longitude, YZ, of some other course, as EA, taken anywhere, is equal to WX -- VX - UV, or equal to the Longitude of the preceding course, minus half its Departure, minus half the Departure of the courst itself, i. e. equal to the Algebraic sum of these three parts, remembering that Westerly Departures are negative, and therefore to be subtracted when the directions are to make an Algebraic addition. To avoid fractions, it will be better to double each of the preceding expressions. We shall then have a GENERAL RULE FOR FINDING DOUBLE LONGITUDES. The Doouble Longitude of the FIRST COURSE is equal to its )De parture. The Double Longitude of the SECOND COURSE is equal to the Dzuble Longitude of the first course, plus the Departure of that course, plus the Departure of the second course. The Double Longitude of the THIRD COURSE is equal to the Double Longitude of the second course, plus the D)eparture of tha, ^ourse, plus the Departure of the course itself. 182 COIPASS SURVEYING. [PART III The Double Longitude of ANY course is equal to the Double Longitude of the preceding course, plus the.Departure of that course, plus the iDeparture of the course itseyf. The Double Longitude of the last course (as well as of the first) is equal to its Departure. Its "' coming out" so, when obtained by the above rule, proves the accuracy of the calculation of all the preceding Double Longitudes. (289) Areas, We will now proceed to find the Area, or Con. tent of a field, by means of the "' Double Longitudes" of its sides, which can be readily obtained by the preceding rule, whatever their number. (290) Beginning with a three-sided field, ABC in the figure, draw a Meridian through A, and draw perpendi- Fig. 194. culars to it as in the last figure. It is 3N plain that its content is equal to the differ- X ence of the areas of the Trapezoid DBCE, " and of the Triangles ABD and ACE. The area of the Triangle ABD is equal A to the product of AD by half of DB, or to H —---. —---- the product of AD by FG; i. e. equal to the product of the Latitude of the 1st course by its Longitude. The area of the Trapezoid DBCE is equal C to the product of DE by half the sum of DB I and CE, or by HJ; i.e. to the product of D the Latitude of the 2d course by its Longitude. The area of the Triangle ACE is equal to the product of AE by half EC, or by KL; i. e. to the product of the Latitude of the 8d course by its Longitude. Calling the products in which the Latitude was North, North Products, and the products in which the Latitude was South, South Products, we shall find the area of the Trapezoid to be a South Product, and the areas of the Triangles to be North Pro* The last course is a "preceding course" to the first course, as will appear on remembering that these two courses join each other on the ground CHAP. VI.] Calculating the Content, 18I ducts. The Difference of the North Pr)oducts acnd the South Products is therefore the desired area of the three-sided field ABC, Using the Double Longitudes, (in order to avoid fractions), in each of the preceding products, their difference will be the double area of the Triangle ABC. (291) Taking now a four-sided field, ABCD in the figure, and drawing a Meridian and Longitudes as be- Fig. l10. fore, it is seen, on inspection, that its area N would be obtained by taking the two Trian- B gles, ABE,ADG, from the figure EBCDGE, or from the sum of the two Trapezoids EBCF and FCDG. l —- -------- The area of the Triangle AEB will be found, as in the last article, to be equal to A the product of the Latitude of the 1st course ---- by its Longitude. The Product will be North. The area of the Trapezoid EBCF will be - - found to equal the Latitude of the 2d course t by its Longitude. The product will be South. The area of the Trapezoid FCDG will be found to equal the product of the Latitude of the 3d course by its Longitude. The product will be South. The area of the Triangle ADG will be found to equal the pioduct of the Latitude of the 4th course by its Longitude. The product will be _North. The difference of the North and South products will therefore be the desired area of the four-sided field ABCD. Using the Double Longitude as before, in each of the preceding products, their difference will be double the area of the field. (292) Whatever the number or directions of the sides of a field, or of any space enclosed by straight lines, its area will always be equal to half of the difference of the North and South Products 184 C0MPASS SURVEYING. [PART IIT arising from multiplying together the Latitude and Double Longi. tude of each course or side. We have therefore the following GENERAL RULE FOR FINDING AREAS. 1. Prepare ten columns, headed as in the example below, and in the first three write the Stations', Bearings and Distances. 2. Find the.Latitudes and Departures of each course, by the Traverse Table, as directed in Art. (2S1), placing them in the four following columns. 3. Balance them, as in Art. (284), correcting them in red ink. 4. Find the Do1uble Longitudes, as in Art. (288), with reference to a Meridian passing through the extreme East or West Station, and place them in the eighth column. 5. Multiply the Double Longitude of each course by the corrected Latitude of that course, placing the North Products in the ninth column, and the South Products in the tenth column. 6. Add up the last two columns, subtract the smaller sum from the larger, and divide the difference by two. The quotient will be the content desired. (293) To find the most Easterly or Westerly Station of a sur. vey, without a plat, it is best to make a rough hand-sketch of the survey, drawing the lines in an approximation to their true directions, by drawing a North an South, and East and West lines, and considering the Bearings as fractional parts of a right angle, or 90~; a course N. 450 E. for example, being drawn about half say between a North and an East direction; a course N. 28~ W. being not quite one-third of the way around from North to West; and so on, drawing them of approximately true proportional lengths. (294) Example 1, given below, refers to the five-sided field, of which a plat is given in Fig. 175, page 151, and the Latitudes and Departures of which were calculated in Art. (282), page 175. Station 1 is the most Westerly Station, and the Meridian will be supposed to pass through it. The Double Longitudes are best CHAP. VI.] Calculating the Cosntent 185 found by a continual addition and subtraction, -STA.. 1 as in the margin, where they are marked D. L. +- 1.55 D. L -+- 1.55 The Double Longitude of the last course comes + 1.28 out equal to its Departure, thus proving the 2 +.38 D. L. + 1.28 work. + 1.86 The Double Longitudes being thus obtained, + 7.52 L. are multiplied by the corresponding Latitudes, 2.00 and the content of the field obtained as directed 4 +7.38 D. L. in the General Rule. 2.69 This example may serve as a pattern for the 5 + 2.69 D. L1 most compact manner of arranging the work. D13- LATITUDES. DEP'TURES. B DOUBLE 1REAS. STTION RI. TANCES. N. + S.- E.+ W.- LONGITUDES. N+ S. - 1 N. 35~ E. 2.70 2.21 1.55 + 1.55 3.4255 2 N. 83 o E. 1.29.15 1.28 + 4.38 0.6570 3 S. 570 E. 2.22 121 1.86 + 7.52 9.0992 4 S. 341~ W. 3.55 2.93 2.00 + 7.38 21.6234 5 N. 560~ W. 3.23 1.78 2.69 -+ 2.69 4.7882 4.14 4.14 4.69 1 4.69 1 18.8707 130.7226 8.8707 (contente lA. OR. 15P. 2)21.8519 Squaire Chains, 10.9259 (295) The Meridian might equally well have STA, been supposed to pass through the most Easterly 4 - 00 D. L. station, 4 in the figure. The Double Longitudes - 2.69 could then have been calculated as in the mar- 5 - 6.69 D. L. gin. They will of course be all West, or minus. - 1s55 The products being then calculated, the sum of 1 - 7.83 D. L. n,, r~VUIMVUN IYVL+ -- 1.55 the North products will be found to be 29.9625, + 1.28 and of the South products 8.1106, and their 2 -5.00 ). L. -\- 1.28 difference to be 21.8519, the same result as be- + 1.86 fore, 3 1.86 (296) A number of examples, with and without answers, will now be given as exercises for the student, who should plat them by some of the methods given in the preceding chapter, using each of them at least once. He should then calculate their content by the method just given, and check it, by also calculating the area of the plat by some of the Geometrical or Instrumental methods given in Part I, Chapter IV; for no single calculation is ever reliable. 186 COIPASS SURVEYING [PART II1, All the examples (except the last) are from the author's actual surveys. 3 Fig. 196..Example 2, given below, is also fully worked out, as anoth- / er pattern for the student, who / need have no difficulty with any possible case if he strictly follows the directions which have been given. The plat is on a scale of 2 chains to 1 inch, (= 1:1584)........_ / DIS- LATITUDES. DEP'TURES. DOUBLE DOUBLE AREAS. STATION. BEARINGS. STATION TANCES.. + iS.- E.+ W.- LONGITUDES. N. + S.1 N. 12~ E. 2.81 2.75.60 + 6.56 18.0400 2 N. 76~ W. 3.20.77 3.11 + 4.05 3.1185 3 S. 24~0 W. 1.14 1.04.47 +.47.4888 4 8.48~ E. 1.53 1.02 1.14 + 1.14 1.1628 5 S. 12A E. 1.12 1.09.24 + 2.52 2.7468 6 S. 77~ E. 1.64.37 1.60 + 4.36 1.6132 35 3.52 3.58 3.58 1 3 21.1585l6.01i 6.0116 Content=OA. 3R. 1P. 2)15.1469 Square Chains, 7.5734.Example 3..Example 4. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 N.52~ E. 10.64 1 S. 21~W. 12.41 2 S. 290 E. 4.09 2 N. 83s 0E. 5.86 3 S. 31~0 W. 7.68 3 N. 12~ E. 8.25 4 N. 610 W. 7.24 4 N. 47~ W. 4.24 Ans. 4A. 3R. 28P. Ans. 4A. 2R. 37P. Example 5. -Example 6. STA. BEARING. DISTANCE. STA. BEARING. _DISTANCE. 1 N. 34' 0E. 2.73 1 N. 350 E. 6.49 2 N. 85~ E. 1.28 2 S. 561o E. 14.15 3 S. 561" E. 2.20 3 S. 340 W. 5.10 4. 34s W. 3.53 4 N. 560 W. 5.84 5 N. 561^ W. 3.20 5 S. 291 W. 2.52 Ans. 1A. OR. 14P. 6 N.48 W. 8'( ECAP. vI.] Calculating the Content. 187 Example 7. Example 8. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1S. 211 W. 17.62 1 S. 651 E. 4.98 2 S. 34o W. 10.00 2 S. 580~. 8.56 3 N. 560 W. 14.15 3 S. 14~ W. 20.69 4 N. 34O E. 9.76 4 S. 47" W. 0.60 5 N. 67~ E. 2.30 5 S. 571 W. 8.98 6 N. 23 E. 7.03 6 N. 56~ W. 12.90 7 N. 181 EF. 4.43 7. 24~ E. 10.00 S. 761~ E. 12.41 8 N. 21} E. 17.62 Example 9. Example 10. STA. BEARING. DISTANCE. STA. BEARING.:DISTANCE. I S. 57~ E2. 5.77 1 N. 63" 51' W. 6.91 2 S. 361~ W. 2.25 2 N. 63~ 44' W 7.26 3 S. 391~ W. 1.00 3 N. 690 35' W. 3.34 4 S. 70~ WN. 1.04 4 N. 77~ 50' W. 6.54 5 N. 683~ W. 1.23 5 N. 31~ 24' E. 14.38 6 N. 56~ W. 2.19 6 N. 31~ 18' E. 16.81 7 N. 33M E. 1.05 7 S. 68~ 55' E. 13.64 8 N. 56~ W. 1.54 8 S. 680 42' E. 11.54 9 N.33 E. 3.18 9 S, 33 45' W. 31.55 Ans. 2A. OR. 32P. Ans. 74 Acres. Example 11. Example 12. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 N. 183 E. 1.93 1 N. 7230 E. 0.88 2 N. 9~ W. 1.29 2 S. 200 E. 0.22 3 N. 140 W. 2.71 S. 630 E. 0.75 4 N. 74 E. 0.95 4 N. 510 E. 2.35 5 S. 481 E. 1.59 5 N. 440 E. 1.10 6 S 1410 E. 1.14 6 N. 252 WN. 1.96 7 S 19~ E. 2.15 7 N. 81~ W. 1.05 8 S. 23~0 W. 1.22 8 S. 290 W. 1.63 9 S. 50 W. 1.40 9 N. 710 W. 0.81 10 S. 300~ W. 1.02 10 N. 13'~ w. 1.17 11 S. 811~ W. 0.69 11 N. 63~. 1.28 12 N. 32 W.. 1.98 12 West. 1.68 13 N 49~ W. 0.80 14 S. 192 E. 6.20 188 COMPASS SURVEYTING [PART m, Example 13. A farm is described in an old Deed, as bounded thus. Beginning at a pile of stones, and running thence twentyseven chains and seventy links South-Easterly sixty-six and a half degrees to a white-oak stump; thence eleven chains and sixteen links North-Easterly twen- Fig. 197. ty and a half degrees to a hickory tree; thence two chains and thirty-five links North-Easterly thirty-six degrees to the South-East-/ 5 erly corner of the homestead; thence nineteen chains and thirty-two links North-Easterly twenty-six degrees to a stone set in 1 the ground; thence twentyeight chains and eighty links North-Westerly sixty-six A, degrees to a pine stump; thence thirty-three chains and nineteen links South-Westerly twenty-two degrees to the place of beginning, containing ninety-two acres, be the same more or less. Required the exact content. (297) laseheroni's Theorm,. The surface of any polygon is equal to half the sum of the products of its sides (omitting any one side) taken two and two, into the sines of the angles which those sides make with each other. Fig. 198. Thus, take any polygon, such as the fivesided one in the figure. Express the angle which the directions of any two sides, as AB, CD, make - with each other, thus (AB/ACD). Then will Athe content of that polygon be, as below; = [AB. BC. sin (AB A BC) + AB. CD. sin (AB / CD) + AB. DE. sin (AB A DE) + BC. CD. sin (BC A CD) -+ BC. DE. sin (BC A DE) + CD. DE. sin (CD A BE)] CHAP. vii.] Vartation of the lagneati Needle. 189 The demonstration consists merely in dividing the polygon into triangles by lines drawn from any angle, (as A); then expressing the area of each triangle by half the product of its base and the perpendicular let fall upon it from the above named angle; and finally separating the perpendicular into parts which can each be expressed by the product of some one side into the sine of the angle made by it with another side. The sum of these triangles equals the polygon. The expressions are simplified by dividing the proposed polygon into two parts by a diagonal, and computing the area of each part separately, making the diagonal the side omitted.* CHAPTER VI1. THE VTRIIATION OF THE 3IVAI gETIC NEEDLE, (298) Defiitions. The Mlagnetic Mlferidian is the Fig. 199 direction indicated by the Magnetic Needle. The True e, Mferidian is a true North and South line, which, if produced, would pass through the poles of the earth. The Variation, or Declination, of the needle is the angle which one of these lines makes with the other.t GA In the figure if NS represent the direction of the True Meridian, and N'S' the direction of the Magnetic Meridian at any place, then is the angle NAN' the Variation of the Needle at that place. (299) Direction of Needle, The directions of these two meridians do not generally coincide, but the needle in most places points to the East or to the West of the true North, more or less The original Theorem is usually accredited to Lhuillier, of Geneva, who published it in 1789. But Mascheroni, the ingenious author of the " Geometry of the Compasses," had published it at Pavia, two years previously. The method is well developed in Prof. Whitlock's " Elements of Geometry.' + " Declination" is the more correct term, and ".Variation" should be reserved for the change in the Declination which will be considered in the next chapter; but custom has established the use of Variation in the sense of Declination. 190 CO0MPASS SURVEYINll [PART Iit according to the locality. Observations of the amount and the direction of this variation have been made in nearly all parts of the world. In the United States the Variation in the Eastern States is Westerly, and in the Western States is Easterly, as will be given in detail, after the methods for determining the True Meridian, and consequently the Variation, at any place, have been explained. TO DETERMINE THE TRUE MERIDIAN. (300) By eqiual shadows of the Sun. On the South side of any level surface, erect an up- Fig. 200. right staff, shown, in horizontal projection, at S. Two or three hours before noon, mark the extremity, A, of its shadow. /'\ Describe an arc of a circle with S, the foot of the staff, for centre, and SA, the distance to the extremity of the shadow, for radius. About as many hours after noon as it had been before noon when the first mark was made, watch for the moment when the end of the shadow touches the arc at another point, B. Bisect the arc AB at N. Draw SN, and it will be the true meridian, or North and South line required. For greater accuracy, describe several arcs before hand, mark the points in which each of them is touched by the shadow, bisect each, and adopt the average of all. The shadow will be better defined, if a piece of tin with a hole through it be placed at the top of the staff, as a bright spot will thus he substituted for the less definite shadow. Nor need the staff be vertical, if from its summit a plumb-line be dropped to the ground, and the point which this strikes be adopted as the centre of the arcs. This method is a very good approximation, though perfectly correct only at the time of the solstices; about June 21st and December 22d. It was employed by the Romans in laying out cities. To get the Variation, set the compass at one end of the True Meridian line thus obtained, sight to the other end of it, and take CHAP. vii.] Variation of the Magnetie Needle. 191 the Bearing as of any ordinary line. The number of degrees in the reading will be the desired variation of the needle. (30I) By the North Star, when in the IerMian. The North Star, or Pole Star, (called by astronomers Alpha Ursce Minonis, or Polaris), is not situated precisely at the North Pole of the heavens. If it were, the Meridian could be at once determined,y sighting to it, or placing the eye at some distance behind a plumbs line so that this line should hide the star. But the North Star is about 1.~ from the Pole. Twice in 24 hours, however, (more precisely 23h. 56m.), it is in the Meri- Fig. 201. dian, being then exactly above or below A the Pole, as at A and C in the figure. To know when it is so, is rendered easy by the aid of another star, easily identified, which B' D at these times is almost exactly above or below the North Star, i. e. situated in the same vertical plane. If then we watch for....^ the moment at which a suspended plumb- line will cover both these stars, they will then be in the Meridian. The other star is in the well known constellation of the Great Bear, called also the Plough, or the Dipper, or Charless Wain. Fig. 202.'Fig. 203. eb T* X 4t 7 Two of its five bright stars (the right-hand ones in Fig. 202) are known as the " Pointers," from their pointing near to the North 192 COMIPASS SURVEYING. [PART IIL Star, thus assisting in finding it. The star in the tail or handle, nearest to the four which form a quadrilateral, is the star which comes to the Meridian at the same time with the North Star, twice in 24 hours, as in Fig. 202 or 203. It is known as Alioth. or Epsilon Ursce 3Jajoris.' To determine the Meridian by this method, suspend a long plumb-line from some elevated point, such as a stick projecting from the highest window of a house suitably situated. The plumbbob may pass into a pail of water to lessen its vibrations. South of this set up the compass, at such a distance from the plumb-line that neither of the stars will be seen above its highest point, i. e. in Latitudes of 400 or 50 notquite asfarfrom the plumb-line as it is long. Or, instead of a compass, place a board on two stakes, so as to form a sort of bench, running East and West, and on it place one of the compass-sights, or anything having a small hole in it to look through. As the time approaches for the North Star to be on the Meridian (as taken from the table given below) place the compass, or the sight, so that, looking through it, the plumb-line shall seem to cover or hide the North Star. As the star moves one way, move the eye and sight the other way, so as to constantly keep the star behind the plumb-line. At last Alioth, too, will be covered by the plumb-line. At that moment the eye and the plumb-line are (approximately) in the Meridian. Fasten down the sight on the board till morning, or with the compass take the bear ing at once, and the reading is the variation.f Instead of one plumb-line and a sight, two plumb-lines may be suspended at the end of a horizontal rod, turning on the top of a pole. The line thus obtained points to the East of the true line when the North Star is above Alioth, and vice versa. The North Star is exactly in the Meridian about 17 minutes after it has been in the same vertical plane with Alioth, and may be sighted to after that interval of time, with perfect accuracy. * The North Pole is very nearly at the intersection of the line from Polaris to Alioth, and a perpendicular to this line from the small star seen to the left of it in Fig. 202. t If a Transit or Theodolite be used, the cross-hairs must be illuminated by throwing the light of a lam! into the telescope by its reflection from white paper CHAP. vII.] Variation of the Magnetic Needle. 193 Another bright star, which is on the opposite side of the Pole, and is known to astronomers as Gammna cassiopeice, also comes on the Meridian nearly at the same time as the North Star, and will thus assist in determining its direction. (302) The time at which the North Star passes the Meridian above the Pole, for every 10th day in the year, is given in the fol. lowing Table, in common clock time.' The upper transit is the most convenient, since at the other transit Alioth is too high to be conveniently observed. MONTH.r 1st DAY. 11th DAY. 21st DAY. H. M. H. MH. M. i January, 6 21 P. M. 5 41 Pi M. 5 02, p. M. February, 4 18 P.M. 3 89 P.M. 3 00 P. M. t arch, 2 28 P. M. 1 49 P. M. 1 09 P.M..X April, 0 26 P. M. ll 47 A. Ai. II 08 A. M., May, 10 28 A. M. 9 49 A. M. 9 10 A. M. I June, 8 27 A. M. 7 48 A. M. 7 08 A. M. July, 6 29 A.. 5 50 A. M. 5 11 A. M. k August, 4 28 A. M. 3 49 A.. 3 09 A. M. September, 2 26 A. M.. 1 47 M. M. 1 07 A. A. x October, 0 28 A. M.11 45 P. M. 11 06 p. rM. I November, 10 22 P. M. 9 43 P. M. 9 04 P. M. December, 8 24 P. M. 7 45 P. M. 7 06 p. M. * To calculate the time of the ime North Star passing the Meridian at its upper cu[ uination: Find in the " American Almanac," (Boston), or the " Astronomical Ephemeris," (Washington), or the " Nautical Almanac," (London), or by interpolation from the data at the end of this note, the right ascension of the star, and from it (increased by twenty-four hours if necessary to render the subtraction possible) subtract the Right ascension of the Sun at mean noon, or the sidereal time at mean noon, for the given day, as found in the " Ephemeris of the Sun," in the same Almanacs. From the remainder subtract the acceleration of sidereal on mean time corresponding to this remainder, (3m. 56s. for 24 hours), and the new remainder is the required mean solar time of the upper passage of the star across the Meridian, in "Astronomical" reckoning, the astronomical day beginning at noon of the common civil day of the same date. The right ascension of the North Star for Jan. 1, 1850, is lh. 05m. 01.4s.; for 1860, lh. 08m. 02.8s.; for 1870, lh. 11m. 16.9s.; for 1880, lh. 14m. 45.1s.; for 1890, lh. 18m. 29.2s.; for 1900, lh. 22m. 31s. 13 194 COIM SS SURVEYING, [PART HIi. To find the time of the star's passage of the Meridian for other days than those given in the Table, take from it the time for the day most nearly preceding that desired, and subtract from this time 4 minutes for each day from the date of the day in the Table to that of the desired day; or, more accurately, interpolate, by saying: As the number of days between those given in the Table is to the number of days from the next preceding day in the Table to the desired day, so is the difference between the times given in the Table for the days next preceding and following the desired day to the time to be subtracted from that of the next preceding day. The first term of the preceding proportion is always ten, except at the end of months having more or less than 30 days. For example, let the time of the North Star's passing the Meridian on July 26th be required. From July 21st to August 1st being 11 days, we have this proportion: 11 days: 5 days:: 43 minutes: 19A6 minutes. Taking this from 5h. 11m. A. M., we get 4h. 511m. A. M. for the time of passage required. The North Star passes the Meridian later every year. In 1860, it will pass the Meridian about two minutes later than in 1854; in 1870, five minutes, in 1880, eight minutes, in 1890, twelve minutes, and in 1900, sixteen minutes, later than in 1854: the year for which the preceding table has been calculated. the times at which the North Star passes the Meridian below the Pole, in its lower Transit, can be found by adding llh. 58m. to the time of the upper Transit, or by subtracting that interval from it.* (303) By the torth Star at its extreme elongatlon. When the North Star is at its greatest apparent angular distance East or West of the Pole, as at B or D in Fig. 201, it is said to be at its extreme Eastern, or extreme Western, Elongation. If it be observed at either of these times, the direction of the Meridian can be easily * The North Star, which is now about lo 28' from the Pole, was 12~ distant from it when its place was first recorded. Its distance is now dimlinishing at the rate of about a third of a minute in a year, and will continue to do so till it apjproaches to within half a degree, when it will again recede. The brightest star in the Northern hemisphere, Alpha Lyra, will be the role Star in about 12,000 years, being then within about 50 of the Pole, though now more than 510 distant from it CHAP. VIi.] Variation of the Magnetic Needle. 195 obtained from the observation. The great advantage of this method over the preceding is that then the star's motion apparently ceases for a short time. (304) The following Table gives the TIMES OF EXTREME ELONGATIONS OF THE NORTH STAR.* MONTH. IST DAY. 11TH DAY. 21ST DAY. EASTERN. WESTERN. EASTERN. WESTERN. EASTERN. WESTERN. H. M. H. 1. H. M. II. M. H. M. H. -. Jan'y, 0 27 P.M. 0 19 A.M. 11 47 A.M. 11 35 P.M. 11 08 A.M. 10 o8 P.J. Feb'y, 10 24A.M. 10 13 P.M. 9 45A.M. 9 33 P.M. 9 06A.M. 8 54 P.i. March, 8 34 A.. 8 22 P.M. 7 55 A.M. 7 43 P.M. 7 15 A.M. 704 P.. April, 632A.M. 6 20 P.M. 553 A.M. 5 41 P.M. 514A.M. 5 02 P.M. May, 4 34 A.M. 4 22 P.. 3 55 A.M. 3 43 P.M. 316A.M. 3 04 P.M. June, 2 3 A.M. 2 21P.M. 1 53A.M. 142 P.M. 1 14 A.. 1 02 P.IM. July, 0 35A.M. 0 23 p.Ml. 1152p.M.1144A.M.1113 P.M. 1105A.MQ. August, 10 30 P.M. 10 22A.M. 9 51 P.M. 943 A.M. 911 P.M. 903.M. Sept'r, 8 28 P.M. 820A.M. 749 P.M. 7 41A.M. 709 PM. 701 sM. Octr, 6 30 P.M. 622A.MI. 551 P.M. 5 43A.M. 512 P.M. 504A,:, Nov'r, 4 28 P.M. 421A.M. 3 49 P.. 3 41A.M. 3 10 P.M. 302A.M. Dec'r, 2 30p.M. 222A.M. 151pP.i. 143A.M. 112P.M. 104A.Y The Eastern Elongations from October to March, and the Western Elongations from April to September, occurring in the day time, they will generally not be visible except with the aid of a powerful telescope. * To calculate the times of the greatest elongation of the North Star: Find in one of the Almanacs before referred to, or from the data below, it Polar dis. tance at the given time. Add the logarithm of its tangent to the logarithm of the tangent of the Latitude of the place, and the sum will be the logarithm of the cosine of the Hour angle before or after the culmination. Reduce the space to time; correct for sidereal acceleration (3m. 56s. for 24 hours) and subtract the result fiom the time of the star's passing the meridian on that day, to get the time of the Eastern elongation, or add it to get the Western. The Polar distance of the North Star, for Tan. 1, 1850, is 1~ 29' 25"; for 1860, 1V 26' 12".7; for 1870, 1~ 23~ 01"; for 1880, 10 12 50".4; for 1890, 1~ 16' 40".7 for 1900, \" 13' 32".2. 196 COMPASS SURVEYING, [PART III The preceding Table was calculated for Latitude 40~. Ths Time at which the Elongations occur vary slightly for other Latitudes. In Latitude 50~, the Eastern Elongations occur about 2 minutes later and the Western Elongations about 2 minutes earlier than the times in the Table. In Latitude 26~, precisely the reverse takes place. The Times of Elongation are continually, though slowly, becoming later. The preceding Table was calculated for July 1st, 1854. In 1860, the times will be nearly 2 minutes later; and in 1900, the Eastern Elongations will be about 15 minutes, and the Western Elongations 17 minutes later than in 1854. (305) Observationso Knowing from the preceding Table the hour and minute of the extreme Elongation on any day, a little before that time suspend a plumb-line, precisely as in Art. (301), and place yourself south of it as there directed. As the North Star moves one way, move your eye the other, so that the plumbline shall continually seem to cover the star. At last the star will appear to stop moving for a time, and then begin to move backwards. Fix the sight on the board (or the compass, &c.) in the position in which it was when the star ceased moving; for the star ~was then at its extreme apparent Elongation, East or West, as the case may be. (306) Azimuths. The angle which the line from the eye to the plumb-line, makes with the True Meridian (i. e. the angle between the meridian plane and the vertical plane passing through the eye and the star) is called the Azimuth of the Star. It is given in the following Table for different Latitudes, and for a number of years to come, For the intermediate Latitudes, it can be obtained by a simple proportion, similar to that explained in detail in Art. (302).' * To calculate this Azimuth From the logarithm of the sine of the Polar dis tance of the star, subtract the logarithm of the cosine of the Latitude of the place; the remainder will be the logarithm of the sine of the angle required, The PF lar distance can be obtained as directed in the last note. CHAP. vii.] Variation of the Magnetic Needle. 197 AZIMUjTHS OF THE NORTH STAR. Latitudes. 1854 1855 1856 1857 1858 1859 1860 1870 500 2 1%6' 20 16%'2016' 20~15' 20 15' 20 14'120 14' 2009L 490 2- 14' 20 13-'2 138-'2012320 122-' 201 20 111220 061 48 20 11'20 11 20 10 20 10' 20 091o'l 2 009' 20 09, 20 04' 470- 20 09' 20 08I'20 08' 20 07/3 20 07' 2~ 06' 320 0Q6 2I 01' 46 20 06' 2 06' 2 051' 22 05' 2 05 2~ 04' 20 041 o 591 450 20 04'/ 20 04' 20 03'j20 031' 20 02-/ 23 02- U20 02' 1o 57' 440 202' 2002' 2 j013 20 011'200 1 2 000 2 000' 1 552 430' 20 00420 00' Io 59 o10 59' io 581 /10 58410 58 1 L53' 420 10 58 10 58' 10 57l' 10 571'o 1 563/to 56/1 56 t 1051' 410 D t 10561 S 56' 10 55x'1055 110 o55/ 1 54l 1 54' 1/ 50 40~ I I 55' to 54110 54' 534 105 5310 ~1 53/ 1~ 521/1~ 481' 390 1053 /10 523 /1 52' 10 52/ 1to 51 1 i10 s 51 51/ 1i 4 380 10 51310 51'a1 51' 1 o 1050' 10 50' 1 493 11 49a 0 45y/ 370 wi 50 to 49 493 1 4 9' 489 10 48' h1 48' lo 44' 360 ao 48 /1i48 48 048 o 10473 1 4 it 47' 10 46/ 10 42 350 10 471 110 47' 1o 463' 10 46 10o 46/ 10 45'1 1 451' 1~ 41' 340 10 46t10 453/104514 lo045/' 1 44'10 144/ 10 44' 1i 404 33o t? 45' 1044 10 44 10 43 1043T1043' 1042i1039 8320 I10 44' 10 43,'10 43' 10 423 11042:' 1042/ 1o 414 10o 38 310 10 423' 10 42'10 42' 10411'10 41' 10 403/ 140' 10 37 30~ 10 411 10 411' 41' 10 4010 40 10 40' 1 3911036' (307) Setting ot a Meridian* When two points in the direction of the North Star at its extreme elongation have been Fig. 204. obtained, as in Art. (305), the True Meridian can be k found thus. Let A and B be the two points. Multiply the natural tangent of the Azimuth given in the Table, by the distance AB. The product will be the length of a line which is to be set off from B, perpendicular to AB, to some point C. A and C will then be points in the True Meridian. This operation may be postponed till morning. Clj'B If the directions of both the extreme Eastern and extreme Western elongations be set out, the line lying midway between them will be the True Meridian. A 198 COIPASS SURVEYITG, [PART IIi (308) Determinng the Variaton, The variation would of course be given by taking the Bearing of the Meridian thus obtained, but;i can also be determined by taking the Bearing of the star at the time of the extreme elongation, and applying the following rules. When the Azimuth of the star and its magnetic bearing are one East and the other West, the sum of the two is the Magnetic Varition, which is of the same name as the Azimuth; i. e. East, if that be East, and West, if it be West. When the Azimuth of the star and its Magnetic Bearing are both East, or both West, their difference is the Variation, which will be of the same name as the Azimuth and Beanig, if the Azimuth be the greater of the two, or of the contrary name if the Azimuth be the smaller. Fig. 205. All these cases are presented together in the 2 NX V figure, in which P is the North Pole; Z the place / of the observer; ZP the True Meridian; S the star at its greatest Eastern elongation; and ZN, ZN',ZN", various supposed directions of the needle. Call the Azimuth of the star, i. e. the angle PZS, 2~ East. Suppose the needle to point to N, and the Bearing of the star, i. e. SZN, to be 5~ West of Magnetic North. The variation PZN will evidently be 7~ East of true North. z Suppose the needle to point to N', and the bearing of the star, i. e. N'ZS, to be 11~ East of Magnetic North. The Variation will be 3~ East of true North, and of the same name as the Azimuth, because that is greater than the bearing. Suppose the needle to point to N" and the bearing of the star, i. e. N"ZS, to be 10~ East of Magnetic North. The Variation will be 8~ West of true North, of the contrary name to the Azimuth, because that is the smaller of the two.O * Algebraically, always subtract the Bearing from the Azimuth, and give the remainder its proper resulting algebraic sign. It will be the Variation; East if plus, and West, if minus. Thus in the first case above, the Variation -- 2~( — 5) += 7~ =- 7 East. In the second case, the Variation = -+- 2~ - ( + 1~) + ~ = = 1~ East. In the third case, the Variation = + 20 -(+ 10 )8~ 83 WVest. CHAP. VII.] Variation of the lagnetic Needle. 199 If the star. was on the other side of the Pole, the rules would Apply likewise. (309) Other Methods. Many other methods of determining the true Meridian are employed; such as by equal altitudes and azimuths of the sun, or of a star; by one azimuth, knowing the time; by observations of circumpolar stars at equal times before and after their culmination, or before and after their greatest elongation, &c All these methods however require some degree of astronomical knowledge; and those which have been expl:ined are abundantly sufficient for all the purposes of the ordinary Land-Surveyor. 4"Burt's Solar Compass" is an instrument by which, "when adjusted for the Sun's declination, and the Latitude of the place, the azimuth of any line from the true North and South can be read off, and the difference between it and the Bearing by the compass will then be the variation." (310) I3agnetc variation n the Unite States. The variation of the AMagnetic needle in any part of the United States, can be approximately obtained by mere inspection of the map at the beginning of this volume.* Through all the places at which the needle in 1850,t pointed to the true North, a line is drawn on the map, and called the Line of ~no Vcariaton. It will be seen to be nearly straight, and to pass in a N.NT.W. direction from a little west of Cape IHatteras, N. C. through the middle of Virginia, about midway between Cleveland, (Ohio), and Erie, (Pa.), and through the middle of Lake Erie and Lake Huron. If followed South-Easteily it would be found to touch the most Easterly point of South America. It is now slowly moving Westward. At all places situated to the East of this line (including the New-England States, New-York,New-Jersey, Delaware, Maryland, nearly all of Pennsylvania, and the Eastern half of Virginia and North Carolina) the Variation is Westerly, i. e. the north end of the needle points to the west of the true North. At all places * Copied (by permission) from one prepared in 1856, by Prof. A. D. Bache, Supt. U. S. Coast Survey, from the U. S. C. S. Observations. The dotted portions of the lines are interpolations due to the kindness of J. E. Hilgard, Assist. U. S. Coast Survey. + A gradual change in the Variation is going on from year to year, as will be ex. plained in the next Chapter. 200 COMPASS SURVEYING. [PART 11i situated to the West of this line (including the Western and South, ern States) the Variation is easterly, i. e. the North end of the needle points to the East of the true North. This variation increases in proportion to the distance of the place on either side of the line of no variation, reaching 21" of Easterly Variation in Oregon, and 180 of Westerly Variation in Maaine. Lines of'equal Fariation are l;nes drawn through all the places which have the same variation. On the map they are drawn for each degree. All the places situated on the line marked 1~, East or West, have 1~ Variation; those on the 2~ line. have 20 Variation, &c. The variation at the intermediate places can be approximately estimated by the eye. These lines all refer to 1840. The lines of equal Variation, if continued Northward, would all meet in a certain point called the 1Mffagnetic Pole, and situated in the neighborhood of 96~ West Longitude from Greenwich, and 70~ of North Latitude. Towards this pole the needle tends to point. Another Magnetic pole is found in the Southern hemisphere; but the farther development of this subject belongs to a treatise on Natural Philosophy. The Variation on the Pacific slope of this country has been very imperfectly ascertained. A few leading points are as below. California; Point Conception, Sept. 1850, 130 49' E. Point Penos, Monterey, Feb. 1851, 14~ 58' E. Presidio, San Francisco, Feb. 1852, 15~ 27' E. San Diego, Mar. 1851, 12" 29' E. Oregon; Cape Disappointment, July, 1851, 200 45' E. Ewing Harbor, Nov. 1851, 18~ 29' E. Wash. Ter'y. Scarboro' Harbor, Aug. 1852, 21~ 30' E. (311) To correct Magnetic Bearings. The Variation at any place and time being known, the Magnetic Bearings taken there and then, may be reduced to their true Bearings, by these Rules. RULE 1. When the Variation is West, as it is in the NorthEastern States, the true Bearing will be the sum of the Variation and a Bearing which is North and West, or South and East; and the difference of the Variation and a Bearing which is North and East, or South and West. To apply this to the cardinal points, a CHAP. vII.] Variation of the Magnetic Needle, 201 North Bearing must be called N. 0~ West, an East Bearing N. 900 E., a South Bearing S. 0~ E., and a West Bearing S. 90~ W.; counting around from N' to N, in the figure, and so onward, wiit the Sun." The reasons for these corrections Fig. 206. are apparent from the Figure, in which N, the dotted lines and the accented let- ters represent the direction of the needie, and the full lines and the uuac- cented letters represent the true North w -- — ~-' -- and South and East and West lines. w' \ When the sum of the Variation and the Bearing is directed to be taken, and comes to more than 90~, the sup- C plement of the sum is to be taken, and the first letter changed. When the difference is directed to be taken, and the Variation is greater than the Bearing, the last letter must be changed. A diagram of the case will remove all doubts. Examples of all these cases are given below for a Variation of S~ West. ztUL~ 2~. ~t~2en the ~-~ricr/tia~ _2 T. U...... MAGNETIC TRUE MAGNETIC TRUE BEARING. BEARING. BEARING. BEARING. North. N. 8~W. South. S. 8~ E. N. 10 E. N. 7 W. S. 2~ W. S. 6" E. N. 40.. 4 0~ E. N. 32~ E. S. 52~ W. East. N. 82~ E. West. S. 82~ W. S. 50~ E. S. 58~ E. N 700 W. N. 78~ W. S. 89" E. N. 8 3~E. N. 83~ W. S. 89~ W. RULE 2. When the Variation is Fig. 207. East, as in the Western and Southern N - States, the preceding directions must' \ be exactly reversed; i. e. the true / Bearing will be the difference of the.... / Variation and a Bearing which is ~ ^=' --- North and West, or South and East; / ALA and the sum of the Variation and a Bearing which is North and East, or South and West. A North Bearing s' 202 COMPASS SURVEYING, EPART III must be called N. 0~ E., a West Bearing N. 900 W., a South Bearing S. 0~ W'., and an East Bearing S. 900 E., counting fromn N' to N, and so onward, " against the sun." The reasons fox these rules are seen in the Figure. Examples are given below, fox a Variation of 50 E. MAGNETIC TRUE MAGNETIC TRUE BEARING. BEARING. BEARING. BEARING. North. N. 50 E. South. S. 5~ W. N. 40~ E. N. 45~ E. S. 60~ W. 8. 65" W. N. 89~ E. S. 86~ E. S. 87~ W...88~ W. East. S. 85~ E. West. N. 85~ W. S. 1 E. S. 40 W. N. 70~ W. N. 650 W. S. 500 E. S. 45~ E. N. 2~ W. N. 30 E. (312) To survey a line with tre Bearings. The compass may be set, or adjusted, by means of the Vernier, (noticed in Arts. (229) and (237), and shown in Fig. 148, page 126) according to the Variation in any place, so that the Bearings of any lines then taken with it will be their true Bearings. To effect this, turn aside the compass plate, by means of the Tangent Screw which moves the Vernier, a number of degrees equal to the Variation, moving the S. end of the Compass-box to the right, (the North end being supposed to go ahead) if the Variation be Westerly, and vice versa; for that moves the North end of the Compass-box in the contrary direction, and thus makes a line which before was N. by the nee. die, now read, as it should truly, North, so many degrees, West if the Variation was West; and similarly in the reverse cmse. CHAP, vI1.9 203 CHAPTER VIII. CHANGES IN THE VARIATION. (813) The Changes in the Variation are of more practical importance than its absolute amount. They are of four kindss Irregular, Diurnal, Annual and Secular. (314) Irregular changes, The needle is subject to sudden and violent changes, which have no known law. They are sometimes coincident with a thunder storm, or an Aurora Borealis, (during which, changes of nearly 1~ in one minute, 22~ in eight minutes, and 10~ in one night, have been observed), but often have no apparent cause, except an otherwise invisible " Magnetic Storm." (315) The DiBnral changee On continuing observations of the direction of the needle throughout an entire day, it will be found, in the Northern Hemisphere, that the North end of the needle moves Westward from about 8 A. AI. till about 2 P. M. over an arc of from 10' to 15', and then gradually returns to its former position.' In the Southern Hemisphere, the direction of this motion is reversed. The period of this change being a day, it is called the Diurnal Fariation. Its effect on the permanent Variation is necessarily to cause it, in places where it is West, to attain its maximum at about 2 P. M., and its minimum at about 8 A. M.; and the reverse where the Variation is East. This Diurnal change adds a new element to the inaccuracies of the compass; since the Bearings of any line taken on the same day, at a few hours interval, might vary a quarter of a degree, which would cause a deviation of the end of the line, amounting to nearly half a link at the end of a chain, and to 35 links, or 23 feet, at the end of a mile. The hour of the day at which any important Bearing is taken should therefore be noted. "A similar but smaller movement takes place daring the night. 204 COMIPASS SURVEYING, [PART In, (316) The Annual change. If the observations be continued throughout an entire year, it will be found that the Diurnal changes vary with the seasons, being about twice as great in Summer as in Winter. The period of this change being a year, it is called the Annual Variation. (317) The Secular change. When accurate observations on the Variation of the needle in the same place are continued for several years, it is found that there is a continual and tolerably regular increase or decrease of the Variation, continuing to proceed in the same direction for so long a period, that it may be called the Secular change of Variation.* The most ancient observations are those taken in Paris. In the year 1541 the needle pointed 7~ East of North; in 1580 the Variation had increased to 112l East, being its maximum; the needle then began to move Westward, and in 1666, it had returned to the Meridian; the Variation then became West, and continued to increase till in 1814 it attained its maximum, being 22~ 34' West of North. It is now decreasing, and in 1853 was 20~ 17' W. In London, the Variation in 1576 was 11~ 15' E.; in 1662, 00; in 1700, 9~ 40' W.; in 1778, 22~ 11' W.; in 1815, 24~ 27' W.; and in 1843, 23" 8' W. In this country the north end of the needle was moving Eastward at the earliest recorded observations, and continued to do so till about the year 1810 (variously recorded as from 1793 to 1819), when it began to move Westward which it has ever since continued to do. Thus, in Boston, from 1708 to 1807 the Varia tion changed from 9~ W. to 6~ 5' W., and from 1807 to 1840, it changed from 6~ 5' W. to 90 18' W. Valuable Tables of the Secular changes of the Variation in van ous parts of the United States have been published by Prof. Loomis In Silliman's "American Journal of Science," Vol. 34, July, 1838, p. 301; Vol. 39, Oct. 1840, p. 42; and Vol. 43, Oct. 1842, p. 107. An abstract of the most reliable of them is here given. Troy and Schenectady are from other sources.' If the term " Declination of the Needle" could be restored to its proper usee itis " Change of Variation" would be properly called the "Variation of the De clination." CHAP. vIII.] Changes in the Variation, 205 PLACE. LATITUDE. LONGITUDE. DATES. AMOTION Burlington, Vt. 440 27' 73~ 10 1811..1834 4'.4 Chesterfield, N. H. 42~ 53' 72~ 20' 1820...1836 6'.4 Deerfield, Mass. 42~ 34' 720 29' 1811...1837 5'.7 Cambridge, Mass. 420 22' 71~ 7' 1810...1840 3'.4 New-Haven, Conn. 41~ 18' 720 58' 1819...1840 4'.6 Keeseville, N. Y. 440 28' 730 32' 1825...1838 5'.4 Albany, N, Y. 420 39' 730 45' 1818...1842 3'.6 "41< ";~" 1842...1854 4'.9 Troy, N.Y. 420 44' 730 40' 1821...1837 6'.2 Schenectady, N. Y. 42~ 49' 730 55' 1829...1841 7'.2 66 C C 1 " " 1841...1854 6'.0 New-York City. 400 43' 740 01' 1824...1837 3'.7 Philadelphia. 390 57' 750 11' 1813...1837 3'.6 Milledgeville, Ga. 330 7' 830 20' 1805...1835 1'.7 Mobile, Al. 300 40' 880 11' 1809...1835 2.2 Cleveland, 0. 410 30' 810 46' 1825...1838 4'.5 Marietta, 0. 390 25' 810 26' 1810...1838 2'.4 Cincinnati, 0. 390 6' 840 27' 1825...1840 2'.0 Detroit, Mich. 420 24' 82~ 58 1822...1840 4'.3 jAlton, Ill. 380 52' 900 12' 1835...1840 3'.0 From these and other observations it appears that at present the lines of equal variation are moving Westward, producing an annual change of variation (increasing the Westerly and lessening the Easterly) which is different in different parts of the country, and is about five or six minutes in the North-Eastern States, three or four minutes in the Middle States, and two minutes in the Southern States. (318) Determination of the change, by lnterpolation. To determine the change at any place and for any interval not found in the recorded observations, an approximation, sufficient for most purposes of the surveyor, may be obtained by interpolation (by a simple proportion) between the places given in the Tables, assuming the movements to have been uniform between the given dates; and also assuming the change at any place not found in the Tables, to have been intermediate between those of the lines of equal variar tion, which pass through the places of recorded observations orn each side of it, and to have been in the ratio of its respective dis 206 COPASS SURVETING. [PART Iii. tances from those two lines; for example, taking their arithmetical mean, if the required place is midway between them; if it be twice as near one as the other, dividing the sum of twice the change of the nearest line, and once the change of the other, by three; and so in other cases; i. e. giving the change at each place, a " weight" inversely as its distance from the place at which the change is to be found. (319) Determination of the change, by old lines. When the former Bearing of any old line, such as a farm-fence, &c. is recorded, the change in the Variation from the date of the original observation to the present time can be at once found by setting the compass at one end of the line and sighting to the other. The difference of the two Bearings is the required change. If one end of the old line cannot be seen from the other, as is often the case when the line is fixed only by a "corner" at each end of it, proceed thus. Run a line from one corner with the old Bearing and with its distance. Measure the distance from the end of this line to the other corner, to which it will be opposite. Multiply this distance by 57.3, and divide by the length of the line. The quotient will be the change of variation in degrees.* For example, a line 63 chains long, in 1827 had a Bearing of Northl10 East. In 1847 a trial line was run from one end of the former line with the same Bearing and distance, and its other end was found to be 125 links to the West of the true corner. The 1.25 x 57.3 change of Variation was therefore - 63 57 = 10.137 = 12 8' Westerly. LXet AB be the original line; AC the trial line, Fig. 208. and BC the distance between their extremities. AB and AC may be regarded as radii of a circle —... and BC as a chord of the arc which subtends their "~a angle. Assuming the chord and arc to coincide (which they will, nearly, for small angles) we have this proportion; Whole cirajmference: arc / BC:: 3600: BAC: or, 2 X AC X 3.1416: BC * 3600: BAC, whence BAC = BC X 57.3; or more precisely 57.29578. AB CHAP. vIII.o Changes in the Variation. 207 (320) Effects of the Secular change. These are exceedingly important in the re-survey of farms Fig. 209. by the Bearings recorded in old'' deeds. Let SN denote the direc- tion of the needle at the time of \ the original Survey, and S'N' its \ / direction at the time of the re-sur- ^ " vey, a number of years later, Suppose the change to have been / \ 30, the needle pointing so much / \\, farther to the west of North. The line SN, which before was due North and South by the needle' $ will now bear N. 30 E. and S. 3~ W; the line AB, which before was N. 40~ E. will now bear N. 43~ E; the line DF which before was N. 40~ W. will now bear N. 37~ W; and the line WE, which before was due East and West, will now bear S. 87~ E. and N. 870 NV. Any line is similarly changed. The proof of this is apparent on inspecting the figure. Suppose now that a surveyor, ignorant or neglectful of this change, should attempt to run out a Fig. 21n. farm by the old Bearings of the deed, none of the old fences or cor- ra, ners -emaining. The full lines in the figure represent the original \ bounds of the farm, and the dotted \ \ \ ^. lines those of the new piece of land which, starting from A, he would unwittingly run out. It would be of the same size and the same shape as \ / the true one, but it would be in the wrong place. None of its lines would agree with the true ones, and / in some places it would encroach on one neighbor, and in other places would leave a gore which belongs to it, between itself and another neighbor. Yet this is often done, and is the source of a great part of the litigation among farmers respecting their " lines." 208 COMPASS SU VEYINGo LPART IIL. (321) To rum out old lines. To succeed in retracing old lines, proper allowance must be made for the change in the varia. tion since the date of the original survey. That date must first be accurately ascertained; for the survey may be much older than the deed, into which its bearings may have been copied from an older one. The amount and direction of the change is then to be ascertained by the methods of Arts. (318) or (319). The beartugs may then be corrected by the followlng RULES. When the North end of the needle has been moving Westerly, (as it has for about forty years), the present Bearings will be the sums of the change and the old Bearings which were North-Easterly or South-Westerly, and the differences of the change and the old Bearings which were North-Westerly or South-Easterly. If the change have been Easterly reverse the preceding rules, subtracting where it is directed to add, and adding where it is directed to subtract. Run out the lines with the Bearings thus corrected. It will be noticed that the process is precisely the reverse of that in Art. (311). The rules there given in more detail, may therefore be used; RULE 1 " when the Variation is West," being employed when the change has been a movement of the N. end of the needle to the East; and RULE 2, "6 when the Variation is East," being employed when the N. end of the needle has been moving to the West. If the compass has a Vernier, it can be set for the change, once for all, precisely as directed in Art. (312), and then the courses can be run out as given in the deed, the correction being made by the instrument. (322) Example. The following is a remarkable case which recently came before the Supreme Court of New-York. The North line of a large Estate was fixed by a royal grant, dated in 1704, as a due East and West line. It was run out in 1715, by a surveyor, whom'we will call Mr. A. It was again surveyed in 1765, by Mr. B. who ran a course N. 87~ 30' E. It was run out for a third time in 1789, by Mr. C. who adopted the course N. 86~ 18' E. In 1845 it was surveyed for the fourth time by CHnAP. vII.] Changes in the Variation. 209 Mr. D. with a course of N. 88~ 30' E. He found old " corners," and' blazes" of a former survey, on his line. They are also found on another line, South of his. Which of the preceding courses were correct, and where does the true line lie? The question was investigated as follows. There were no old records of variation at the precise locality, but it lies between the lines of equal variation which pass through New-York and Boston, its distance from the Boston line being about twice its distaneh from the New-York line. The records of those two cities (referred to in Art. (317)) could therefore be used in the manner explained in Art. (3S8). For the later dates, observations at New-Haven could serve as a check. Combining all these, the author inferred the variation at the desired place to have been as follows: In 1715, Variation 80 02' West. In 1765, 6 50 32' " Decrease since 1715, 20 30'. In 1789, " 50 05''; Decrease since 1765, 0 027'. In 1845, " 70 23' " Increase since 1789, 20 18'. We are now prepared to examine the correctness of the allowances made by the old surveyors. The course run by Mr. B. in 1765, N. 870 30' E. made an allowance of 20 30' as the decrease of variation, agreeing precisely with our calculation. The course of Mr. C. in 1789, N. 860 18' E., allowed a change of 1~ 12', wlich was wrong by our calculation, which gives only about 27', and was deduced from three different records. Mr. D. in 1845, ran a course of N. 880 30' E, calling the increase of variation since 1789, 23 12'. Our estimate was 20 18'^ the difference being comparatively small. Our conclusion then is this: the second surveyor retraced correctly the line of the first: the third surveyor ran out a new and incorrect line: and the fourth surveyor correctly retraced the line of the third, and found his marks, but this line was wrong originally and therefore wrong now. All the surveyors ran their lines on the supposition that the original "; due East and West line " meant East anrd West as the needle pointed at the time of the original survey. The preponderance of the testimony as to old land marks agreed with the results of the above reasoning, and the decision of the court was in accordance therewith. 14 210 COMPASS SURVEYING. [PART m ^N F- 2Fig. 211 In the above figure the horizontal and vertical lines represent true East and North lines; and the two upper lines running from left to right represent the two lines set out by the surveyors and in the years, there named. (323) Remedy for the evils of the Secular change. Tho only complete remedy for the disputes, and the uncertainty of bounds, resulting from the continued change in the variation, is this. Let a Meridian, i. e. a true North and South line, be estab. lished in every town or county, by the authority of the State; monuments, such as stones set deep in the ground, being placed at each end of it. Let every surveyor be obliged by law to test his compass by this line, at least once in each year. This he could do as easily as in taking the Bearing of a fence, by setting his instrument on one monument, and sighting to a staff held on the other. Let the variation thus ascertained be inserted in the notes of the survey and recorded in the deed. Another surveyor, years or centuries afterwards, could test his compass by taking the Bearin'g of the same monuments, and the difference between this and the former Bearing would be the change of variation. He could thus determine with entire certainty the proper allowance to be made (as in Art. (321)) in order to retrace the original line, no matter how uch, or ho w uch r irregularly, the variation may have changed, or how badly adjusted was the compass of the original survey. Any permanent line employed in the same manner as the meridian line. would answer the same purpose, though less conveniently, and every surveyor should have such a line at least, for his own use.*'T'his remedy seems to have been first suggested by Rittenhouse. It has since been recommended by T. Sopwith, in 1822; by E. F. Johnson, in 1831, and by W. Roberts, of Troy, in 1839. The errors of re-surveys, in which the change is neglected, were noticed in the " Philosophical Transactions," as long ago as 1679 PART IV. TRANSIT AND THEODOLITE SURVEYING: By the Third Method. CHAPTER I. THE INSTRUMIENTS (321) THE TRANSIT and THE THEODOLITE (figures of which are given on the next two pages) are Goniometers, or Angle-Measurers. Each consists, essentially, of a circular plate of metal, supported in such a manner as to be horizontal, and divided on its outer circumference into degrees, and parts of degrees. Through the centre of this plate passes an upright axis, and on it is fixed a second circu-,ar plate, which nearly touches the first plate, and can turn freely around to the right and to the left. This second plate carries a Telescope, which rests on upright standards firmly fixed to the plate, and which can be pointed upwards and downwards. By the combination of this motion and that of the second plate around its axis, the Telescope can be directed to any object. The second plate has some mark on its edge, such as an arrow-head, which serves as a pointer or index for the divided circle, like the hand of a clock. When the Telescope is directed to one object, and then turned to the right or to the left, to some other object, this index, wlwich moves with it and passes around the divided edge of the other plate, points out the arc passed over by this change of direction, and thus measures the angle made by the lines imagined to pass from the centre of the instrument to the two objects. 212 TRANSIT AD TiIEODOLTE IURVEYEI.G [PART ia THE TRANSIT. Fig 212 / / ~~1~ j~~~~~~~~~~~~~ \ / CHAP. I.] The Instruments, 213 THE THEODOLITE. Fig. 213 0F~~~ 214 TRANSIT AND THEODOLITE SURVEYING. [PART IV (325) Distnctaon. The preceding description applies to both the Transit and the Theodolite. But an essential difference between them is, that in the Transit the Telescope can turn corn pletely over, so as to look both forward and backward, while in the Theodolite it cannot do so. Hence the name of the Transit.* This capability of reversal enables a straight line to be prolonged from one end of it, or to be ranged out in both directions froni any one point. The Telescope of the Theodolite can indeed be taken out of the Y shaped supports in which it rests, and be replaced end for end, but this operation is an imperfect substitute for the revolution of the Telescope of the Transit. So also is the turning half way around of the upper plate which carries the Telescope. The Theodolite has a level attached to its Telescope, and a vertical circle for measuring vertical angles. The Transit does not usually have these, though they are sometimes added to it. The instrument may then be named a Transit-Theodolite. It then corresponds to the altitude and azimuth instrument of Astronomy. As the greater part of the points to be explained are common to both the Transit and the Theodolite, the descriptions to be given may be regarded as applicable to either of the instruments, except when the contrary is expressly stated, and some point peculiar to either is noticed. (326) The great value of these instruments, and the accuracy of their measurements of angles are due chiefly to two things; to the Telescope, by which great precision in sighting to a point is obtained; and to the Vernier Scale, which enables minute portions of any arc to be read with ease and correctness. The former assists the eye in directing the line of sight, and the latter aids it in reading off the results. Arrangements for giving slow and steady motion to the movable parts of the instruments add to the value of the above. A contrivance for Repeating the observation of angles still farther lessens the unavoidable inaccuracies of these observations. * It is sometimes called the "Engineers' Transit," or " Railroad Transit," to distinguish it from the Astronomical Transit-instrument. In this country it has nlmost entirely supplanted the Theodolite. CHAP. i.] The Instruments. 21; The inaccurate division of the limb of the instrument is also averaged and thus diminished by the last arrangement. Its want of true' centring," is remedied by reading off on opposite sides of the circle. Imperfections m the parallelism and perpendicularity of the parts of the instrument in which those qualities are required, are corrected by various " adjustments," made by the various screws whose heads appear in the engravings. The arrangements for attaining all these objects render necessary the numerous parts and apparent complication of the instrument. But this complication disappears when each part is examined in turn, and its uses and relations to the rest are distinctly indicated. This we now propose to do, after explaining the engravings. (327) In the figures of the instruments, given on pages 212 and 213, the same letters refer to both figures, so far as the parts are common to both.' L is the limb or divided circle. V is the index, or " Vernier," which moves around it. In the Transit, only a small portion of the divided limb is seen, the upper circle (which in it is the movable one) covering it completely, so that only a short piece of the arc is visible through an opening in the upper plate. S, S, are standards, fastened to the upper plate and supporting the telescope, EO. G is a compass-box, also fastened to the upper plate. c is a clamp-screw, which presses together the two plates, and prevents one from moving over the other. t is a tangentscrew, or slow-motion screw, which gives a slow and gentle motion to one plate over the other. C is a clamp-screw which fastens the lower plate to the body of the instrument, and thus prevents it from moving on its own axis. T is the tangent-screw to give this part a slow-motion. P and P' are parallel plates through which pass four screws, Q, Q, Q, Q, by which the circular plate L is made level, * The arrangements of these instruments are differently made by almost every maker; but any form of them being thoroughly understood, any new one will cause no difficulty. The figure of the Transit was drawn from one made by W. & L. E. Gurley, of Troy, N. Y. to the latter of whom the Author is indebted foi some valuable information respecting the details of the instrument. The Theodo lits is of the favorite English form. 216 TRANSIT AND THEODOLITE SURTEYgl NG [PART IV, as determined by the bubbles in the small spirit levels, B, B, of which there are two at right angles to each other. In the figure of the Theodolite, the large level b, and the semicircle NN are for' the purposes of Levelling, and of measuring Vertical angles. They will therefore not be described in this place, (S28) As the value of either of these instruments depends greatly on the accurate fitting and bearings of the two concentric vertical axes, and as their connection ought to be thoroughly understood, a vertical section through the body of the instrument is given in Fig. 214, to half the real size. The tapering spindle or Fig. 24. / / inverted frustum of a cone, marked AA, supports the upper plate BB, which carries the index, or Verniers, V, V, and the Telescope. The whole bearing of this plate is at C, C, on the top of the hollow inverted cone EE, in which the spindle turns freely, but steadily. This interior position of the bearings preserves them from dust and injury. This hollow cone carries the lower or graduated plate, und it can itself turn around on the bearings D, D, carrying with it the lower circle, and also the upper one and all above it. The Vernier scales V, V, are attached to the upper plate, but lie in the same plane as the divisions L, L, of the lower plate, (so that the two can be viewed together, without parallax,) and are CHAP. I.] The Instruments. 217 covered with glass, to exclude dust and moisture. In Fig. i15 the figure the hatchings are drawn in different directions l on the parts which move with the Vernier, and on those which move only with the limb. (329) The Telescope. This is a combination of lenses, placed in a tube, and so arranged, in accordance with the laws of optical science, that an image of any object to which the Telescope may be directed, is formed within the tube, (by the rays of light coming from the object and bent in passing through the object-glass) and there magnified by an Eye-glass, or Eye-piece, composed of several lenses. The arrangement of these lenses are very various. Those two combinations which are preferred for surveying instruments, will be here explained. Fig. 215 represents a Telescope which inverts objects. Any object is rendered visible by every point of it sending forth rays of light in every direction. In this figure,, t the highest and lowest points of the object, which here is an arrow, A, are alone considered. Those of the rays proceeding from them, which meet the object-glass, 0, form a cone. The centre line of each cone, and its extreme upper and lower lines are alone shown in the figure. It will be seen that these rays, after passing through the object-glass, are refracted, or bent', by it, so as to cross one another, and thus to form at B an inverted image of the object. This would be rendered visible, if a piece of ground glass, or other semi-transparent substance, was placed at the point B, which is called the focus of the object-glass. The rays which form this image continue onward and pass through the two lenses C and D, which act like one magnifying glass, so that the rays, after being refracted by them, enter the eye at such angles as t) form there a magnified and inverted image of the object. This combination of the two plano-convex lenses, C and D, is known as "1 Ramsden's Eye-piece." 218 ATRANST AND TIEODOLITE SURVEYIl', [PARt, IV This Telescope, inverting objects, shows them upside. down, and the right side on the left. They can be shown erect by adding one or two more lenses as in the marginal figure. But as these lenses absorb light and lessen the distinctness of vision, the former arrangement is preferable for the glasses of a Transit or a Theodolite. A little practice makes it equally convenient for the observer, who soon becomes accustomed to seeing his flagmen standing on their heads, and soon learns to motion them to the right when he wishes them to go to the left, and vice versa. Figure 216 represents a Telescope which shows objects erect. Its eye-piece has four lenses. The eyepiece of the commlon terrestrial Telescope, or spy-glass, L has three. Many other combinations may be used, all intended to show the object achromatically, or free from false coloring, but the one here shown is that most generally preferred at the present day. It will be seen that an inverted image of the object A, is formed at B, as before, but that the rays continuing onward are so refracted in passing through the lens C as to again cross, and thus, after farther refraction by the lenses D and E, to form, at F9 an erect image, which is magni- fied by the lens G. In both these figures, the limits of the page render it necessary to draw the angles of the rays very much out of proportion. (330) C'ross-halrs Since a considerable field of view is seen in looking through the Telescope, it is necessary to provide means for directing the line of sight to the precise point which is to be observed. This could be effected by placing a very fine point, such as that of a needle, within the Telescope, at some place where it could be distinctly seen. In practice this fine point is obtained by the intersection of two very fine Lines, placed in the common focus of the object-glass and 1 CRAP. i.] The Instruments. 219 of the eye-piece. These lines are called the cross-hairs, or cross. wires. Their intersection can be seen through the eye-piece, at the same time, and apparently at the same place, as the image of the distant object. The magnifying powers of the eye-piece will then detect the slightest deviation from perfect coincidence. " This applicartion of the Telescope may be considered as completely annihilating that part of the error of observation which might otherwise arise from an erroneous estimation of the direction in which an object lies from the observer's eye, or from the centre of the instrument. It is, in fact, the grand source of all the precision of modern Astronomy, without which all other refinements in instrumental workmanship would be thrown away." What Sir John Herschel here says of its utility to Astronomy, is equally applicable to Surveying. The imaginary line which passes through the intersection of the cross-hairs and the optical centre of the object-glass, is called the line of collimation of the Telescope.* The cross-hairs are attached to a ring, or short thick tube of brass, placed within the Tele- Fig. 217. scope tube, through holes in Pi which pass loosely four screws, B (their heads being seen, at a, a, a, in Figs. 212 and 213), whose threads enter and take hold of the ring, behind or in V,7 / front of the cross-hairs, as shown (in front view and in,/g? section) in the two figures in li ihe margin. Their movements will be explained in Chapter III, Usually, one cross-hair is horizontal, and the Fig. 218 other vertical, as in Fig. 217, but sometimes they are arranged as in Fig. 218, which is thought to \ J enable the object to be bisected with more preci- t sion. A horizontal hair is sometimes added. The cross-hairs are best made of platinum wire, drawn out very fine by being previously enclosed * From the Latin word Collimo, or Collineo, meaning to direct one thing to wards another in a straight line, or to aim at. The line of aim would express the meaning. 220 TANSIT AND THEODOLITE SURVEYING. [PART lV. in a larger wire of silver, and the silver then removed by nitric acid. Silk threads from a cocoon are sometimes used. Spiders' threads are, however, the most usual. If a cross-hair is broken, the ring must be taken out by removing two opposite screws, and inserting a wire with a screw cut on its end, or a stick of suitable size, into one of the holes thus left open in the ring, it being turned sideways for that purpose, and then removing the other screws The spider's threads are then stretched across the notches seen in the end of the ring, and are fastened by gum, or varnish, or beeswax. The operation is a very delicate one. The following plan has been employed. A piece of wire is bent, as in the figure, so as to leave an opening a little wider than the Fr;. 219. ring of the cross-hairs. A cobweb is cho- ~/\ sen, at the end of which a spider is hanging, and it is wound around the bent wire, as in the figure, the weight of the insect keeping it tight and stretching it ready for use, each part being made fast by gum, &c. When a cross-hair is wanted, one of these is laid across the ring and there attached. Another method is to draw the thread out of the spider, persuading him to spin, if he sulks, by tossing him from hand to hand. A stock of such threads must be obtained in warm weather for the winter's wants. A piece of thin glass, with a horizontal and a vertical line etched on it, may be made a substitute. (331) Instrumental Parallax. This is an apparent movement of the cross-hairs about the object to which the line of sight is directed, taking place on any slight movement of the eye of the observer. It is caused by the image and the cross-hairs not being precisely in the common focus, or point of distinct vision of the eye-piece and the object glass. To correct it, move the eye-piece out or in till the cross-hairs are seen clearly and sharply defined against any white object. Then move the object glass in or out till the object is also distinctly seen. The cross-hairs will then seem to be fixed to the object, and no movement of the eye will cause them to appear to change their place. CHAP. I.] The Instrumentse 221 (332) The milled-headed screw seen at M, passing into thle tele scope has a pinion at its other end entering a Fig. 220. toothed rack, and is used to move the object glass, 0, out and in, according as the object looked at is nearer or farther than the one last observed. Short distances require a long tube: long distances a short tube. The eye-piece, E, is usually moved in and out by hand, but a s;imilar arrangement to the preceding is a great improvement. This movement is necessary in order to obtain a distinct view of the "cross-hairs. Short-sighted persons require the& eye-piece to be pushed farther in than persons of ordinary sight, and old or longsighted persons to have it drawn further out. (333) Supports, The Telescope of the Transit is supported by a hollow axis at right angles to it, which itself rests at each end, on two upright pieces, or standards, spreading at their bases so as to increase their stability. In the Theodolite, the telescope rests at each end in f6rked supports, called ys, from their shape. These ys are themselves supported by a cross-bar, which is carried by an axis at right angles to it and to to the telescope. This axis rests on standards similar to those of the Transit. The Telescope of the Theodolite can be taken out of the Ys, and turned " end for end." This is not usual in the Transit. Either of the above arrangements enables the Telescope to be raised or depressed so as to suit the height of the object to which it is directed. A telescope so disposed is'called a "plunging telescope." In some instruments there is an arrangement for raising or lowering one end of the axis..This is sometimes required for reasons to be given in connection with " Adjustments." (334) The IndexesO The supports, or standards, of the telescope just described are attached to the upper, or index-carrying circle.d This, as has been stated, can turn freely on the lower or graduated circle, by means of its conical axis moving in the hollow conical axis of the latter circle. This upper circle carries the index, V, X In some instruments this circle is the under one. In our figures it is the upper one, and we will therefore always speak of it as such. 222 TRANSIT AND THEODOLITE SURTVEING, [PART IV which is an arrow-head or other mark on its edge, or the zero-point of a Vernier scale. There are usually two of these, situated exactly opposite to each other, or at the extremities of a diameter of the upper circle, so that the readings on the graduated circle pointed out by them differ, if both are correct, exactly 1800. The object of this arrangement is to correct any error of eccentricity, arising from the centre of the axis which carries the upper circle, (and with which it and its index pointers turn), not being precisely in the centre of the graduated circle. In the figure, let C Fig 21. be the true centre of the graduated cir- cle, but C' the centre on which the plate //' carrying the indexes turns. Let AC'B represent the direction of a sight taken/... \ to one object, and D'C'E' the direction when turned to a second object. The angle subtended by the two objects at the centre of the instrument is required. Let DE be a line passing through C, and parallel to D'E'. The angle ACD equals the required angle, which is therefore truly measured by the arc AD or BE. But if the arc shown by the index is read, it will be AD' on one side, and BE' on the other; the first being too small by the arc DD' and the other too large by the equal arc EE'. If however the half-sum of the two arcs AD' and BE' be taken, it will equal the true arc, and therefore correctly measure the angle. Thus if AD' was 19~, and BE' 21~ their half sum, 200, would be the correct angle. Three indexes, 120~ apart, are sometimes used. They have the advantage of averaging the unavoidable inaccuracies and inequali ties of graduation on different parts of the limb, and thus diminish ing their effect on the resulting angle. Fig. 222. Four were used on the large Theodolite of A the English Ordnance Survey, two, A and B, opposite to each other, and two, C and D, 1200 from A and from each other. The half-sum or arithmetical mean, of A and B was taken, then c the mean of A, C, and D, and then the mean of these two means. But this was wrong, for' -IJ~ V~LI- II~N IL ILB CHAP. I.] the Instruments. 223 it gave too great value t tthe reading of A, and also to B, though in a less degree; since the share of each Vernier in the final mean was as follows: A = 5, B = 3, C -=- 2, D= 2. This results from the expression for that mean, A= ( ~ B + -A + + D) (5 A + 3 B + 2 C + 2 D). (335) The gradaated circle. This is divided into three hundred and sixty equal parts, or Degrees, and each of these is subdivided into two or three parts or more, according to the size of the instrument. In the first case, the smallest division on the circle will of course be 30'; in the second case 20'. More precise reading, to single minutes or even less, is effected by means of the Vernier of the index, all the varieties of which will be fully explained in the next chapter. The numbers run from 00 around to 3600, which number is necessarily at the same point as the 0, or zero-point.' Each tenth degree is usually numbered, each fifth degree is distinguished by a longer line of division, and each degree-division line is longer than those of the sub-divisions. A magnifying glass is needed for reading the divisions with ease. In the Theodolite engraving this is shown at mn. It should be attached to each Vernier. (33g) Iiovemecntso When the line of sight of the telescope is directed to a distant well-defined point, the unaided hand of the observer cLnot move it with sufficient delicacy and precision to make the intersection.of the —cross hairs exactly -cover or "bisect" that point. To effect this, a clamp, and a Tangent, or slow-motion, screw are required. This arrangement, as applied to the movement of the upper, or Vernier plate, consists of a short piece of brass,, D which is attached to the Vernier plate, and through which passes a long and fine-threaded' Tangent-screw," t. The other end of this screw enters into and carries the clamp. This consists of two pieces of brass, which, by turning the clamp-screw e, which passes through them on the outside, can be made to take * In some instruments there is another concentric circle on which the degrees are also numbered from 0~ to 900 as on the compass circle. 224 TRANSIT AND THEeDOLITE SURVEYuNi LPART Iv hold of and pinch tightly the edge of the lower circle, which lies between them on the inside. The upper circle is now prevented from moving on the lower one; for, the tangent-screw, passing through hollow screws in both the clamp and the piece D, keeps them at a fixed distance apart, so that they cannot move to or from one another, nor consequently the two circles to which they are respectively made fast. But when this tangent-screw is turned by its milled-head, it gives the clamp and with it the upper plate a smooth and slow motion, backward or forward, whence it is called the "' Slow motion screw," as well as " Tangent-screw" from the direction in which it acts. It is always placed at the south end of the compass-box. A little different arrangement is employed to give a similar motion to the lower circle (which we have hitherto regarded as immovable) on the body of the instrument. Its axis is embraced by a brass ring, into which enters another tangent-screw, which also passes through a piece fastened to the plate P. The clampscrew, C, causes the ring to pinch and hold immovably the axis of the lower circle, while a turn of the Tangent-screw, T, will slowly move the clamp ring itself, and therefore with it the lower circle. When the clamp is loosened, the lower circle, and with it every thing above it, has a perfectly free motion. A recent improvement is the employment for this purpose of two tangent screws, pressing against opposite sides of a piece projecting from the clamp-ring. One is tightened as the other is loosened, and a very steady motion is thus obtained. (337) Levels. Since the object of the instrument is to measure horizontal angles, the circular plate on which they are measured must itself be made horizontal. Whether it is so or not is known by means of two small levels placed on the plate at right angles to each other. Each consists of a glass tube, slightly curved upward in its middle and so nearly filled with alcohol, that only a small bubble of air is left in the tube. This always rises to the highest part of the tubes. They are so " adjusted" (as will be explained in chapter III) that when this bubble of air is in the middle of the tubes, or its ends equidistant from the central mark, the plate CHAP. I.] The Instruments. 225 on which they are fastened shall be level, which way soever it may be turned. The levels are represented in the figure of the Transit, on page 212, as being under the plate. They are sometimes placed above it. In that case, the Verniers are moved to one side, between the feet of the standards, and one of the levels is fixed between the standards above one of the Verniers, and the other on the plate at the south end of the compass-box. (338) Parallel Plates, To raise or lower either side of the circle, so as to bring the bubbles into the centres of the tubes, requires more gentle and steady movements than the unaided hands can give, and is attained by the Parallel Plates P, P', (so called because they are never parallel except by accident), and their four screws Q, Q, Q, Q, which hold the plates firmly apart, and, by being turned in or out, raise or lower one side or the other of the upper plate P', and thereby of the graduated circle. The two plates are held together by a ball and socket joint. To level the instrument, loosen the lower clamp and turn the circle till each level is parallel to the vertical plane passing through a pair of opposite screws. Then take hold of two opposite screws and turn them simultaneously and equally, but in contrary directions, screwFig. 223. ing one in and the other out, as shown by the arrows in the figures. A rule easily remembered is that both thumbs must turn in, or both out. The movements represented in the first of these figures would raise the left-hand side of the circle and lower the right-hand side. The movements of the second figure would produce the reverse effect. Care is needed to turn the opposite screws equally, so that they shall not become so loose that the instrument will rock, or so tight as to be cramped. When this last occurs, one of the other pair should be loosened. 1.5 22 TRANSIT AND TlEODOLITE SURVEYING. [PART IT Sometimes one of each pair of the screws is replaced by'a strong spring against which the remaining screws act. The French and German instruments are usually supported by only three screws. In such cases, one level is brought parallel to one pair of screws and levelled by them, and the other level has its bubble brought to its centre by the third screw. If there is only one level on the instrument, it is first brought parallel to one pair of screws and levelled, and is then turned one quarter around so as to be perpendicular to them and over the third screw, and the operation is repeated. (339). Watch Telescope. A second Telescope is sometimes attached to the lower part of the instrument. When a number of angles are to be observed from any one station, direct the upper and principal Telescope to the first object, and then direct the lower one to any other well-defined point. Then make all the desired observations with the upper Telescope, and when they are finished, look again through the lower one, to see that it and therefore the divided circle has not been moved by the movements of the Vernier plate. The French call this the Witness Telescole, (Lunette temoin). (340) The Compass. Upon the upper plate is fixed a compass. Its use has been fully explained in Part III. It is little used in connection with the Transit or Theodolite, which are so incomparably more accurate, except as a " check," or rough test of the accuracy of the angles taken, which should about equal the difference of the magnetic bearings. Its use will be farther noticed in Chapter IV, on " Field Work." (341) The Surveyors Transit. In this instrument (so named by its introducers, Messrs. Gurley, and shown in Fig. 224), the Vernier-plate, which carries the standards and telescope, is under the plate which carries the graduated circle, and the compass is attached to the latter. By this arrangement, when the Vernier is set at any angle, the line of sight of the telescope will make that angle with the N. and S. lines of the compass. Consequently, this instrument can be used precisely like the Vernier compass CHAP. I.] The Instruments 227 to allow for magnet- Fig.224. ic variation, and thus to run out a line with true bearings, as in Art. (312), or to run out old lines, allowing for the secular variation, as in The instrument may also be used like the comn onEngineer's Transit. The compass, however, will then not give the bearings of the o lines surveyed, but they can easily be deduced from that of any one line. (3 2) Gonasmometre. A very compact in- Fig. 224 strument to which the above name has been given in France, where it is much used, is shown i e spt in the figure. The upper half of the cylinder is i movable on its lower half. The observations 1 may be taken through the slits, as in thle Survey- o ot or's Cross, or a Telescope may be added to it. w lrb Readings may be taken both from the compass, ai s and from the divided edge of the lower half of li the cylinder, by means of a Vernier on the upper half.' * The proper care of instruments must not be overlooked. If varnished, they should be wiped gently with fine and clean linen. If polished with oil, they should be rubbed more strongly. The parts neither varnished nor oiled, should be cleaned with Spianish white and alcohol.'Varnished wood, when spotted should be wiped with very soft linen, moistened with a little olive oil or alcohol. Unpainted wood is cleaned with sand-paper. Apply olive oil where steel rubs against brass; and wax softened by tallow where brass rubs against brass.Clean the glasses with kid or buck skin. Wash them, if dirtied, with alcohol. 228 [PART IV CHAPTER II VERNIERS. (343) Definition. A Vernier is a contrivance for measuring smaller portions of space than those into which a line is actually divided. It consists of a second line or scale, movable by the side of the first, and divided into equal parts, which are a very little shorter or longer than the parts into which the first line is divided. This small difference is the space which we are thus enabled to measure. The Vernier scale is usually constructed by taking a length equal to any number of parts on the divided line, and then dividing this length into a number of equal parts, one more or one less than the number into which the same length on the original line is divided. (344) Illustration. The figure represents (to twice the real size) a scale of inches divided into tenths, with a Vernier scale beside it, by which hundredths of an inch can be measured. The Fig. 225. Vernier is made by setting off on it 9 tenths of an inch, and dividVernier is made bPy setting off on it 9 tenths of an inch, and dividing that length into 10 equal parts. Each space on the Vernier is therefore equal to a tenth of nine-tenths of an inch, or to ninehundredths of an inch, and is consequently one-hundredth of an inch shorter than one of the divisions of the original scale. The X The Vernier is so named from its inventor, in 1631. The name " Nonius," often improperly given to it, belongs to an entirely different contrivance for a similar object. CHAP. II.] Verniers. 229 first space of the Vernier will therefore fall short of, or be overlapped by, the first space on the scale by this one-hundredth of an inch; the second space of the Vernier will fall short by two-hundredths of an inch; and so on. If then the Vernier be moved up by the side of the original scale, so that the line marked 1 coincides, or forms one straight line, with the line of the scale which was just above it, we know that the Vernier has been moved onehundredth of an inch. If the line marked 2 comes to coincide with a line of the scale, the Vernier has moved up two-hundredths of an inch; and so for other numbers. If the pcsition of the Fig. 226.- I t I I l l l _ I l1'. Vernier be as in this figure, the line marked 7 on the Vernier corresponding with some line on the scale, the zero line of the Vernier is 7 hundredths of an inch above the division of the scale next below this zero line. If this division be, as in the figure, 8 inches and 6 tenths, the reading will be 8.67 inches.' A Vernier like this is used on some levelling rods, being engraved on the sides of the opening in the part of the target above its middle line. The rod being divided into hundredths of a foot, this Vernier reads to thousandths of a foot. It is also used on some French Mountain Barometers, which are divided to hundredths of a metre, and thus read to thousandths of that unit. (345) General rales. To find what any Vernier reads to, i. e. to determine how small a distance it can measure, observe how many parts on the original line are equal to the same number increased or diminished by one on the Vernier, and divide the * The student will do well to draw such a scale and Vernier on two slips of ihick paper, and move one beside the other till he can read them in any possible position; and so with the following Verniers. 230 TRANSIT AND THEODOLITE SURVEYING, [PART IV length of a part on the original line by this last number. It will give the required distance.* To read any Vernier, firstly, look at the zero line of the Ver nier, (which is sometimes marked by an arrow-head), and if it coincides with any division of the scale, that will be the correct reading, and the Vernier divisions are not needed. But if, as usually happens, the zero line of the Vernier comes between any two divisions of the scale, note the nearest next less division on the scale, and then look along the Vernier till you come to some line on it which exactly coincides, or forms a straight line, with some line (no matter which) on the fixed scale. The number of this line on the Vernier (the 7th in the last figure) tells that so many of the sub-divisions which the Vernier indicates, are to be added to the reading of the entire divisions on the scale. When several lines on the Vernier appear to coincide equally with lines of the scale, take the middle line. When no line coincides, but one line on the Vernier is on one side of a line on the scale, and the next line on the Vernier is as far on the other side of it, the true reading is midway between those indicated by these two lines. (346) Retrograde Verniers. The spaces of the Vernier in modern instruments, are usually each shorter than those on the scale, a certain number of parts on the scale being divided into a larger number of parts on the Vernier.t In the contrary case,4 there is the inconvenience of being obliged to number the lines of the Vernier and to count their coincidences with the lines of the scale, in a retrograde or contrary direction to that in which the numbers on the scale run. We will call such arrangements retrograde Verniers. * In Algebraic language, let s equal the length of one part on the original line, and v the unknown length of one part on the Vernier. Let m of the former m + 1 of the latter. Then ns = (m + 1) v. v = -- s. s-v = sIf m ( ) v the v- -~$ —S - --—, s=. If ms = {m ~- I) v, then v s. rn~1 rn n1 rnm-i In m e7Im mI t i. e. Algebraically, v = - s. i. e. When v = -—'. m-~l m —rn CHAP. II.] Verniers. 231 (347) Illustration. In this figure, the scale, as before, represents (to twice the real size) inches divided into tenths, but the Vernier is made by dividing 11 parts of the scale into 10 equal Fig. 227. parts, each of which is therefore one-tenth of eleven-tenths of an inch, i. e. eleven-hundredths of an inch, or a tenth and a hundredth. Each space of the Vernier therefore overlaps a space on the scale by one-hundredth of an inch. The manner of reading this Vernier is the same as in the last one, except that the numbers run in a reverse direction. The reading of the figure is 30.16. This Vernier is the one generally applied to the common Barometer, the zero point of the Vernier being brought to the level of the top of the mercury, whose height it then measures. It is also employed for levelling rods which read downwards from the middle of the target. (348) The figure below represents (to double size) the usual scale of the English Mountain Barometer.' The scale is first divided into inches. These are subdivided into tenths by the Fig. 228.. This figure, and others in this chapter, are from Bree's " Present Pracice." 232 TRANSIT AND THEODOLITE SURVEYINTG [PART IV longer lines, and the shorter lines again divide these into half tenths, or to 5 hundredths. 24 of these smaller parts are set off on the Vernier, and divided into 25 equal parts, each of which is 24 x.05 therefore = 25 =.048 inch, and is shorter than a division of the scale by.050 -.048 =.002, or two thousandths of an inch, a twenty-fifth part of a division on the scale, to which minuteness the Vernier can therefore read. The leading in the figure is 80.686, (30.65 by the scale and.036 by the Vernier), the dotted line marked D showing where the coincidence takes place. (349) Circle divided into etegree' The following illustrations apply to the measurements of angles, the circle being variously divided. In this article, the circle is supposed to be divided into degrees. If 6 spaces on the Vernier are found to be equal to 5 on the circle, the Vernier can read to one-sixth of a space on the circle, i. e. to 10'. If 10 spaces on the Vernier are equal to 9 on the circle, the Vernier can read to one-tenth of a space on the circle, i. e. to 6'. If 12 spaces on the Vernier are equal to 11 on the circle, the Vernier can read to one-twelfth of a space on the circle, i. e. to 5'. Fig. 229. i,,~ The above figure shows such an arrangement. The index, or zero, of the Vernier is at a point beyond 8580, a certain distance, which the coincidence of the third line of the Vernier (as indicated CHAP. ii.] Verners. 233 by the dotted and crossed line) shows to be 15'. The whole reading is therefore 358~ 15'. If 20 spaces on the Vernier are equal to 19 on the circle, the Vernier can read to one-twentieth of a division on the circle, i. e. to 3'. English compasses, or "6 Circumferentors," are sometimes thus arranged. If 60 spaces on the Vernier are equal to 59 on the circle, the Vernier can read to one-sixtieth of a division on the circle, 1. e. to I'. (350) Circle divided to 30'. Such a graduation is a very common one. The Vernier may be variously constructed. Suppose 30 spaces on the Vernier to be equal to 29 on the circle. Each space on the Vernier will be = 2 -- 29', 30 and will therefore be less than a space of the circle by 1', to which the Vernier will then read. Fig. 230. The above figure shows this arrangement. The reading is 0~, or 360~. In the following figure, the dotted and crossed line shows what divisions coincide, and the reading is 20" 10'; the Vernier being the same as in the preceding figure, and its zero being at a point of the circle 10' beyond 20. 28S4 ThANSIT $ ND T HE 1ODOLI TE SliRVEYING1. 1PAR'IV. Fig. 231. y the yner to be 10'.. 3 32' at a0 p A I f^ I I ^ t~~~~~~~~~~~~~~~~~a ol I CHAP. II.1 Vernlers. 235 Sometimes 30 spaces on the Vernier are equal to 31 on the circle, Each space on the Vernier will therefore be - 31 =30 31' and 30 - will be longer than a space on the circle by 1', to which it will therefore read, as in the last case, but the Vernier will be "; retrograde." This is the Vernier of the compass, Fig. 148. The peculiar manner in which it is there applied is shown in Fig. 239. If 15 spaces on the Vernier are equal to 16 on the circle, each 16 x 30' space on the Vernier will be =- 1 = - 32', and the Vernier will therefore read to 2'. (351) Circle divided to 20'. If 20 spaces on the Vernier are equal to 19 on the circle, each space of the latter will be 19 x 20' 20 - 19', and the Vernier will read to 20'- 19' ='. If 40 spaces on the Vernier are equal to 41 on the circle, each 41 x 20' space on the Vernier will be - 40 = 20'; and the Ver nier will therefore read to 20' - 20' 30". It will be retro. grade. In the following figure the reading is 360~, or 0~; and it will be seen that the 40 spaces on the Vernier (numbered to whole minutes) are equal to 13~ 40' on the limb, i. e. to 41 spaces, eacl of 20'. Fig.233. 3 __ 10 15 If 60 spaces on the Vernier are equal to 59 on the circle, each 59 x 20' of the former will be = 19' 40", and the Vernier 236 TRANSIT AND THEODOLITE SIRVEYING. [PART IV will therefore read to 20'- 19' 40" - 20". The following figure shows such an arrangement. The reading in that position would be 40" 46' 20". Fig. 234. 50 J l ___i^ ___I I I 1io 9 8 7 5 4 3 2 1t %.. ($52) Circle divided to 15'. If 60 spaces on the Vernier are ecual to 59 on the circle, each space on the Vernier will be = 59 x 15' ~_ 0 _ == 14' 45", and the Vernier will read to 15". In the following figure the reading is 10~ 20' 45", the index pointing to 100 15', and something more, which the Vernier shows to be 5' 45' Fig. 235. \l/ X1t ~ -I l.I/l..... II I III,_l.1.1 I..I_ I.. 10 9 8 7 5 - 3 2 A CHAI. Ix.] Verniers. 237 (353) Circle divided to 10'. If 60 spaces on the Vernier be equal to 59 on the limb, the Vernier will read to 10". In the following figure, the reading is 7~ 25' 40", the reading on the circle being 7~ 20', and the Vernier showing the remaining space to be 5' 40"' Fig. 236. -~ 9 8 7~ --- 5 t 3 2 1 i (354) Reading backwards. When an index carrying a Vernier is moved backwards, or in a contrary direction to that in which the numbers on the circle run, if we wish to read the space which it has passed over in this direction from the zero point, the Vernier must be read backwards, (i. e. the highest number be called 0), or its actual reading must be subtracted from the value of the smallest space on the circle. The reason is plain; for, since the Vernier shows how far the index, moving in one direction, has gone past one division line, the distance which it is from the next division line (which it may be supposed to have passed, moving in a contrary direction), will be the difference between the reading and the value of one space. Thus, in Fig. 229, page 282, the reading is 358~ 15'. But, counting backwards from the 360", or zero point, it is 1~ 45'. Caution on this point is particularly necessary in using snall angles of deflection for railroad curves. 238 TRANSIT AND THEODOLITE $URIVEYING. [PART I,. (355) Are of excess, On the sextant and similar instruments, the divisions of the limb are carried onward a short distance beyond the zero point. This portion of the limb is called the " Arc of excess." When the index of the Vernier points to this arc, the reading must be made as explained in the last article. Thus, in the figure, the reading on the arc from the zero of the limb to the Fig. 237'ii 10 9 8 716 5 4 3 1 zero of the Vernier is 4~ 20', and something more, and the reading of the Vernier from 10 towards to the right, where the lines coincide, is 3' 20" (or it is 10'- 6' 40" = 3' 20"), and the entire reading is therefore 4~ 23' 20". (35) Double Verniers. To avoid the inconveniences of reading backwards, double Verniers are sometimes used. The figure below shows one applied to a Transit. Each of the Verniers is Fig. 238. 30 101) 4 9 210 __10_ill_ I ) ill li < IL, CHAP. ii.] Verniers. 239 like the one described in Art. (350), Figs. 230, 231, and 232. When the degrees are counted to the left, or as the numbers run, as is usual, the left-hand Vernier is to be read, as in Art. (350) but when the degrees are counted to the right, from the 360~ line, the right-hand Vernier is to be used. (357) Cempass-Vernier. Another form of double Yernier, often applied to the compass, is shown in the following figure. The Fig. 239. 3 2 10 5~ ~30 R15 / _____ I _I f limb is divided to half degrees, and the Vernier reads to minutes, 30 parts on it being equal to 31 on the limb. But the Vernier is only half as long as in the previous case, going only to 15', the upper figures on one half being a sort of continuation of the lower figures on the other half. Thus in moving the index to the right, read the lower figures on the left hand Vernier (it being retrograde) at any coincidence, when the space passed over is less than 15'; but if it be more, read the upper figures on the right hand Vernier: and vice versa. 240 [PART IV CHAPTER III. ADJUSTMENTI S (358) THE purposes for which the Transit and Theodolite (as well as most surveying and astronomical instruments) are to be used, require and presuppose certain parts and lines of the instrument to be placed in certain directions with respect to others; these respective directions being usually parallel or perpendicular. Such arrangements of their parts are called their Adjustments. The same word is also applied to placing these lines in these directions. In the following explanations the operations which determine whether these adjustments are correct, will be called their Ferifications; and the making them right, if they are not so, their Reetifieations.* (359) In observations of horizontal angles with the Transit or the Theodolite,t it is required, 10 That the circular plates shall be horizontal in whatever way they may be turned around. 2~ That the Telescope, when pointed forward, shall look in precisely the reverse of its direction when pointed backward, i. e. that its two lines of sight (or lines of collimation) forward and backward shall lie in the same plane. 3~ That the Telescope in turning upward or downward, shall move in a truly vertical plane, in order that the angle measured between a low object and a high one, may be precisely the same as would be the angle measured between the low object and a point exactly under the high object, and in the same horizontal plane as the low one. * It has been well said, that " in the present state of science it may be laid down as a maxim, that every instrument should be so contrived, that the observer may easily examine and rectify the principal parts; for, however careful the instrument-maker may be, however perfect the execution thereof, it is not possible that any instrument should long remain accurately fixed in the position in which it came out of the maker's hands."-Adams'' Geometrical and Gradhical Essays," 1791. tThe Theodolite adjustments which relate only to levelling, or to measuring vertical angles, will not be here discussed. CHAP. III.] Adustments. 241 We shall see that all these adjustments are finally resolvable into these; 1st. Making the vertical axis of the instrument perpendicular to the plane of the levels; 2d. Making the line of collimation perpendicular to its axis; and 3d. Making this axis parallel to the plane of the levels. They are all best tested by the invaluable principle of "6 Reversion." We have now, firstly, to examine whether these things are sc, that is, to " verify" the adjustments; and, secondly, if we find that they are not so, to makce them so, i. e. to "' rectify," or " adjust" them correctly. The above three requirements produce as many corresponding adjustments. (360) First adjustment. To cause the circle to be horizontal in every position. TVerification.-Turn the Vernier plate which carries the levels, till one of them is parallel to one pair of the parallel plate screws. The other will then be parallel to the other pair. Bring each bubble to the middle of its tube, by that pair of screws to which it is parallel. Then turn the vernier plate half way around, i. e. till the index has passed over 180~. If the bubbles remain in the centres of the tubes, they are in adjustment. If either of them runs to one end of the tube, it requires rectification. Rectification.-The fault which is to be rectified is that the plane of the level (i. e. the plane tangent to the highest point of the level tube) is not perpendicular to the vertical axis, AA in. figure 214, on which the plate turns. For, let AB represent this Fig. 240. Fig. 241. A-C0 A- - li~~~~~~~~ I:~I 39 D plane, seen edgeways, and CD the centre line of the vertical axis, ~his applies equally to the Transit and the Theodolite. 16 242 TIB SIT AND THEODOLITE SURVEgNG, [PART IV which is here drawn as making an acute angle with this plane on the right hand sid.e. The first figure represents the bubble brought to the centre of the tube. The second figure represents the plate turned half around. The centre line of the axis is sups posed to remain unmoved. The acute angle will now be on the left hand side, and the plate will no longer be horizontal. Consequently the bubble will run to the higher end of the tube. The rectification necessary is evidently to raise one end of the tube and lower the other. The real error has been doubled to the eye by the reversion. Half of the motion of the bubble was caused by the tangent plane not being perpendicular to the axis, and half by this axis not being vertical. Therefore raise or lower one end of the level by the screws which fasten it to the plate, till the bubble comes about half way back to the centre, and then bring it quite back by turning its pair of parallel plate screws. Then again reverse the vernier plate 180~. The bubble should now remain in the centre. If not, the operation should be repeated. The same must be done with the other level if required. Then the bubbles will remain in the centre during a complete revolution. This proves that the axis of the vernier plate is then vertical; and as it has been fixed by the maker perpendicular to the plate, the latter must then be horizontal. It is also necessary to examine whether the bubbles remain in the centre, when the divided circle is turned round on its axis. If not, the axes of the two plates are not parallel to each other. The defect can be remedied only by the maker; for if the bubbles be altered so as to be right for this reversal, they will be wrong for the vernier plate reversal (361) Second adjustment. To cause the line of collimation to revolve in aplane.' Verification. Set up the Transit in the middle of a level piece of ground, as at A in the figure. Level it carefully. Set a stake, with a nail driven into its head, or a chain pin, as far from the instrument as it is distinctly visible, as at B. Direct the telescope I his adjustment is not the same in the Transit and in the Theodolite. That for dte Transit will be first given, and that for the Theodolite in the next article. OAPe III.] Adjustmentse 243 Fig. 242 C to its and fix the intersection of the cross-hairs very precisely upon it. Clamp the instrument. Measure from A. to B. Then turn over the telescope, and set another stake at an equal distance from the Transit and also precisely in the line of sight. If the line of collimation has not continued in the same plane during its half-revolution, this stake will not be at E, but to one side, as at C. To discover the truth, loosen the clamp and turn the vernier plate half around without touching the telescope. Sight to B, as at first, and again clamp it. Then turn over the telescope, and the line of sight will strike, as at D in the figure, as far to the right of the point, as it did before to its left. Rectification. The fault which is to be rectified, is that the line of collimation of the telescope is not perpendicular to the horizontal axis on which the telescope revolves. This will be seen by the figures, which represent the position of the lines in each of the four Fig. 243. B ---— ~13.......... 0'9 Fig. 244. T Fig. 245. B --- Fig. 246. s observations which have been made. In each of the figures the long thick line represents the telescope, and the short one the axis on which it turns. In Fig. 243'the line of sight is directed to B. 244 TRANSIT AND THEODOLLTE SURVET IN. [PART Iv. In Fig. 244 the telescope has been turned over, and with it the axis, so that the obtuse angle, marked O in the first figure, has taken the place, 0', of the acute angle, and the telescope points to C instead of to E. In Fig. 245 the vernier plate has been turned half around so as to point to B again, and the same obtuse angle has got around to 0". In Fig. 246 the telescope has been turned overi the obtuse angle is at 0"', and the telescope now points to D. To make the line of collimation perpendicular to the axis, the former must have its direction changed. This is effected by moving the vertical hair the proper distance to one side. As was explained in Art. (330), and represented in Fig. 217, the crosshairs are on a ring held by four screws. By loosening the lefthand screw and tightening the right-hand one, the ring, and with it the cross-hairs, will be drawn to the right; and vice versa. Two holes at right angles to each other pass through the outer heads of thie screws. Into these holes a stout steel wire is inserted, and the screws can thus be turned around. Screws so made are called' capstan-headed.' One of the other pair of screws may need to be loosened to avoid straimng the threads. In some French instruments, one of each pair of screws is replaced by a spring. To find how much to move this vertical hair, measure from C to D, Fig. 242, page 243. Set a stake at the middle point E, and set another at the point F, midway between D and E. Move the vertical hair till the line of sight strikes F. Then the instrument is adjusted; and if the line of sight be now directed to E, it will strike B, when the telescope is turned over; since the hair is moved half of the doubled error, DE. The operation will generally require to be repeated, not being quite perfected at first. It should be remembered, that if the Telescope used does not invert objects, its eye-piece will do so. Consequently, with such a telescope, if it seems that the vertical hair should be moved to the left, it must be moved to the right, and vice versa. An invertig telescope does not invert the cross-hairs. If the young surveyor has any doubts as to the perfection of his rectification, he may set another stake exactly under the instrument by means of a plumb-line suspended from its centre; and then, in like manner, set his Transit over B or E. He will find that the CHAP. iii.] Adjustments. 245 other two stakes, A and the extreme one, are in the same straight line with his instrument. In some instruments, the horizontal axis of the telescope can be taken out of its supports, and turned over, end for end. In such a case, the line of sight may be directed to any well defined point, and the axis then taken out and turned over. If the line of sight again strikes the same point, this line is perpendicular to the axis. If not, the apparent error is double the real error, as appears from the figures, the obtuse angle 0 coming to 0', and the desired perFig. 247. ----------- Fig. 248. B.o —-' —---- ---- pendicular line falling at C midway between B and B'. The rectification may be made as before; or, in some large instruments, in which th thtelescope is supported on Ys, by moving one of the ys laterally. (362) The Theodolite must be treated differently, since its telescope does not reverse. One substitute for this reversal, when it is desired to range out a line forward and backward from one station, is, after sighting in one direction, to take the telescope out of the ys and turn it end for end, to sight in the reverse direction. This it can be made to do by adjusting its line of collimation as explained in the last article. Another substitute is, after sighting in one direction, and noting the reading, to turn the vernier plate around exactly 180~. But this supposes not only that the graduation is perfectly accurate, but also that the line of collimation is exactly over the centre of the circle. To test this, after sighting to a points and noting the reading, take the telescope out of the ys and turn it end for end, and then turn the vernier plate around exactly 180~. If the line of sight again strikes the same point, the latter condition exists. If not, the maker must remedy 246 TRUiNSiT AND TiEODOLITE SJRVEYIG. [PART Iv, the defect. This error of eccentricity is similar to that explained with respect to the compass, in the latter part of Art. (226). (363) Tbh'd adjustment. To cause the line of collimation to revolve in a vertical plane.* Verification. Suspend a long plumb-line from some high point. Set the instrument near this line, and level it carefully. Direct the telescope to the plumb-line, and see if the intersection of the cross-hairs follows and remains upon this line, when the telescope is turned up and down. If it does, it moves in a vertical plane. The angle of a new and well-built house will firm an imperfect substitute for the plumb-line. Otherwise; the instrument being set up and levelled as above, place a basin of some reflecting liquid (quicksilver being the best, though molasses, or oil, or even water, will answer, though less perfectly,) so that the top of a steeple, or other point of a high object, can be seen in it through the telescope by reflection. Make the intersection of the cross-hairs cover it. Then turn up the telescope, and if the intersection of the cross-hairs bisects also the object seen directly, the line of sight has moved in a vertical plane. If a star be taken as the object, the star and its reflection will be equivalent (if it be nearly over head) to a plumb-line at least fifty million million miles long. Otherwise; set the instrument as close as possible to the base of a steeple, or other high object; level it, and direct Fig. 249 it t6 the top of the steeple, or to some other elevated $ and well defined point. Clamp the plates. Turn down the telescope, and set up a pin in the ground precisely "i in line." Then loosen the clamp, turn over the telescope, and turn it half-way around, or so far as to again sight to the high point. Clamp the plates, and again turn down the telescope. If the line of sight again strikes the pin, the telescope has moved in a vertical plane. If not, the apparent P P 17I error is double the real error. For, let S be the top of the steeple, *This applies to both the Transit and the Theodolite, with the exception of the method of verification by the steeple and pin, which applies only to the Transit. CHAP. III.] Adjustments, 247 (Fig.249) and P' the pin; then the plane in which Fig. 250. the telescope moves, seen eclgewise, is SP'; and. 5 after being turned around, the line of sight moves in the plane SP', as far to one side of the vertical plane SP, as SP' was on the other side of it. Rectificction. Since the second adjustment causes the line of sight to move in a plane perpendicular to the axis on which it turns, it will move in a vertical plane if that axis be horizontal. It may be made so by filing off the feet of the standards which support the higher end of the axis. This will be best done by the T maker. In some instruments one end of the I L axis can be raised or lowered. (364) Ctentrg eye-paice. In some in- struments, such as that of which a longitudinal F section is shown in the margin, the inner end / of the eye-piece may be moved so that the Ad B cross-hairs shall be seen precisely in the cen- tre of its field of view. This is done by means of four screws, arranged in pairs, like those of ^ the cross-hair-ring screws, and capable of mov- ing the eye-piece up and down, and to right or left, by loosening one and tightening the opposite one. Two of them are shown at A, A, in the figure; in which B, B, are two of the [ cross-hair screws. (3t5) Centring object-glass. In some instruments four screws, similarly arranged, two of which are shown at C, C, can move, in any direction, the inner end of the slide which carries the object-glass. The necessity for'' such an arrangement arises from the impossi- 248 TRANSIT AND THEODOLITE SURTVEYIN, [PART IT bility of drawing a tube perfectly straight. Consequently, the line of collimation, when the tube is drawn in, will not coincide with the same line when the tube is pushed out. If adjusted for one position, it will therefore be wrong for the other. These screws, however, can make it right in both positions. They are used as follows. Sight to some well defined point as far off as it can be distinctly seen. Then revolve the telescope half around in its supports; i. e. turn it upside down. If the line of collimation was not in the imaginary axis of the rings or collars on which the telescope rests, it will now no longer bisect the object sighted to. Thus, if the horizontal hair was too high, as in Fig. 251, tins line of Fig. 251. --------- collimation would point at first to A, and after being turned over, it would point to 1. The error is doubled by the reversion, and it should point to C, midway between A and B. Make it do so, by un screwing the upper capstan-headed screw, and screwing in the lower one, till the horizontal hair is brought half way back to the point. Remember that in an erecting telescope, the cross-hairs are reversed, and vice versa. Bring it the rest of the way by means of the parallel plate screws. Then revolve it in the Ys back to its original position, and see if the intersection of the cross-hairs now bisects the point, as it should. If not, again revolve, and repeat the operation till it is perfected. If the vertical hair passes to the right or to the left of the point when the telescope is turned half around, it must be adjusted in the same manner by the other pair of cross-hairs screws. One of these adjustments may disturb the other, and they should be repeated alternately. When they are perfected, the intersection of the cross-hairs, when once fixed on a point, will not move from it when the telescope is revolved in its In Theodolites, the Telescope is revolved in the Ys. I Transits, the maker, by whom this adjustment is usually performed, revolves the Telescope, in the same manner, before it is fixed in its cross-bar. CHAP. IIi.] Adjustments. 249 supports. This double operation is called adjjusting the line of collicmation.* This line is now adjusted for distant objects. It would be so for near ones also, if the tube were perfectly straight. To test this, sight to some point, as near as is distinctly visible. Then turn the telescope half over. If the intersection does not now bisect the point, bring it half way there by the SCrews C, C, of Fig, 250, moving only one of the hairs at a time, as before. Then repeat the former adjustment on the distant object. If this is not quite perfect, repeat the operation. This adjustment, in instruments thus arranged, should precede the first one which we have explained. It is usually performed by the maker, and its screws are not visible in the Transit, being enclosed in the ball seen where the telescope is connected with the cross-bar.i All the adjustments should be meddled with as little as possible, lest the screws should get loose; and when once made right they should be kept so by careful usage. * This "acdjustment of the line of collimation" has merely brought the intersection of the cross-hairs (which fixes the line of sight) into the line joining the centres of the collars on.which the telescope turns in the Ys; bt the maker is supposed to have originally fixed the optical axis of thetelescope (i. e. the line joining the optical centres of the glasses), in the same line. tThe adjustment of " Centring the olject-glass is the invention of Messre Gurley, of Troy. 250 FPABT 17 CHAPTER IV, THE FIfELD-WOl KE (366) lo measire a horizontal angle. Set up the instrument so that its centre shall be Fig. 252. exactly over the angu- A lar point, or in the intersection of the two lines whose difference of A direction is to be measured; as at B in the figure. A plumb line must be suspended from under the centre. Dropping a stone is an imperfect substitute for this. Set the instrument so that its lower parallel plate may be as nearly horizontal as possible. The levels will serve as guides, if the four parallel-plate screws be first so screwed up or down that equal lengths of them shall be above the upper plate. Then level the instrument carefully, as in Art. (338). Direct the telescope to a rod, stake, or other object, A in the figure, on one of the lines which form the angle. Tighten the clamps, and by the tangent-screw, (see Art. (336)), move the telescope so that the intersection of the crosshairs shall very precisely bisect this object. Note the reading of the vernier, as explained in the preceding chapter. Then loosen the clamp of the vernier, and direct the telescope on the other line (as to C) precisely as before, and again read. The difference of the two readings will be the desired angle, ABC. Thus, if the first reading had been 40~ and the last 190~, the angle would be 1500. If the vernier had passed 360~ in turning to the second object, 360~ should be added to the last reading before subtracting. Thus, if the first reading had been 300~, and the last reading 90~, the angle would be found by calling the last reading, as it really is, 360~ + 90~ == 450~, and then subtracting 300~. It is best to sight first to the left hand object and then to the right hand one, turning " with the sun," or like the hands of a watch, since the numbering of the degrees usually runs in that direction. CHAP. iv.] The Field-work. 251 It is convenient, though not necessary, to begin by setting the vernier-at zero, by the upper movement (that of the vernier plate on the circle) and then, by means of the lower motion, (that of the whole instrument on its axis), to direct the telescope to the first object. Then fasten the lower clamp, and sight to the second object as before. The reading will then be the angle desired. An objection to this is that the two verniers seldom read alike.* After one or more angles have been observed from one point, the telescope must be directed back to the first object, and the reading to it noted, so as to make sure that it has not slipped. A watch-telescope (see Art. 339) renders this unnecessary. The error arising from the instrument not being set precisely over the centre of the station, will be greater the nearer the object sighted to. Thus a difference of one inch would cause an error of only 3" in the apparent direction of an object a mile distant, but one of nearly 3' at a distance of a hundred feet. (367) ledltoion of high and low objects. When one of the objects sighted to is higher than the other, the' plunging telescope" of these instruments causes the angle measured to be the true horizontal angle desired; i. e. the same angle as if a point exactly under the high object and on a level with the low object (or vice versa) had been sighted to. For, the telescope has been caused to move in a vertical plane by the 3d adjustment of Chapter II, and the angle measured is therefore the angle between the vertical planes which pass through the two objects, and which " project" the two lines of sight on the same horizontal plane. This constitutes the great practical advantage of these instruments over those which are held in the planes of the two objects observed, such as the sextant, and the "' circle" much used by the French. * The learner will do wvell to gauge his own precision and that of the instrument (and he may rest assured that his own will be the one chiefly in fault) by measur. ing, from any station, the angles between successive points all around him, till he gets back to the first point, beginning at different parts of the circle for each angle. The sum of all these angles should exactly equal 3600. He will probably find quito a difference from that. 2 TRANSIT AND T DIEEODOLITE VEYING [PART TV (3S8) Notation of angles, The angles observed may be noted in various ways. Thus, the observation of the angle ABC, in Fig. 252, may be noted " At B, from A to C, 150~," or better, "- At B, between A and C, 150~." In column form, this becomes Between A 150 ~and C. At B When the vernier had been set at zero before sighting to the first object, and other objects were then sighted to, those objects, the readings to which were less than 180~, will be on the left of the first line, and those to which the readings were more than 1800, will be on its right, looking in the direction in which the survey is proceeding, from A to B, and so on.* (391) Probable error. When a number of separate observations of an angle have been made, the mean or average of them all, (obtained by dividing the sum of the readings by their number,) is taken as the true reading.' The " Probable error" of this mean, is the quantity, (minutes or seconds) which is such that there is an even chance of the real error being more or less than it. Thus, if ten measurements of an angle gave a mean of 35~ 18', and it was an equal wager that the error of this result, too much or too little, was half a minute, then half a minute would be the " Probable error" of this determination. This probable error is equal to the square root of the sum of the squares of the errors (i. e. the differences of each observation from the mean) divided by the number of observations, and multiplied by the decimal 0.674489. The same result would be obtained by using what is called The weight" of the observation. It is equal to the square of the number of observations divided by twice the sum of the squares of the errors. The " Probable error" is equal to 0.476936 divided by the square root of the weight. These rules are proved by thq "Theory of Probabilities." (370) To repeat an angle, Begin as m Art. (36;), an measure the angle as there directed. Then unclamp below, and turn the circle around till the telescope is again directed to the first object, and made to bisect it precisely by the lower tan* This is very useful in preventing any ambiguity in the field-notes CHAP. IV.] The Field-work. 253 gent screw. Then unclamp above and turn the vernier plate till the telescope again points to the second object, the first reading remaining unchanged. The angle will now have been measured a second time, but on a part of the circle adjoining that on which it was first measured, the second arc beginning where the first ended. The difference between the first and last readings will therefore be twice the angle. This operation may be repeated a third, a fourth, or any number of times, always turning the telescope back to the first object by the lower movement, (so as to start with the reading at which the preceding observation left off) and turning it to the second object by the upper movement. Take the difference of the first and last readings and divide by the number of observations. The advantage of this method is that the errors of observation (i. e. sighting sometimes to the right and sometimes to the left of the true point) balance each other in a number of repetitions; while the constant error of graduation is reduced in proportion to this number. This beautiful principle has some imperfections in practice, probably arising from the slipping and straining of the clamps. (371) Bngles of deflectton. The angle of deflection of one line from another, is the Fig. 253. angle which one line' makes with the other line produced. Thus, in... the figure, the angle of deflection of BC from AB, is B'BC. It is evidently the supplement of the angle ABC. To measure it with the Transit, set the instrument at B, direct the telescope to A, and then turn it over. It will now point in the direction of AB produced, or to B', if the 2d adjustment of Chapter II, has been performed. Note the reading. Then direct the telescope to C. Note the new reading, and their (lifference will be the required angle of deflection, B'BC. If the vernier be set at zero, before taking the first observation, the readings for objects on the right of the first line will be less than 254 TRANSIT AND THEODOLITE STRVEING, [PART IV 180~, and more than 180~ for objects on the left conversely to Art. (3X8). (372) Line surveying, The survey of a line, such as a road, &c., can be made by the Theodolite or Transit, with great precision; measuring the angle which each line makes with the preceding line, and noting their lengths, and the necessary offsets on each side. Short lines of sight should be avoided, since a slight inaccuracy in setting the centre of the instrument exactly over or under the point previously sighted to, would then much affect the angle, as noticed at close of Art. (366). Very great accuracy can be obtained by using three tripods. One would be set at the first station and sighted back to from the instrument placed at the second station, and a forward sight be then taken to the third tripod placed at the third' station. The instrument would then be set on this third tripod, a back sight taken to the tripod remaining on the second station, and a foresight taken to the tripod brought from the first station to the fourth station; to which the instrument is next taken: and so on. This is especially valuable in surveys of mines. The field-notes may be taken as directed in Chapter III of Compass Surveying, pages 149, &c., the angles taking the place of the Bearings. The " Checlks by intersecting Bearings," explained in Art. (246), should also be employed. The angles made on each side of the stations may both be measured, and the equality of their sum to 3600, would at once prove the accuracy of the work. If the magnetic Bearing of any one of the lines be given, and that of any of the other lines of the series be required, it can be deduced by constructing a diagram, or by modifications of the rules given for the reverse object, in Art. (243). (373) Traversing: Or Surveying by the back-angle. This is a method of observing and recording the different directions of successive portions of a line, (such as a road, the boundaries of a farm, &c.,) so as to read off on the instrument, at each station, the angle which each line makes-not with the preceding line, but-with the first line observed. This line is, therefore, called the meridian of that survey. CHAP. iv.] The Field-work, 255 Fig. 254. A B, - t —Set up the instrument at the first angle, or second station, (B, in the figure), of the line to be surveyed. Sight to A and then to C. Clamp the vernier, and take the instrument to C. Loosen the lower clamp, and direct the telescope to B, the reading remaining as it was at B. Clamp below, loosen above, and sight to D. The reading of the instrument will be the angle which the line CD makes with the first line, or Meridian, AB. Take the instrument to D. Sight back to C, and then forward to E, as before directed, and the reading of the instrument will be the angle which DE makes with AB. So proceed for any number of lines. When the Transit is used, the angles of deflection of each line from the first, obtained by reversing the telescope, may be used in "Traversing," and with much advantage when the successive lines do not differ greatly in their directions. A 00 A 0 B 200~ The survey represented in the figure, B 200 -C 50~ is recorded in the first of the accompa- C 500 D 180~ nying Tables, as observed with the The- D 00 E 300 o 0 E {000 F 2100 oodolite; and in the second Table, as F 83 G 250~ observed with the Transit. G 250 The chief advantage of this method is its greater rapidity in the field and in platting, the angles being all laid down from one meridian, as in Compass-surveying. This also increases the accuracy of the plat, since any error in the direction of one line does not affect the directions of the following lines. (374) Use of the Compass, The chief use of the Compass attached to a Transit or Theodolite, is as a check on the observations; for the difference between the magnetic Bearings of any * If there are two verniers; take care always to read the degrees from the same vernier. Mark it A. 256 TRANSIT AND THEODOLITE SRVERYIllNGo [PART iv. two lines should be the same, approximately, as the angle between them, measured by the more accurate instruments. The Bearing also prevents any ambiguity, as to whether an angle was taken to the right or to the left. The instrument may also be used like a simple compass, the telescope taking the place of the sights, and requiring similar tests of accuracy. A more precise way of taking a Bearing is to turn the plate to which the compass box is attached, till the needle points to zero, and note the reading of the vernier; then sight to the object, and again read the vernier. The Bearing will thus be obtained more minutely than the divisions on the compass box could give it. (375) IMeasur'ig distances with a telescope and red, On the cross-hair ring, described in Art. (330), stretch two more horizontal spider-threads at equal distances above and below the original one; or all may be replaced by a plate of thin Fig 255. glass, placed precisely in the focus, with the necessary lines, as in the figure, etched by fluoric acid. Let a rod, 10 or 15 feet long, be heldup at 1000 feet off, and __ let there be marked on it precisely the length which the distance between two of these lines covers. Let this be subdivided as minutely as the spaces, painted alternately white and red and numbered, can be seen. If ten subdivisions are made, each will represent a distance of 100 feet off, and so on. Continue these divisions over the whole length of the rod. It is now ready for use. The French call it a stadia. When it is held up at any unknown distance, the number of divisions on it intercepted between the two lines, will indicate the distance with considerable precision. It should be tested at various distances. A "Levelling-rod," divided into feet, tenths and hundredths, may be used as a stacdia, with less convenience but more precision. Experiments must previously determine at what distances the space between the lines in the telescope covers one foot, &c. Then, at any unknown distance, let the sliding "6 target" of the rod be moved till one line bisects it, and its place on the rod be read off; let the target be then moved so that the other line bisects it and let CHAP. Iv.] The Field-work, 257 its place be again noted. Then the required distance will be equal to the difference of the readings on the rod, in feet, multiplied by the distance at which a foot was intercepted between the lines. One of the horizontal hairs may be made movable, and its distance from the other, when the space between them exactly covers an object of known height, can be very precisely measured by counting the number of turns and fractions of a turn, of a screw by which this movable hair is raised or lowered. A simple proportion will then give the distance. On sloping ground a double correction is necessary to reduce the slope to the horizon and to correct the oblique view of the rod. The horizontal distance is, in consequence, approximately equal to the observed distance multiplied by the scqure of the cosine of the slope of the ground. The latter of the above two corrections will be dispensed with by holding the rod perpendicular to the line of sight, with the aid of a right angled triangle, one side of which coincides with the rod at the height of the telescope, and the other side of which adjoining the right angle, is caused, by leaning the rod, to point to the telescope. Other contrivances have been used for the same object, such as a Binocular Telescope with two eye-pieces inclined at a certain angle; a Telescope with an object-glass cut into two movable parts; &c. (376) Rangiag out lines. This is the converse of Surveying lines. The instrument is fixed over the first station with great precision, its telescope being very carefully adjusted to move in a vertical plane. A series of stakes, with nails driven in their tops, or otherwise well defined, are then set in the desired line as far as the power of the instrument extends. It is then taken forward to a stake three or four from the last one set, and is fixed over it, first by the plumb and then by sighting backward and forward to the first and last stake. The line is then continued as before. A good object for a long sight is a board painted like a target, with black and white concentric rings, and made to slide in grooves cut in the tops of two stakes set in the ground about in the line. It 17 258 TRANSIT AND THEODOLITE $TERVElN[. LPART IV is moved till the vertical hair bisects the circles (which the eye can determine with great precision) and a plumb-line dropped from their centre, gives the place of the stake. 6"Mason & Dixon's Line" was thus ranged. If a Transit be used for ranging, its " Second Adjustment" is most important to ensure the accuracy of the reversal of its Telescope. If a Theodolite be used, the line is continued by turning the vernier s180, or by reversing the telescope in its ys, as noticed in Arts. (325) and (362). (377) Far m Sirveyng, &e. A large farm can be most easily and accurately surveyed, by measuring the angles of its main boun' daries (and a few main diagonals, if it be very large,) with a Theodolite or Transit, as in Arts. (~66) or (3TI), and filling up the interior details, as fences, &c., with the Compass and Chain. If the TheocZolite be used, Fig. 056. keep the fielcd on the left C hanl, as in following the or- O der of the letters in this figure, and turn the telescope, 2o00~ around 4 with the sun," and \ ~ L the gangles measured as in D Art. (~66), will be the interior angles of the field, as noted in the figure. The accuracy of the work will be proved, as alluded to in Art. (257), if the sum of all the interior angles be equal to the pro" duct of 1800 by the number of sides of the figure less two. Thus in the figure, the sum of all the interior angles = 540~ = 180~ x (5 - 2). The sum of the exterior angles would of course equal 1800 x (5 + 2) = 1260". If the Transit be used, the farm should be kept on the right hand, and then the angles measured will be the supplements of the interior angles. If the angles to the right be called positive, and those to the left negative, their algebraic sum should equal 360~. If the boundary lines be surveyed by " Traversing," as in Art. (373), the reading, on getting back to the last station and looking back to the first line, should be 860~, or 0~. CHAP. IV.] The Field-work' 259 The content of any surface surveyed by " Traversing" with the Transit can be calculated by the Traverse Table, as in Chapter VI, of Part III, by the following modification. When the angle of deflection of any side from the first side, or Meridian, is less than 90~, call this angle the Bearing, find its Latitude and Departure, and call them both plus. When the angle is between 90~ and 180~ call the difference between the angle and 180~ the Bearing, and call its Latitude minus and Its Departure plus. When the angle is between 180~ and 270~, call its difference from 180~ the Bearing, and call its Latitude minus and its Departure minus. When the angle is more than 270~, call its difference from 360~ the Bearing, and call its Latitude plus and its Departure mginus. Then use these as in getting the content of a Compass-survey. The signs of the Latitudes and Departures follow those of the cosines and sines in the successive quadrants. Town-Surveying would be performed as directed in Art. (261), substituting I angles" for "Bearings."' Traversing" is the best method in all these cases. Inaccessible areas would be surveyed nearly as in Art. (134), except that the angles of the lines enclosing the space would be measured with the instrument, instead of with the chain. (378) Plattlng Any of these surveys can be platted by any of the methods explained and characterized in Chapter IV, of the preceding Part. A circular Protractor, Art. (264), may be regarded as a Theodolite placed on the paper. " Platting Bearings," Art. (2X5), can be employed when the survey has been made by " Traversing." But the method of " Latitudes and De. partures," Art. (~28), is by far the most accurate. IPART V, TRIANGULAR SURVEYING; OR By the Fourth Method. (379) TRIANGULAR SURVEYING is founded on the Fourth Method of determining the position of a point, by the intersection of two known lines, as given in Art. (8). By an extension of the principle, a field, a farm, or a country, can be surveyed by measuring only one line, and calculating all the other desired distances, which are made sides of a connected series of imaginary Triangyes, whose angles are carefully measured. The district surveyed is covered with a sort of net-work of such triangles, whence the name given to this kind of Surveying. It is more commonly called " Trigonometrical Surveying;" and sometimes "6 Geodesic Surveying," but improperly, since it does not necessarily take into account the curvature of the earth, though always adopted in the great surveys in which that is considered. (380) Outline of operations. A base line, as long as possible, (5 or 10 miles in surveys of countries), is measured with extreme accuracy. From its extremities, angles are taken to the most distant objects visible, such as steeples, signals on mountain tops, &c. The distances to these and between these are then calculated by the rules of Trigonometry. The instrument is then placed at each of these new stations, and angles are taken from them to still more distant stations, the calculated lines being used as new base lines. This process is repeated and extended till the whole district is embraced by these " primary triangles" of as large sides as possible, PART v.] TRIANGTLAR SURVEYITlGo 261 One side of the last triangle is so located that its length can be obtained by measurement as well as by calculation, and the agreement of the two proves the accuracy of the whole work. Within these primary triangles, secondary or smaller triangles are formed, to fix the position of the minor local details, and to serve as starting points for common surveys with chain and compass, &c. Tertiary triangles may also be required, The larger triangles are first formed, and the smaller ones based on them, in accordance with the important principle in all surveying operations, always to work from the whole to the parts, and from greater to less. Each of these steps will now be considered in turn, in the following order: 1. The Base; articles (381), (382). 2. The Triangulation; articles (383) to (390). 3. Modifications of the method; articles (391) to (395). (381) Ie1asuring a Base. Extreme accuracy in this is necessary, because any error in it will be multiplied in the subsequent work. The ground on which it is located must be smooth and nearly level, and its extremities must be in sight of the chief points in the neighborhood. Its point of beginning must be marked by a stone set in the ground with a bolt let into it. Over this a Theodolite or Transit is to be set, and the line " ranged out" as directed in Art. (376). The measurement may be made with chains, (which should be formed like that of a watch,) &c. but best with rods. We will notice in turn their Materials, Supports, Alinement, Levelling, and Contact. As to Materials, iron, brass and other metals have been used, but are greatly lengthened and shortened by changes of temperature. Wood is affected by moisture. Glass rods and tubes are preferable on both these accounts. But wood is the most convenient. Wooden rods should be straight-grained white pine, &c.; well seasoned, baked, soaked in boiling oil, painted and varnished. They may be trussed, or framed like a mason's plumb-line level, to prevent their bending. Ten or fifteen feet is a convenient length. Three are required, which may be of different colors, to prevent 262 TRIANGULAR SURVEYIN [PART v mistakes in recording. They must be very carefully compared with a standard measure., Supports must be provided for the rods, in accurate work. Posts set in line at distances equal to the length of the rods, may be driven or sawed to a uniform line, and the rods laid on them, either directly, or on beams a little shorter. Tripods, or trestles, with screws in their tops to raise or lower the ends of the rods resting on them, or blocks with three long screws passing througl them and serving as legs, may also be usei. Staves, or legs, for the rods have been used; these legs bearing pieces which can slide up and down them and on which the rods themselves rest. The Alinement of the rods can be effected, if they are laid on the ground, by strings, two or three hundred feet long, stretched between the stakes set in the line, a notched peg being driven when the measurement has reached the end of one string, which is then taken on to the next pair of stakes; or, if the rods rest on supports, by projecting points on the rods being alined by the instrument. The Levelling of the rods can be performed with a common mason's level; or their angle measured, if not horizontal, by a slope-level." The Contacts of the rods may be effected by bringing them end to end. The third rod must be applied to the second before the frst has been removed, to detect any movement. The ends must be protected by metal, and should be rounded (with radius equal to length of rod) so as to touch in only one point. Round-headed nails will answer tolerably. Better are small steel cylinders, horizontal on one end and vertical on the other. Sliding ends, with verniers, have been used. If one rod be higher than the next one, one must be brought to touch a plumb-line which touches the other, and its thickness be added. To prevent a shock from contact, the rods may be brought not quite in contact, and a wedge be let down between them till it touches both at known points on its graduated edges. The rods may be laid side by side, and lines drawn across the end of each be made to coincide or form one line. This is more accurate. Still better is a "' visual contact," a double microscope with cross-hairs being used, so placed that one tube bisects a dot at the end of one rod, and the other tube bisects a dot at the end PART V.] TRLNULAR SURVEYING. 26S of the next rod. The rods thus never touch. The distance between the two sets of cross-hairs is of course to be added. A Base could be measured over very uneven ground, or ever water, by suspending a series of rods from a stretched rope by rings in which they can move, and levelling them and bringing them into contact as above. (382) C'rrectlB ns f Base. If the rods were not level, their length must be reduced to its horizontal projection. This would be the square root of the difference of the squares of the length of the rod (or of the base) and of the height of one end above the other; or the product of the same length by the cosine of the angle which it makes with the horizon.@ If the rods were metallic, they would need to be cr:lected for temperature. Thus, if an iron bar expands o o o-o o of its length for 1~ Fahrenheit, and had been tested at 32~, and a Base had been measured at 720 with such a bar 10 feet long, and found to contain 3000 of them, its apparent length would be 80,000 feet, but it, real length would be 8.4 feet more. An iron and a brass ba can be so combined that the difference of their expansion, causes two points attached to their ends to remain at the saim distance at all temperatures. Such a combination is used on the U. S. Coast Survey. (383) Chlice of Stations, The stations, or "' Trigonometrical points," which are to form the vertices of the triangles, and to be observed tc and from, must be so selected that the resulting triangles may be " well-conditioned," i.. may have such sides and angles that a small error in any of the measured quantities will cause the least possible errors in the quantities calculated from them. The higher Calculus shows that the triangles should be as nearly equilateral as possible. This is seldom attainable, but no angle should be admitted less than 30~, or more than 120~.t * More precisely, A being this angle, and not more than 2~ or 3", the differ ance between the inclined and horizontal lengths, equals the inclined or real iength multiplied by the square of the minutes in A, and that by the decimal 0.00000004231; as shewn in Appendix B. In a Geodesic survey, the base would also be required to be reduced to the level of file sea. t When two angles only are observed, as is often the case in the secondary. triargulation, the unobserved angle ought to be nearly a right angle. 264 TRmilGULAR $SURVEYING. [PART v. To extend the triangulation, by continually increasing the sides of the triangles, without introducing " ill-conditioned"' triangles, may be effected as in the figure. AB is the measured base, Fig. 257. C and D are the nearest stations. In the triangles ABC and ABD, all the angles being observed and the side AB known, the other sides can be readily calculated. Then in each of the triangles DAC and DBC, two sides and the contained angles are given to find DC, one calculation checking the other. DC then becomes a base to calculate EF; which is then used to find GH; and so on. The fewer primary stations used, the better; both to prevent confusion and because the smaller number of triangles makes the correctness of the results more " probable.' The United States Coast Survey, under the superintendence of Prof. A. D. Bache, displays some fine illustrations of these principles, and of the modifications they may undergo to suit various localities. The figure on the opposite page represents part of the scheme of the primary triangulation resting on the Massa, chusetts base and including some remarkably well-conditioned triangles, as well as the system of quadlrilaterals which is a valuable feature of the scheme when the sides of the triangles are extended to considerable lengths, and quadrilaterals, with both diagonals determined, take the place of simple triangles. The engraving is on a scale of 1:1200,000. PART s1 TRITNGULAR SURVEYING 26G Fig. 258. \ \l ^^\^Bt /~b 266 TRIANIGLAi SlTRVEl iNG, [PART v (384)' Sigals. They must be high, conspicuous, and so made that the instrument can be placed precisely under them. Three or four timbers framed into a Fig. 259. pyramid, as in the figure, with a long mast projecting above, fulfil the first and last conditions. The mast may be made vertical by directing two theodolites to it and ad- justing it so that their telescopes follow it up and down, their lines of sight being at D... right angles to each other. Guy ropes may be used to keep it vertical. A very excellent signal, used on the Massachusetts State Survey, by Mr. Borden, is represented in the three following figures. It Fig. 260. Fig. 261. Fig. 262 consists merely of three stout sticks, which form a tripod, framed with the signal staff, by a bolt passing through their ends and its middle. Fig. 260 represents the signal as framed on the ground; Fig. 261 shews it erected and ready for observation, its base being steadied with stones; and Fig. 262 shews it with the staff turned aside, to make room for the Theodolite and its pro- Fig. 263 tecting tent. The heights of these signals varied between 15 and 80 feet. Another good signal consist3 of a stout post let into the ground, with a mast fastened to it by a bolt below i and a collar above. By opening the collar, the mast can be turned down and the Theodolite set exactly under the former summit of the signal, i. e. in its vertical axis. Signals should have a height equal to at least Jw- of their dic PART V.] TRIATGULA SURVEINT. 267 tance, so as to subtend an angle of half a minute, Lvhich experience has shown to be the least allowable. To make the tops of the signal-masts conspicuous, flags may be attached to them; white and red, if to be seen against the ground, and red and green if to be seen against the sky.* The motion of flags renders them visible, when much larger motionless objects are not. But they are useless in calm weather. A disc of sheet iron, with a hole in it, is very conspicuous. It should be arranged so as to be turned to face each station. A barrel, formed of muslin sewn together four or five feet long, with two hoops in it two feet apart, and its loose ends sewn to the signal-staff, which passes through it, is a cheap and good arrangement. A tuft of pine boughs fastened to the top of the staff, will be well seen against the sky. In sunshine, a number of pieces of tin nailed to the staff at different angles, will be very conspicuous. A truncated cone of burnished tin will reflect the sun's rays to the eye in almost every situation. But a " heliotrope," which is a piece of looking-glass, so adjusted as to reflect the sun directly to any desired point, is the most perfect arrangement. For night signals, an Argand lamp is used; or, best of all, Drum. mond's light, produced by a stream of oxygen gas directed through a flame of alcohol upon a ball of lime. Its distinctness is exceedingly increased by a parabolic reflector behind it, or a lens in front of it. Such a light was brilliantly visible at 66 miles distance. (g85) 0bei'rations of the Angles, These should be repeated as often as possible. In extended surveys, three sets, of ten each, are recommended. They should be taken on different parts of the circle. In ordinary surveys, it is well to employ the method of' Traversing," Art. (373). In long sights, the state of the atmosa Fig. 264. * To determine at a station A, z whether its signal can "be seen from B, projected against the -/9 W sky or not, measure the vertical,,- ~ angles BAZ and ZAC. If their "_,. sum equals or exceeds 180", A B - will be thus seen from B. If - not, the signal at A must be rais. ed till this sum equals 180~. 268 TRIANGULAR S$RVEYING, [PART V. phere has a very remarkable effect on both the visibility of the signals, and on the correctness of the observations. When many angles are taken from one station, it is important to record them by some uniform system. The form given below is convenient. It will be noticed that only the minutes and seconds of the second vernier are employed, the degrees being all taken from the first. Observations at STATION READINGS. MEAN RIGHT OR LEFT OF OBSERYED TO VERNIER A. VERNIER B. READING. PRECED G OBJ'T. REDIARKS. A 700 19' 0" 18' 40" 700 18' 50" B 1030 32' 20" 32' 40" 103~ 32' 30" R. a 115~ 14' 20"' 14' 50" 115~ 14' 35" R. When the angles are "'repeated," Art. (370), the multiple arcs will be registered under each other, and the mean of the seconds shewn by all the verniers at the first and last readings be adopted. (386) Reduction to the centre, It is often impossible to set the instrument precisely at or under the signal which has been observed. In such cases pro- Fig. 25. ceed thus. Let C be the cen- tre of the signal, and RCL the desired angle, I being the right hand object and L the left hand one. Set the instrument at D,.EAR as near as possible to C, and measure the angle RDL. It may be less than RCL, or greater than it, or equal to it, according as D lies without the circle passing through C, L and I, or within it, or in its circumference. The instrument should be set as nearly as possible in this last position. To find the proper correction for the observed angle, observe also the angle LDC, (called the angle of direction), counting it from 0~ to 360~, going from the left-hand object toward the left; and measure the distance DC. Calculate the distances CR and CL with the angle RDL instead of RGL, since they are sufficiently nearly equal. Then PART V.] TRINGULIR SURVEYING, 269 RL DL + CD. sin. (RDL + LDC) CD. sin. LDC, RCL RDL + ~C. sin. I" CL. sin.l" The last two terms will be the number of seconds to be added or subtracted. The Trigonometrical signs of the sines must be attended to. The log. sin. 1" =4. 6855749. Instead of dividing by sin. 1", the correction without it, which will be a very small fraction, may be reduced to seconds by multiplying it by 206265. Example. Let RDL = 22 20' 1".06; LDO = 101~ 15' 32".4; CD = 0.9; CR = 35845.12; CL = 29783.1. The first term of the correction will be + 3".750, and the second term —6'.113. Therefore, the observed angle RDL must be diminished by 2".363, to reduce it to the desired angle RCL. Much calculation may be saved by taking the station D so that all the signals to be observed can be seen from it. Then only a single distance and angle of direction need be measured. It may also happen that the centre, C, of the Fig 2 signal cannot be seen from D. Thus, if the signal be a solid circular tower, set the Theodolite at D, and turn its telescope so that its line of sight be T comes tangent to the tower at T, T'; measure on these tangents equal distances DE, DF, and direct the telescope to the middle, G, of the line EF. It D will then point to the centre, C; and the distance DC will equal the distance from D to the tower plus the radius obtained by measuring the circumference. If the signa.' be rectangular, measure DE, DF. 27 Take any point G on DE, and on DF set off DHI DF E = G DE. Then is GH parallel to EF, (since DG: DH:: DE: DF) and the telescope directed Gj to its middle, K, will point to the middle of the DE diagonal EF. We shall also have PC = DIK -G. Any such case may be solved by similar methods. For the investigation, see Appendix B. 270 TRIANGULAR SURVEYINGo [PART V The.'Phase" of objects is the effect produced by the sun shining on only one side of them, so that the telescope will be directed from a distant station to the middle of that bright side instead of to the true centre. It is a source of error to be guarded against. Its effect may however be calculated. (37S) Coa r'ctton of the angles. When all the angles of any triangle can be observed, their sum should equal 180.* If not they must be corrected. If all the observations are considered equally accurate, one-third of the difference of their sum from 180~, is to be added to, or subtracted frorl, each of them. But if the angles are the means of unequal numbers of observations, their errors may be considered to be inversely as those numbers, and they may be corrected by this proportion; As the sum of the reciprocals of each of the three numbers of observations Is to the whole error, So is the reciprocal of the number of observations of one of the angles To its correction. Thus if one angle was the mean of three observations, another of four, and the third of ten, and the sum of all the angles was 180~ 3', the first named angle must be diminished by the fourth term of this proportion; I + X + 1: 3':: 1' 27".8. The second angle must in like manner be diminished by 1' 5".9; and the third by 26".3. Their corrected sum will then be 180~. It is still more accurate but laborious, to apportion the total error, or difference from 180~, among the angles inversely as the " Feights," explained in Art. (369). On the U. S. Coast Survey, in six triangles measured in 1.844 by Prof. Bache, the greatest error was six-tenths of a second. (388) Calaulation and platting. The lengths of the sides of the triangles should be calculated with extreme accuracy, in two ways if possible, and by at least two persons. Plane Trigonometry may be used for even large surveys; for, though these sides are really arcs and not straight lines, the difference will be only one If the triangles were very large, they would have to be regarded as spherical, and the sumn of their angles would be more than 180~; but this " spherical ex eess" would be only 1" for a triangle containing 76 square miles, 1 for 4500 squa:e miles, &c.; and may therefore be neglected in all ordinary surveying ope. rations. PART V.] TRIA gULa SURVEYIG. 271 twentieth of a foot in a distance of 11i miles; half a foot in 28 miles; a foot in 341 miles, &c. The platting is most correctly clone by constructing the triangles, as in Art. (90), by means of the calculated lengths of their sides. If the measured angles are platted, the best method is that of chords, Art. (275). If many triangles are successively based on one another, they will be platted most accurately, by referring all their sides to some one meridian line by means of " Rectangular Coordinates," the Method of Art. (6), and platting as in Art. (277.) In the survey of a country, this Meridian would be the true North and South line passing through some well determined point. (389) Base of Verilcation. As mentioned in Art..(380), a side of the last triangle is so located that it can be measured, as was the first base. If the measured and calculated lengths agree, this,proves the accuracy of all the previous work of measurement and calculation, since the whole is a chain of which this is the last link, and any error in any previous part would affect the very last line, except by some improbable compensation. How near the agreement should be, will depend on the nicety desired and attained in the previous operations. Two bases 60 miles distant differed on one great English survey 28 inches; on another one inch; and on a French triangulation extending over 500 miles, the difference was less than two feet. Results of equal or greater accuracy are obtained on the U. S. Coast Survey. (390) tnstertor Sillng ap. The stations whose positions have been Jetermined by the triangulation are so many fixed points, from which more minute surveys may start and interpolate any other points. The Trigonometrical points are like the observed Latitudes and Longitudes which the mariner obtains at every opportunity, so as to take a new departure from them and determine his course in the intervals by the less precise methods of his coirn pass and log. The chief interior points may be obtained by " Secondary Triangulation," and the minor details be then filled in by any of the methods of surveying, with Chain, Compass, or Transit, already explained, or by the Plane Table, described in Part VIII. 272 TRIANGULAR SURVEYING. FPART V. With the Transit, or Theodolite, " Traversing" is the best mode of surveying, the instrument being set at zero, and being then directed from one of the Trigonometrical points to another, which line therefore becomes the " Meridian" of that survey. On reaching this second point, in the course of the survey, and sighting back to the first, the reading should of course be 00 as explained in Art. (377). (391) Radiating Triangulation. This name may be given to a method shown in the figure. Choose Fig. 268. a conspicuous point, 0, nearly in the'~ centre 9f the field or farm to be sur- / \ veyed. Find other points, A, B, C,/ D, &c. such that the signal at O can be G.'\/0. i seen from all of them, and that the tri- l.. angles ABO, BCO, &c, shall be as nearly equilateral as possible. Mea- sure one side, AB for example. At A measure the angles OAB, and OAG; at B B measure the angles OBA and OBC; and so on, around the polygon. The correctness of these measurements may be tested by the sum of the angles, as in Art. (377). It may also be tested by the Trigonometrical principle that the product of the sines of every alternate angle, or the odd numbers in the figure, should equal the product of the sines of the remaining angles, the even numbers in the figure.' The calculations of the unknown sides are readily made. In the triangle ABO, one side and all the angles are given to find AO and BO. In the triangle BCO, BO and all the angles are given to find BC and CO; and so with the rest. Another proof of the accuracy of the work will be given by the calculation of the length of the side AO in the last triangle, agreeing with its length as obtained in the first triangle. (392) Farm Triangulation. A Farm or Field may be surveyed by the previous methods, but the following plan will often be more For the demonstration, see Appendix B. PART v.] TRINGULAR SUVIEYING. 273 convenient. Chooseabase, asXY, within Fig. 269. the field, and from its ends measure the'A angles between it and the direction of / each corner of the field, if the Theodo-.F..'. —ot rid lite or Transit be used, or take the - \ / bearing of each, if the Compass be used. i Consider -first the triangles which have D XY for a base, anc the corners of the field, A, B, C, &c., for vertices. In each of them one side and the angles will be known to find the other sides, XA, XB, &c. Then consider the field as made up of triangles which have their vertices at Xo In each of them two sides and the included angle will be given to find its content, as in Art. ({5). If Y be then taken for the common vertex, a test of the former work will be obtained. The operation will be somewhat simplified by taking for the base line a diagonal of the field, or one of its sides. (393) Inaccesslble Areas. A field or farm may be surveyed, by this "' Fourth Method," without entering Fig, 270. it. Choose a base line XY, from which all the corners of the field can be seen. Take their Bearings, or the angles between the \ Base line and their directions. The dis- \ \ be calculated as in the last article. The l ///I\c /' / figure will then shew in what manner the >' content of the field is the difference between /, - the contents of the triangles, having X (or Y) for a vertex, which lie outside of it, and those which lie partly within the field and partly outside of it. Their contents can be calculated as in the last articles and their difference will be the desired content. If the figure be regarded as generated by the revolution of a line one end of which is at X, while its other end passes along the boundaries of the field, shortening and lengthening accordingly, and if those triangles generated by its movement in one direction be called plus and those generated by the contrary movement be called minus, their algebraic sum will be the content. 18 274 TRIlNGULAR SURVEYING, [PART v (394) Inversion of the Fourth Mlethod. In all the opera. tions which have been explained, the position of a point has been determined, as in Art. (8), by taking the angles, or bearings, of two lines passing from the two ends of a Base line to the unknown point. But the same determination may be effected inversely, by taking from the point the bearings, by compass, of the two ends of the Base line, or of any two known points. The unknown point will then be fixed by'platting from the two known points the opposite bearings, for it will be at the intersection of the lines thus determined. (395) Defects of the Method of Intersection. The determination of a point by the Fourth Method (enunciated in Art. (8), and developed in this Part) founded on the intersection of lines, has the serious defect that the point sighted to will be very indefinitely determined if the lines which fix it meet at a very acute or a very obtuse angle, which the relative positions of the points observed from and to, often render unavoidable. Intersections at right angles should therefore be sought for, so far as other considerations will permit. PART VT. TRILINEAR SURVEYING; By the Fifth Method. (396) TRILINEAR SURVEYING is founded on the Fifth Method of determining the position of a point, by measuring the angles betwen three lines conceived to pass from the required point to three known points, as illustrated in Art. (O1). To fix the place of the point from these data is much more difficult than in the preceding methods, and is known as the " Problem of the three points." It will be here solved Geometrically, Instrumentally and Analytically. (397) Geometrical Solution. Let A, B and C be the known Fig. 271. Tic Im \ \^^-__^J objects observed from S, the angles ASB and BSC being there measured. To fix this point, S, on the plat containing A, B and C, draw lines from A and B, making angles with AB each equal 276 TRILINEIR lRlVEIING [PARI Vi to 90~-ASB. The intersection of these lines at 0 will be the centre of a circle passing through A and B, in the circumference of which the point S will be situated.' Describe this circle. Also, draw lines from B and C, making angles with BC, each equal to 90~-BSC. Their intersection, 0, will be the centre of a circle passing through B and C. The point S will lie somewhere in its circumference) and therefore in its intersection with the former circumference. The point is thus determined. In the figure the observed angles, ASB and BSG, are supposed to have been respectively 400 and 600~ The angles set off are therefore 500 and 30~. The central angles are consequently 80~ and 120~, twice the observed angles. The dotted lines refer to the checks explained in the latter part of this article. When one of the angles is obtuse, set off its difference from 90~ on the opposite side of the line joining the two objects to that on which the point of observation lies. When the angle ABC is equal to the supplement of the sum of the observed angles, the position of the point will be indeterminate for the two centres obtained will coincide, and the circle described from this common centre will pass through the three points, and any point of the circumference will fulfil the conditions of the probleam. A third angle, between one of the three points and a fourth point, should always be observed if possible, and used like the others, to serve as a check. Many tests of the correctness of the position of the point determined may be employed. The simplest one is that the centres of the circles, 0 and 0', should lie in the perpendiculars drawn through the middle points of the lines AB and BC. Another is that the line BS should be bisected perpendicularly by the line 00'o A third check is obtained by drawing at A and C perpendiculars to AB and CB, and producing them to meet BO and BO' produced, * For, the arc AB measures the angle AOB at the centre, which angle- 180~ 2 (900 - ASB)-=2 ASB. Therefore, any angle inscribed in the circumference and measured by the same arc is equal to ASB PART VI.] TRILUNER SJRVEYING. 277 in D and E. The line DE should pass through S; for, the angles BSD and BSE being right angles, the lines DS and SE form one straight line. The figure shews these three checks by its dotted lines. (398) Insttumental Solition. The preceding process is tedious where many stations are to be determined. They can be more readily found by an instrument called a Station-pointer, or Chorograph. It consists of three arms, or straight-edges, turning about a common centre, and capable of being set so as to make with each other any angles desired. This is effected by means of graduated arcs carried on their ends, or by taking off with their points (as with a pair of dividers) the proper distance from a scale of chords (see Art. (274)) constructed to a radius of their length. Being thus set so as to make the two observed angles, the instrument is laid on a map containing the three given points, and is turned about till the three edges pass through these points. Then their centre is at the place of the station, for the three points there subtend on the paper the angles observed in the field. A simple and useful substitute is a piece of transparent paper, cr ground glass, on which three lines may be drawn at the proper angles and moved about on the paper as before. (399) Analytical Solution. The distances of the required point from each of the known points may be obtained analytically. Let AB =; BC a; ABC = B; ASB = S; BSC =S'. Also, make T = 360~- S-S'- B. Let BAS = U; BCS = Ve Then we shall have (as will be shewn in Appendix B) Cot. U= cot. T (+ 1) a sin. S. cos.T V=T-U c. sin. U a. sin. V sin. S or,= sin. 8 SA =e. sin. ABS a.sin. CBS sin. S sin. S' 278 TRILINEAR SURVEYINGe [PART vi Attention must be given to the algebraic signs of the trigonometrical functions. _Example. ASB - 33~ 45'; BSC= 22~ 30'; AB = 600 feet; BC = 400 feet; AC - 800 feet. Required the distances and directions of the point S from each of the stations. In the triangle ABC, the three sides being known, the angle ABC is found to be 104~ 28' 39". The formula then gives the angle BAS = U = 105" 8' 10"; whence BCS is found to be 94" 8' 11"; and SB = 1042.51; SA = 710.193; and SC = 934.291. (400) Miarlitme SBrveying. The chief application of the Trilinear Method is to Mitaritime or Hydrograp1hcal Surveying, the object of which is to fix the positions of the deep and shallow points in harbors, rivers, &c., and thus to discover and record the shoals, rocks, channels and other important features of the locality. To effect this, a series of signals are established on the neighboring shore, any three of which may be represented by our points A, B, C. They are observed to from a boat, by means of a sextant, and the position of the boat is thus fixed as just shewn. The boat is theni rowed in any desired direction, and soundings are taken at regular intervals, till it is found convenient to fix the new position of the boat as before. The precise point where each sounding was taken can now be platted on the map or chart. A repetition of this pro. cess will determine the depths and the places of each point of the bottom. PART VII OBSTAOLES IN ANGULAR SURVEYING. (401) THE obstacles, such as trees, houses, hills, vallies, rivers, &c., which prevent the direct alinement or measurement of any desired course, can be overcome much more easily and precisely (vith any angular instrument than with the chain, methods for using which were explained in Part II, Chapter V. They will however be taken up in the same order.' As before, the given and measured lines are drawn with fine full lines; the visual lines with broken lines; and the lines of the result with heavy full lines. CHAPTER I. PERPENDICULARS AND PARALLELS. (402) Erecting Perpendiculars. To erect a perpendicular to a line at a given point, set the instrument at the given point, and, if it be a Compass, direct its sights on the line, and then turn them till the new Bearing differs 90~ from the original one, as explained in Art. (243). A convenient approximation is to file notches in the Compass-plate, at the 90~ points, and stretch over them a thread, sighting across which will give a perpendicular to the direction of the sights. The Transit or Theodolite being set as above, note the reading of the vernier and then turn it till the new reading is 90~ more or less than the former one. The Demonstrations of the Problems which require them, and from which they can conveniently be separated, will be found in Appei lix B. 280 OBSTACLES IT A NiULAR SiRvTEYXG. [PART VIEI (403) To erect a eperendicular to an inaccessible line, at a given point of it. Let AB be the line Fig. 272. and A the point. Calculate the distance A., -B from A to any point C, and the angle CAB, by the method of Art. (430). Set — =- _ the instrument at C, sight to A, turn an angle = CAB, and measure in the direc- p~ ---- ~ tion thus obtained a distance CP = CA cos. CAB. PA will be the required perpendicular. (404) Letting fall perpendicalaiSo To let fall a perpendis cular to a line from a given point. With the Compass, take the Bearing of the given line and then from the given point run a line, with a Bearing differing 90~ from the original Bearing, till it reaches the given line. With the Transit or Theodolite, set it at any point of the given line, as A, and observe the angle between this Fig. 273. line and a line thence to the given point,,bB P. Then set at P, sight to the former posi- tion of the instrument, and turn a number of degrees equal to what the observed angle at A wanted of 90~. The instrument will then point in the direction of the required perpendicular PB. (405) To let fall a perpendicular to a line from an inaccessible point. Let AB be the line and P the Fig. 274. point. Measure the angles PAB, and P PBA. Measure AB. The angles APC and BPC are known, being the complements of the angles measured. Then is A C C-= AB. tn tan. APO tan. APC + tan. BPC' CHAP. i.] Perpendiculars and Parallels,. 281 (406) To let fall a perpendicular to an inaccessible line from a given point. Let C be the point and Fig. 275. AB the line. Calculate the angle CAB. __Eby the method of Art. (430). Set the instrument at C, sight to A, and turn an = = angle = 90- CAB. It will then point in the direction of the required perpendicular CE. (407) iRuning Parallels. To trace a line through a given point parallel to a given line. With the Compass, take the Bearing of the given line, and then, from the given point, run a line with the same Bearing. With the Transit or Theodolite, set it at any convenient p)oint of the given line, as A, direct Fig. 276. it on this line, and note the read- A, ing. Then turn the vernier till the cross-hairs bisect the given _ point, P. Take the instrument to p Q this point and sight back to the former station, by the lower motion, without changing the reading. Then move the vernier till the reading is either the same as it was when the telescope was directed on the given line, or is 1800 different. It will then be directed (forward or backward) on PQ, a parallel to AB, since equal angles have been measured at A and P. The manner of reading them is similar to the method of "Traversing," Art. (373). (408) To trace a line through a given point parallel to an %iaceessible line. Let C be the given Fig. 277. point, and AB the inaccessible line. A,- - Find the angle CAB, as in Art. (430). ~ - Sct the instrument at C, direct it to A, -_ and then turn it so as to make an angle c MI with CA equal to tte supplement of the angle CAB. It will then point in a direction, CE, parallel to AB. 282 OBSTACLES IN ANGULAR SURVEYING. [PART VIL CHAPTER IL OBSTACLES TO ALINEMENT. A. To PROLONG A LINE. (409) The instrument being set at the farther end of a line, and directed back to its beginning, the sights of the Conpass, if that be used, will at once give the forward direction of the line. They serve the purpose of the rods described in Art. (169). A distant point being thus obtained, the Compass is taken to it and the process repeated. The use of the Transit or Theodolite, for this purpose, was fully explained in Art. (376). (410) By perpemdiculars. When a tree, or house, obstructing the line, is met with, place the instru- Fig. 278. ment at a point B of the line, and set A bly off there a perpendicular, to C; set off ~~ ~- ~'~ another at C to D, a third at D to E, l D making DE = B3, and a fourth at E, which last will be in the direction of AB prolonged. If perpendiculars cannot be conveniently used, let BC and DE make any equal angles with the line AB, so as to make CD parallel to it. (411) By an equilateral triangle. Fig. 279. At B, turn aside from the line at an. B _ I3 angle of 60~, and measure some con- venient distance BC. At C, turn 60~ in the contrary direction, and mea- c sure a distance CD = BC. Then will D be a point in the line AB prolonged. At D, turn 600 from CD prolonged, and the new direction will be in the line of AB prolonged. This method re quires the measurement of one angle less than the preceding. CHAP. II.] Obstacles to Alineient. 283 (412) By triangulation. Let Fig. 80. AB be the line to be prolonged. A. __ o Choose some station C, whence can be seen A, B, and a point \,' beyond the obstacle. Measure.\. AB and the angles A and B, of the triangle ABC, and thence calculate the side AC. Set the instrument at C, and measure the angle ACD, CD being any line which will clear the obstacle. Let E be the desired point in the lines AB and CD prolonged. Then in the triangle ACE, will be known the side AC and its including angles, whence CE can be calculated. Measure the resulting distance on the ground, and its extremity will be the desired point E. Set the instrument at E, sight to C, and turn an angle equal to the supplement of the angle AEC, and you will have the direction, EF, of AB prolonged (413) when the Hne to be prolonged is inacessible. In this case, before the preceding method can be applied, it will be necessary to determine the lengths of the lines AB and AC, and the angle A, by the method given in Art. (430). (414) To prolong a line with only an angular istrument. This may be done when no means of measuring any distance can be obtained. Let AB be the line Fig. 281. to be prolonged. Set the in- C strument at B and deflect angles of 450 in the directions C A. and D. Set at some point, C, on one of these lines and deflect from CB 45, and mark the point D where this direction intersects the direction BD. Also, at C, deflect 90~ from CB. Then, at D, deflect 90~ from DB. The intersections of these last directions will fix a point E. At E deflect 135~ from EC or ED, and a line EF, in the direction of AB will be obtained and may be continued.* This ingenious contrivance is due to a former student, Mr. R. Hood, in whose practice, while running an air line for' a railroad, the necessity occurred. 284 OBSTACLES IN ANGULAR SITUVEYNLG. [PART VII. B. To INTERPOLATE POINTS IN A LINE. (415) The instrument being set at one end of a line and directed to the other, intermediate points can be found as in Art. (177), &c. If a valley intervenes, the sights of the Compass, (if the Compass-plate be very carefully kept level cross-ways), or the tele. scope of the Transit or Theodolite, answer as substitutes for the plumb-line of Art. (179). (.11E) By a random line. When a wood, hill, or other obstacle, prevents one end of the line, Z, Fig. 282. from being seen from the other, A, run a random line AB with the Compass or Ar Zt- / Transit, &c., as nearly in the desired B direction as can be guessed, till you arrive opposite the point Z. Measure the error, BZ, at right angles to AB, as an offset. Multiply this error by 57-;-, and divide the product by the distance AB. The quotient will be the degrees and decimal parts of a degree, contained in the angle BAZ. Add or subtract this angle to or from the Bearing or reading with which AB was run, according to the side on which the error was, and start from A, with this corrected Bearing or reading, to run another line, which will come out at Z, if no error has been committed.i Example. A random line was run, by compass, with a Bearing of S. 80~ E. At 20 chains' distance a point was reached opposite to the desired point, and 10 links distant from it on its right. Required the correct Bearing. Ans. By the rule, < 570 = 00.2865 = 17'. The cor2000 rect Bearing is therefore S. 80~ 17' E. If the Transit had been used, its reading would have been changed for the new line by the same 17'. A simple diagram of the case will at once shew whether the correction is to be added to the original Bearing or angle, or subtracted from it. 4 This rule is substantially identical with that of Art. (319), where its reason in given. CHAP. II.] Obstacles to Alinement. 285 If Trigonometrical Tables are at hand, the correction will be more precisely obtained from this equation; Tan. BAZ B_. AB' BZ 10 In this example, -005 = tan. 17'. AB 2000 The 57~.3 rule, as it is sometimes called, may be variously modified. Thus, multiply the error by 86", and divide by one and a half times the distance; or, to get the correction in minutes, multiply by 3438 and divide by the distance; or, if the error is given in feet and the distance in four-rod chains, multiply the former by 52 and divide by the distance, to get the correction in minutes. The correct line may be run with the Bearing of the random line, by turning the vernier for the correction, as in Art. (312). (1S7) By Latitudes and Departures. When Fi 283. a single line, such as AB, cannot be run so as to 4 come opposite to the given point Z, proceed thus, I Z with the Compass. Run any number of zig-zag courses, AB, BC, CD, DZ, in any convenient J (,'D direction, so as at last to arrive at the desired point. Calculate the Latitude and Departure of each of i c these courses and take their algebraic sums. The, / Li sum of the Latitudes will be equal to AX, and that }, B of the Departures to XZ. Then is Tan. ZAX; ^ i. e. the algebraic sum of the Departures divided by the algebraic sum of the Latitudes is equal to the tangent of the Bearing.' (118) When the Transit or Theodolite is used, any line may be taken as a Meridian, i. e. as the line to which the following lines are referied; as in "Traversing," Art. (373), page 254, all the successive lines were referred to the first line. In the figure, on the next page, the same lines as in the preceding figure are repre~ The length of the line AZ can also be at once obtained since it is equal to the square root of the sum of the squares of AX and XZ; or to the Latitude divided by the cosine of the Bearing. 286 OBSTACLES IN ANGULAR $URVEYING. [PART VIi, sented, but they are referred to the first course, Fig. 284. AB, instead of to the Magnetic Meridian as Z before, and their Latitudes are measured along its produced line, and its Departures perpen- F dicular to it. As before, a right-angled triangle ~/[^'> \ will be formed, and the angle ZAY will be Y i the angle at A between the first line AB and the desired line AZ. This method of operation has many usefuli applications, such as in obtaining data for running Railroad Curves, &c., and the student should master it thoroughly. The desired angle (and at the same time the distance) can be obtained, approximately, in this and the preceding case, by finding in a Traverse Table, the final Latitude and Departure of the desired line (or a Latitude and Departure having the same ratio) and the Bearing and Distance corresponding to these will be the angle and distance desired. (419) By similar triangles. Fig 285. Through A measure any line CD. - E Take a point E, on the line CB, | beyond the obstacle, and fromit A set off a parallel to CD, to some c point, F, in the line DB. Measure D EF, CD, and CA. Then this proporcion, CD: CA:: EF: EG, will give the distance EG, from E to a point in the line AB. So for other points. (420) By triangulation. When Fig. 286. obstacles prevent the preceding me-;; B thods being used, if a point, C, can be c/ found, from which A and B are accessible, measure the distances CA, CB, c and the angle ACB, and thence calculate the angle CAB. Then observe any angle ACD, beyond the obstacle. In the triangle ACD; a side and its including angles are known, to find CD. Mea sure it, and a point, D, in the desired line, will be obtained. cHAp. mI.] Obstacles to Ieasurement. 287 CHAPTER III. OBSTAILES TO lIEASUREMENTO A. WHEN BOTH ENDS OF THE LINE ARE ACCESSIBLE, (421) The methods given in the preceding Chapter for prolonging a line and for interpolating points in it, will generally give the length of the line by the same operation. Thus, in Fig. 278, the inaccessible distance BE is equal to CD; in Fig. 279, BD = BC = CD; in Fig. 280, the distance BE can be calculated from the same data as CE; in Fig. 282, AZ = V(AB2 + BZ2); in Fig. 283, AZ = V(AX2 + XZ2); in Fig. 284, AZ -- (AY2 + YZ2) in Fig. 285 AG GB (C- EG); in Fig. 286, the triangle ACD will give the distance AD. The method of Latitudes and Departures, Arts. (417) and (418), is very generally applicable. So is the following. (422) By triangulation. Let AB Fig. 287. be the inaccessible distance. From A- N. any point, C, from which both A and - \/x B are accessible, measure CA, CB, and the angle ACB. Then in the triangle ABC two sides and the included angle are known to find the side AB. If all the angles can be measured, they-may be corrected, as in Art. (387).* (423) A broken Base, When the angle C is very obtuse, the preceding problem may be modified as follows. Naming the lines as is usual in Trigonometry, by small letters corresponding to the " In this figure, and the followingones,the angular point enclosed in a circle indicates the place at which the instrument is set. 288 OBSTACLES IN AINGULR SURTEYING. [PART VIi capital letters at the angles to which they are opposite, and letting K = the number of minutes in the supplement of the angle C, we Fig. 288. A-11 shall have AB c = a + b 0.000000042308 x.ab a b+~ b This formula is chiefly used in the case of what is called in Triangular Surveying "A broken Base;" such as above; AC and CB being measured and forming very nearly a straight line, and the length of AB being required. Log. 0.000000042308 = 2.6264222 — 10. (424) By angles to nown points, The length of a line, both ends of which are accessible, may also be determined by angles measured at its extremities between it and the directions of two or more known points. But as the methods of calculation involve subsequent problems, they will be postponed to Articles (435), (436) and (437). B. WHEN ONE END OF THE LINE IS INACCESSIBLE. (425) By perpendicfars. Many of the methods given for the chain, in Part II, Chapter V, may be still more advantageously employed with angular instruments, which can so much more easily and precisely set off the Perpendiculars required in Articles (191), (192), (19m), &c. (426) By eqal angles. Let AB Fig. 289. be the inaccessible line. At A set off -D A ___.... AC, perpendicular to AB, and as'. )) nearly equal to it, by estimation, as am the ground will permit. At C: mea- " l sure the angle ACB, and turn the C sights, or vernier, till ACD = ACB. Find the point, D, at the intersections of the lines CD and BA produced. Then is AD -- AB. CHAP. iii.] Obstacles to Ileasurement 289 (427) By traagglatlsono Measure a distance Fig. 290. AC, about equal to AB. Measure the angles at A j1 A and C. Then in the triangle ABC, two angles/ ) and the included side are known, to find another I side, AB AC sin. ACB I i side, AB == ~ sin. ABO' When the compass is used, the angles between ) the lines will be deduced from their respective Bearings, by the principles of Art. (21B). If the angle at A is 90~ AB = AC. tang. ACB. If A = 90~, and C = 45~, then AC = AB; but this position could not easily be obtained, except by the use of the Sextant, a reflecting instrument, not described in this volume. (428) When one point cannot be seen from the other,Choose two points, C and D, in the line Fig. 291. of A, and such that from C, A and B can c // be seen, and from, A and B. Measure. AC, AD, and the angles C and D. Then, A I- B in the triangle BCD, are known two an- / ( W gles and the included side, to find CB. Then, in the triangle ABC, are known two sides and the included angle, to find the third side, AB. (429) To Hid the distance from a given point to an inacces sible ilie, In Fig. 275, Art. (406), the required distance is CE. The operations therein directed give the line CA and the angle CAB, or CAE. The recuired distance CE = CA. sin. CAE. 19 290 OBSTACLES IN ANGLLAR SURVTING. [PART VII C. WHEN BOTH ENDS OF THE LINE ARE INACCESSIBLE. (430) General 1RIethod. Let Fig. 292 AB be the inaccessible line.;,B Measure any convenient distance, / CD, and the angles ACD, BCD,.- / \ ADC, BDC. Then, in the triangle CDA, Ad\ / two angles and the included side L are given, to find CA. In the,-~~,, triangle CDB, two angles and the /. included side are given, to find i_/. \ CB. Then, in the triangle ABC, two sides and the included angle c0 __D are given, to find AB. The work may be verified by taking another set of triangles, and finding AB from the triangle ABD instead of ABC. The following formulas will however give the desired distance with less labor. sin. ADO. sin. CBD Find an angle K, such that tang. K =sin. AD. sin. BD sin. CAD. sin. BDC Then find the difference of the unknown angles in the triangle CAB from the formula Tang. 1 (CAB —ABC) = tang. (45~- K). cot. I ACB. Then is CAB = (CAB -ABC) + ( (CAB + ABC). sin. BD. sin. ACB Finally AB _ CD sin. CBD. sin. CAB Example. Let CD = 7106.25 feet; ACD = 95~ 17' 20"; BCD = 61~ 41' 50"; ADC = 39 38' 40"; BDC = 780 35' 10"; required AB. The figure is constructed with these data on a scale of 5000 feet to 1 inch = 1:60000. By the above formulas, K is found to be 30~ 26' 5"; CAB = 1130 55' 37"; and lastly AB = 6598.32. Both the methods may be used as mutual checks in any im portant case. CHAP. III.] Obstacles to leasurement, 291 Fig. 293. If the lines AB and CD crossed f... each other, as in Fig. 293, instead of \\ -\ being situated as in the preceding \ _\ _ —— \~ figure, the same method of calcula- \ tion would apply. (431) Problem, To measure an inaccessible distance, AB when a point, C, in its line can be obtained. Set the instrument at a point, D, from which A, B Fi 294. and C can be seen, and measure C \ \\} A the angles CDA and AIDB. \/\\ l" ~ ~" Measure also the line DC and the angle C. Then in the triangle ACD two angles and the \ \ included side are given to find AD. In the triangle DAB, the D angle DAB is known, (being equal to ACD - CDA), and AD having been found, we again have two angles and the included side to find AB. (432) Probles 2 To measure an inaccessible distance, AB, when only one point, C, can be found from which both ends of the line can be seen. Consider CA Fig. 295. and CB as distances to be deter- A -—. —---------- B J/ \, t mined, having one end accessible. - _ / Determine them, as in Art. (427), ---— ___ by choosing a point D, from which = C and A are visble, and a point E: W, from which C and B are visible.,~'~ ~c 4~ At C observe the angles DA,ACB and BCE. Mleasure the distances CD and CE. Observe the angles ADC and BEC. Then in the triangle ADC, two angles and the included side are given, to find CA; and the same in the triangle CBE, to find CB. Lastly, in the triangle ACB.wo sices and the included angle are known, to find AB. 292 OBSTACLES IN ANGULAR SURVEYINU. [PART VII (133) Problems To measure an inaccessible distance, AB: when no point can be found from which the two ends can be seen. Let C be a point from which A is Fig. 296. visible, and D a point from which __..... B is visible, and also C. Measure - CD. Find the distances CA and / \/ DB, as in the preceding problem; / \:, W i. e. choose a point E, from which A C and C are visible, and another point, F, from which D and B are visible. Measure EC and DF. Observe the angles AEC, ECA, BDF and DFB; and at the same time the angles ACD and CDB, for the subsequent work. Then CA and DB will be found, as were CA and CB in the last problem. Then in the triangle CDB, two sides and the included angle are known to find CB and the angle DCB; and, lastly, in the triangle. ACB, two sides and the included angle (the difference of ACD and DCB) to find AB. (431) Probleim To interpolate a Base, Four inaccessible objects, A, B, C, D, being in a right Fig. 297. line, and visiblefrom onlyone point, A __ B\ ( C.X E, it is required to determine the dis- \\- / tance between the middle points, B ~ and C, the exterior distances, AB ~and CD, being known. ", Let AB == a, CD b, BC - x; AEB - P, AEC =Q, AED = R. Calculate an auxiliary angle, K, such that tang.2 K - ^4ab sin. Q sin. ( — P) tang.~- K -- (a - b)" sin. P. sin. (R- Q) a+b a-b Thlen is x =- - fi a b 2 2. cos. K' Of the two values of x, the positive one is alone to be taken. This problem is used in Triangular Surveying wvlen a portion of a Base line passes over water, &c. CHAP. III.] Obstacles to Mleasurement. 293 (435) Problem, Given the angles observed, at the ends of a line which cannot be measured, between it and the ends of a line of known length but inaccessible, required the length of the formeer line. This Problem is the converse of that given in Arto (430). Its figure, 292, may represent the case, if the distance AB be regarded as known and. CD as that to be found. Use the first and second formulas as before, and invert the last formula, obtaining CD = AB sin. CBD. sn. CAB sin. BDC sin. ACB' This problem may also be solved, indirectly, by assuming any length for CD, and thence calculating as in the first part of Art. (430), the length of AB on this hypothesis. The imaginary figure thus calculated is similar to the true one; and the true length of CD will be given by this proportion; calculated length of AB: true length of AB:: assumed length of CD: true length of CD. The length of CD can also be obtained Fig. 29,. graphically. Take a line of any length,' as C'D', and from C' and D' lay off angles equal to those observed at C and D, and \ / thus fix points A, B'. Produce AB' till it ^< \ equals the given distance AB, on any de- Ad/ X a I sired scale. From B draw a parallel to -. \ B'D', meeting AD' produced in D; and -. from D draw a parallel to D'C' meeting AC' produced in C. Then CD will be the squired distance to the same scale as AB,(418) ProTblem Three points, A, B, C, being gwen by their distances from each other, and two other points, P and Q, being so situated that from each of them two of the three points can be seen and the angles APQ, BPQ, CQP, BQP, be measured, it is required to determine the positions of P and Q. * See Article (458) for a solution of this problem by the Plane-Table. 294 OBSTACLES IN ANGULAR SURVEING. [PART vII. CONSTRUCTION. Begin, Fig. 299. as in Art. (3l7), by describ- -" r ^_ ing a circle passing through / A and B, and having the cen-' tral angle subtended by AB, -- - -. equal to twice the given an- ---- gle APB, and thus contain- \ / / ing that angle. The point ". ^- P will lie somewhere in its -- _ circumference. Describe another circle passing through B and C, and having a central angle subtended by BC equal to twice the given angle BQC. The point Q will lie somewhere in its circumference. From A draw a line making with AB an angle = BPQ, and meeting at X the circle first drawn. From C draw a line making with CB an angle - BQP, and meeting the second circle in Y. Join XY and produce it till it cuts the circles in points P and Q, which will be those required; since BPX - BAX BPQ; and BQY = BCY=BQ P. CALCULATION. In the triangle ABC, the sides being given, the angle ABC is known. In the triangle ABX, a side and all the angles are known, to find BX. In the triangle CBY, BY is similarly found. By subtracting the angle ABC from the sum of the angles ABX and CBY, the angle XBY can be obtained. Then in the triangle XBY, the sides BX, BY, and the included angle are given to find the other angles. Then in the triangle' BPX are known all the angles and the side BX to find BP. In the triangle BQy, BQ is found in like manner. Finally, in the triangle BPQ, PQ can then be found. If desired, we can also obtain AP in the triangle APB; and CQ in the triangle CBQ. (437) ProbFlem Four points, A, B, C, D, being given in position, ~by their mutual distances and directions, and two other points, P and Q, being so situated that from each of them two of the four points ean be seen and the angles APB, APQ, PQC and PQD measured, it is required to determine the position of P and Q. CAxPo III.] Obstacles to leasurement. 29s Fig. 300. P ^Q,' /^/ ",^'......' \ \- ",- ^^ / / /'"i \'\,.;"/-^;:. \ \,,, CONSTRUCTION. Begin as in the last artiche, by describing on AB the segment of a circle to contain an angle equal to APB. From B draw a chord BE, making an angle with BA equal to the supplement of the angle APQ. On CD describe another segment to contain an angle equal to CQD. From C draw a chord CF, making an angle with CD equal to the supplement of the angle DQP. Draw the line EF, and it will cut the two circles in the required points P and Q.$ CALCULATION. To obtain PQ - EF - EP - QF, we proceed to find those three lines thus. In the triangle ABE, we know the side AB, the angle ABE, and the angle AEB = APB; whence to find EB. In the same way, the triangle CFD gives FC, In the triangle EBO are known EB and BC, and the angle 1BC = ABC-G ABE; whence EC and the angle ECB are found. In the triangle ECF are known EC, FC, and the angle EGF = BGD -ECB —FCD; whence we find EF, and the angles CEF and CFE. In the triangle BEP, we have EB, the angle BEP ==BEC + CEP, and the angle BPE = BPA + APE; to find EP and PB. In the triangle QOF, we have CF, and the angles CQF and CFQ, to find QC and QF. Then we know PQ = EF —EP - QF. For, the angle APQ in the figure equals the measured angle APQ, because the supplement of the former, EPA, equals the supplement of the latter, since it is measured by the same arc as the angle ABE, equal to that supplement by construction. So too with the angle DQP. 296 OBSTACLES IN AGULAR SURVEYING. [PART vnI The other distances, if desired, can be easily found from the above data, some of the calculations, not needed for PQ, being made with reference to them. In the triangle ABP, we know AB, BP, and the angle BAP, to find the angle A anand AP. In the triangle QDC we know QC, CD, and the angle CQD, to find the angle QCD and QD. In the triangle PBC, we know PB, BC, and the angle PBC = ABC - ABP, to find PC. Lastly, in the t riangle QCB, we know QC, CB, and the angle QCG = DOBD-DQ, to find QB. The solution of this problem includes the two preceding; for, let the line BC be reduced to a point so that its two ends come together and the three lines become two, and we have the problem of Art. (436); and let the line AB be reduced to a point, B, and CD to a point, C, and we have but one line, and the problem becomes that of Art. (435). In these three problems, if the two stations lie in a right line with one of tho given points, the problem is indeterminate. (438) Problem of the eight pour pots o ints, A, B, C, D, are inaccessible, but visible Fig. 301. from four other points, E, F, G, H; it is required to D find the respective distances of these eight points; the _ only data being the obser- - vation, from each of the, -~points of the second sys- tem, of the angles under 0 which are seen the points ) oJ the first system. 0 This problem can be solved, but the great length and complication of the investigation and resulting formulas render it more a matter of curiosity than of utility. It may be found in Puissant's Topographie," page 55; Lefevre's' Trigonometrie9, p. 90, and Lefevre's "Arpentayge" No. 387. CHAP. IV.] To Supply Omissions 297 CHAPTER IV. TO SUPPLY OMI8SS0N3 (439) Any two omissions in a closed survey, whether of the direction or of the length, or of both, of one or more of the sides bounding the area surveyed, can always be supplied by a suitable application of the principle of Latitudes and Departures, as was stated in Art. (283); although this means should be resorted to only in cases of absolute necessity, since any omission renders it impossible to "Test the survey," as directed in Art. (282). In the following articles the survey will be considered to have been made with the Compass. All the rules will however apply to a Transit or Theodolite survey, the angles being referred to any line as a meridian, as in " Traversing." To save unnecessary labor, the examples in the varlous cases now to be examined, will all be taken Fig. 302. from the same survey, a plat of which A - D is given in the margin on the scale of 40 chains to 1 inch (1:31,680), and the Field-notes of which, with the Latitudes and Departures carried out to five decimal places, are given on the following page.* " The teacher can make any number of examples for his own use by taking a tolerably accurate survey, striking out the bearing and distance of any ene course, and calculating it precisely as in Case 1, given below. He can then omit any two quantities at will, to be supplied by the student by means of the rules aow to be given. 298 OBSTACLES IN ANGULAR SURVEYING. [PART VII. STA. BEARING. B lDIST". _ LATITUDES. DEPARTURE. IN LINKS. N.. W A l North. 1284 1284.00000 0 0 B t N. 32~ E. 1782 1511.22171 944.31619 C i N. 800 E. 2400 416.75568 2363.53872 D S. 48B E. 2700 1806.652'2i 2006.49096 E S. 18~ W. 2860 2720.02159 883.78862 J N.. 730 8' 21" W. 4621 1314.69682 4430.55725 - 4526.67421 4526.67421 5314.34587 5314.34587 CASE 1. When the length and the.Bearing of any one side are wanting. (440) FinI the Latitudes and the Departures of the remaining sides. The difference of the North and South Latitudes of these lines, is the Latitude of the omitted line, and the difference of their Departures is its Departure.. This Latitude and Departure are two sides of a right angled triangle of which the omitted line is the hypothenuse. Its length is therefore equal to the square root of the sum of their squares, and the quotient of the Departure divided by the Latitude is the tangent of its Bearing; as in Art. (417). In the above survey, suppose the course from F to A to have been omitted or lost. The difference of the Latitudes of the remaining courses will be found to be 1314.69682, and the difference of the Departures to be 4430.55725. The square root of the sum of their squares is 4621.5; and the quotient of the Departure divided by the Latitude is the tangent of 73~ 28' 21". The deficiencies were in North Latitude and West Departure; and the omitted course is therefore N. 73~ 28' 21" W., 4621.5 CASE 2. When the length of one side and the Bearing of another are wantingc, (441) Wlien tAe defcient sies adjoin each other Find, as in Case I, the length and Bearing of the line joining the ends of the remaining courses. This line and the deficient lines will form a triangle, in which two sides will be known, and the angle between the calculated side and the side whose Bearing is given can be found by Art. (243). The parts wanting can then be obtained by the common rules of Trigonometry. CHAP. iv.] To Supply Omissions, 299 In the figure, let the length of EF, Fig. 3 and the Bearing of FA be the omitted D parts. The difference of the sums of f the N. and S. Latitudes, and the E. and W. Departures of the complete B - courses from A to E, are respectively - 1405.32477 North Latitude, and a 314:.245g87 East Departure. The course, EA, corresponding to this de- fciency we find, by proceeding as in case 1, to be S. 75 11t' 15 V., 5497.026. The angle AEF is therefore ==75 11' 15" 18~ 57" 11' 15". Then in the triangle AEF are given the sides AE, AF, and the angle AEF to find the remaining parts; viz. the angle AFE-91~ 28' 21", whence the Bearing of FA -91~ 28' 21" - 18~ N. 730 28' 21" W.; and the side EF c 2860. (442), When the &le:eint sides are separated from each others A modification of the preceding method will still apply. In this figure let the omissions be the Bearing Fig. 304. of FA and the length of CD. Imagine:) the courses to change places without " / changing Bearings or lengths, so as to bring the deficient lines next to each B E other, by transferring CD to AG, AB to GH, and BC to HD. This will not A affect their Latitudes or Departures. Join GF. Then in the figure DEFGH, F the Latitudes and Departures of all the sides but FG are known, whence its length and Bearing can be found as in Case 1. Then the triangle AGF may be treated like the triangle AEF in the last article, to obtain the length of AG = CD, and the Bear* mg of FA. (441) Otheruise, by changiny the Meridian. Imagine the field to turn around, till the side of which the distance is unknown, becomes the Meridian, i. e. comes to be due North and South' 300 OBSTACLES IN ANULAR SURVEYINg. [PART vII. all the other sides retaining their relative positions, and continuing to make the same angles with each other. Change their Bearings, accordingly, as directed in Art. (244). Find the Latitudes and Departures of the sides in their new positions. Since the side whose length was unknown has been made the Meridian, it has no Departure, whatever may be its unknown length; and the difference of the columns of Departure will therefore be the Departure of the bide whose Bearing is unknown. The length of this side is given. It is the hypothenuse of a right angled triangle, of which the Departure is one side. Hence the other side, which is the Latitude, can be at once found; and also the unknown Bearing. Put this Latitude in the Table in the blank where it belongs. Then add up the columns of Latitude, and the difference of their sums will be the unknown length of the side which had been made a Meridian.* Let the omitted quantities be, as in the last article, the length of CD and the IBearingo of FA. STA. OLD BEARING. NEW BEARING.. A~ ~~ North. NMake CD the Meridian. The changA North. N. 80~ W. AB No. 3 E. N. 480 W. ed Bearings will then be found by B N. 3~ E. N. 48~ W. C N. 800 E. NorAlh. Art. (214) to be as in the margin. D S. 48~ E. N. 52~ E. To aid the imagination, turn the E S. 1S~ W. S. 62~ E. book around till CD points up and _____~~~_I_______- edown, as North lines are usually placed on a map. Then obtain the Latitudes of the courses with their new Bearings and old distances, and proceed as has been directed. CASE 3. When the lengths of two sides are wantinc. (4i4) When the declient sides ajoin each other~ Find the Latitudes and Departures of the other courses, and then, by Case I, find the length and Bearing of the line joining the extremities of the deficient courses. Then, in the triangle thus formed, are known one side and all the angles (deduced from the Bearings) to ind the lengths of the other two sides. " This conception of thus changing the Bearings is stated to be due to Pro; Robert Patterson, of Philadelphia, by whom it was communicated to Mr. John iunmmere, and published by him, in 1814, in his "Treatise on Surveying" CHAP. IV.] To Supply OmissIonso 301 Thus, in Fig. 303, page 299, let EF and FA be the sides whose lengths are unknown. EA is then to be calculated, and its length will be found, as in Art. (4i1), to be 5497.026, and its bearing S. 75" 11' 15" W., whence the angle AEF 75~ 11' 15~"-18" = 57~ 11' 15"; AFE = 18~ + 730 28' 21" = 91~ 28' 21"; and EAF = 31~ 20' 24"; whence can be obtained EF =2860 and FA = 4621.5. (445) When the defcient sides are separated from each other Let the lengths of BC and DE be those Fig. 305. omitted. Again imagine the courses D to change places, so as to bring the /\ deficient lines together, DE being,/ \ transferred to CG, and CD to GE. By — Join BG. Then in the figure ABGEFA, are known the Latitudes and Departures of all the courses except BG, whence its length and Bearing F can be found as in Case 1. Then in the triangle BCG, the angle CBG can be found from the Bearings of GB and BG, and the angle CGB from the Bearings of BG and GC. Then all the angles of the triangle are known and one side, BG, whence to find the required sides, BC - 1782, and CG = DE == 2700. (416) Otherwise, by changing the Mieridian. As in Art. (443), imagine the field to turn around, till one of the sides whose length is wanting, becomes a Meridian or due North and South. Change all the Bearings correspondingly. Find the Latitudes and Departures of the changed courses. The difference of the columns of Departure will be the Departure of the second course of unknown length, since the course made Meridian has now no Departure. The new Bearing of this second course being given, in the right angled triangle formed by this course (as an hypothenuse) and its Departure and Latitude, we know one side, the Departure, and the acute angles, which are the Bearing and its complement. The length of the course is then readily calculated; and also its Lati. tude. This Latitude being inserted in its proper place, the iffer 302 OBSTACLES IN ANGTLAR ST RVEYIN,. [PART VIL, ence of the columns of Latitude will be the length of that wanting side which had been made a Meridian. Thus, let the lengths of BC and DE be wanting, as in the preOLD ~BE G. - ~ ceding example. Make BC STA. OLD BEARING. NEW BEAlRING. ~.__, __ NEW BE_ _ _RI__. a Meridian. The other BearA North. N.s 32~0 V. ^ins are then changed as in B N. 32~ E. North. C N:. 80~ E. N;. 48~ E. the margin. Calculate new D S 48~ E. S. 80~ E. Latitudes and Departures. E S. 0~ W S. S 140 E. The difference of the DeparF IN. 73~ 28' 21" W.S. 74~ 31' 39 W. tures will be the Departure tures will be the Departure of DE, since BC, being a Meridian, has no Departre. Hence the length and Latitude of DE are readily obtained. This Latitude being put in the table, and the columns of Latitude then added un, their difference will be the length of BC. CASE 4. When the Bearings of two sides are wanting. (447) Wien thi de cleint sides adjoin each othier, Find the Latitudes and Departures of the other sides, and then, as in Case 1, find the length and bearing of the line joining the extremities of the deficient sides. Then in the triangle thus formed we have the three sides to find the angles and thence the Bearings. (448) Whlien the declient sides are separated from each other Change the places of the sides so as to bring the deficient ones next to each other. Thus, in the Fig. 306. figure, supposing the Bearings of CD, ID and EF to be wanting, transfer EF to / / DG, and DE to GF. Then calculate, as in Case 1, the length and Bearing \ of the line joining the extremities of / / the deficient sides, CG in the figure. A' This line and the deficient sides form a triangle in which the three sides are V given to determine the angles and thence the required Bearings. * The fullest investigation of this subject, developing many curious points, will be found in Mascheroni's "Probl&ees de Ggeometrie pour les Arpenteurs," and LhU1. illier's "Polygonoomtrie." The method of Arts. (442), (445), and (448) is new. ]PART VIIL. PLANE TABLE SURVEYING. (449) Tnrl Plane Table is in substance merely a drawing board fixed on a tripod, so that lines may be drawn on it by a ruler placed so as to point to any object in sight. All its parts are mere additions to render this operation more convenient and precise.* Such an arrangement may be applied to any kind of " Angular Surveying"; such as the Third Method, 6 Polar Surveying," in its two modifications of Radiation and Progreession, (characterized in Art. (220)), and the Fourth Method, by Intersections. Each of these will be successively explained. The instrument is very convenient for filling in the details of a survey, when the principal points have been determined by the more precise method of " Triangular Surveying," and can then be platted on the paper in advance. It has the great advantage of dispensing with all notes and records of the measurements, since they are platted as they are made. It thus saves time and lessens mistakes, but is wanting in precision. (450) The Tableo It is usually a rectangular board of well seasoned pine, about 20 inches wide and 30 long. The paper to be drawn upon may be attached to it by drawing-pins, or by clamping plates fixed on its sides for that purpose, or by springs pressed upon it, or it may be held between rollers at opposite sides of the table. Tinted paper is less dazzling in the sun. Cugnot's joint, described on page 134, is the best for connecting it with its tripod, though a pair of parallel plates, like those of the Theodolite, are often used. A detached level is placed on the board to test its horizontality; though a smooth ball, as a marble, will answer the same purpose approximately. * The Plane Table is not a Goniometer, or Angle-measure, like the Compass, Transit, &.; but a Gonigraoph, or Angle-drawer, 304 PLANE TABLE SURTEYINGE [PART VIIL A pair of sights, like those of the compass, are sometimes placed under the board, serving, like a " Watch Telescope," (Art. (339), to detect any movement of the instrument. To find what point on the lower side of the board is exactly under a point on the upper side, so that by suspending a plumb-line from the former the latter may be exactly over any desired point of ground, a large pair of " callipers," or dividers with curved legs, may be used, one of their points being placed on the upper point of the board, and their other point then determining the corresponding under point; or a frame forming three sides of a rectangle, like a slate frame, may be placed so that one end of one side of it touches the upper point, and the end of the corresponding side is under the table precisely below the given point, so that from this end a plumb-line can be dropped. A compass is sometimes attached to the table, or a detached compass, consisting of a needle in a narrow box, (called a Declinator), is placed upon it, as desired. The edges of the table are sometimes divided into degrees, like the " Drawing board Protractor,' Art. (273). It then becomes a sort of Goniometer, like that of Art. (213). (451) The Alidade. The ruler has a fiducial or feather edge, which may be divided into inches, tenths, &c. At each end it carries a sight like those of the compass. Two needles would be tolerable substitutes. The sights project beyond its edge so that their centre lines shall be precisely in the same vertical plane as this edge, in order that the lines drawn by it may correspond to the lines sighted on by them. To test this, fix a needle in the board, place the ruler against it, sight to some near point, draw a line by the ruler, turn it end for end, again place it against the needle, again sight to the same point, and draw a new line. If it coincides with the former line, the above condition is satisfied. The ruler and sights together take the name of Alidade. If a point should be too high or too low to be seen with the alidade, a plumb-line, held between the eye and the object, will remove the difficulty. A telescope is sometimes substituted for the sights, being supported above the ruler by a standard, and capable of pointing upward or downward. It admits of adjustments similar in principle PART iii.] PLANE TABLE SURVEYING. 305 to the 2d and 3d adjustments of the Transit, Part IV, Chapter 3, pages 242 and 246. But even without these adjustments, whether of the sights or of the telescope, a survey could be made which would be perfectly correct as to the relative position of its parts, however far the line of sight might be from lying in the same vertical plane as the edge of the ruler, or even from being parallel to it; just as in the Transit or Theodolite the index or vernier need not to be exactly under the vertical hair of the telescope, since the angular deviation affects all the observed directions equally. (452) Method of adiation, This is the simplest, though not the best, method of surveying with the Plane-table. It is especially applicable to survey- Fig. 307. ing a field, as in the figure. B In it and the following fi- A gures, the size of the Table / / >C is much exaggerated. Set the instrument at any conve- nient point, as O; level it G and fix a needle (having a \ / head of sealing-wax) in the board to represent the sta- F D tion. Direct the alidade to any corner of the field, as A, the fiducial edge of the ruler touching the needle, and draw an indefinite line by it. Measure OA, and set off the distance, to any desired scale, from the needle point, along the line just drawn, to a. The line OA is thus platted on the paper of the table as soon as determined in the field. Determine and plat in the same way, OB, OC, &c., to b, c, &c. Join ab, be, &c., and a complete plat of the field is obtained. Trees, houses, hills, bends of rivers, &c., may be determnied in the same manner. The corresponding method with the Compass or Transit, was described in Articles (258) and (391). The table may be set at one of the angles of the field, if more convenient. If the alidade has a telescope, the method of measuring distances with a stadia, described in Art. (375), may be here applied with great advantage. 20 306 PLANE TABLE SURtVEYING, [PART vIII (453) I4ethod of Progression, Let ABCD, &c., be the 1mn to be surveyed. Fig. 308. Fix a needle at a / r convenient point E of the Plane-table, A. \ near a corner so as to leave room for the plat, and ~ D set up the table at B{ B, the second an- gle of the line., so that the needle, whose point repre- sents B, and which should be named b, shall be exactly over that station. Sight to A, pressing the fiducial edge of the ruler against the needle, and draw a line by it. Measure BA, and set off its length, to the desired scale, on the line just drawn, from b to a point a, representing A. Then sight to C, draw an indefinite line by the ruler, and on it set off the length of BC from b to c. Fix the needle at c. Set up at C, the point c being over this station, and make the line cb of the plat coincide in direction with CB on the ground, by placing the edge of the ruler on cb, and turning the table till the sights point to B. The compass, if the table have one, will facilitate this. Then sight forward from C to D, and fix CD,'?d on the plat, as be was fixed. Set up at D, make do coincide with DC, and proceed as before. The figure shews the lines drawn at each successive station. The Table drawn at A shews how the survey might be commenced there. In going around a field, the work would be proved by the last line " closing" at the starting point;and, during the progress of the survey, by any direction, as from C to A on the ground, coinciding with the corresponding line, ca, on the plat. This method is substantially the same as the method cf surveying a line with the Transit, explained in Art. (372). It requires all the points to be accessible. It is especially suited to the sur vey of a road, a brook, a winding path through woods, &c. The offsets required may often be sketched in by eye with sufficient precision. PART TIII.] PLINE TABLE SURVEYIXN4 307 When the paper is filled, put on a new sheet, and begin by fixing on it two points, such as C and D, which were on the former sheet, and from them proceed as before. The sheets can then be afterwards united, so that all the points on both shall be in their true relative positions. ( 4~) teaod of tersectios, This is the most usual and the most rapid method of using the Plane-table. The principle was referred to in Articles (259) and (392). Set up the instrument at any convenient point, as X in the figure, and sight to all Fig. 309.,/ / /^',/ \.'~ \/!__x Y I-r~_2=..__~.J ~X'~Y the desired points A, B, C, &c., which are visible, and draw indefinite lines in their directions. Mieasure any line XY, Y being one of the points sighted to, and set off this line on the paper to any scale. Set up at Y, and turn the table till the line XY on the paper lies in the direction of XY, on the ground, as at C in the last method. Sight to all the former points and draw lines in their directions, and the intersections of the two lines of sight to each point will determine them, by the Fourth Method, Art. (8). Points on the other side of the line XY could be determined at the same time. In surveying a field, one side of it may be taken for the base XY. Very acute or obtuse intersections should be avoided. 80~ and 150~ should be the extreme limits. The impossibility of always doing this, renders this method often deficient in precision. When the paper is filled, put on a new sheet, by fixing on it two known points, as in the preceding method. 308 PLANE TABLE SURVEYINGo [PART VIIL, (455) Method of Resection. This method (called by the French Becoupement) is a modification of the preceding method of Inter Fig. 310. Att section. It requires the measurement of only one distance, but all the points must be accessible. Let AB be the measured distance. Lay it off on the paper as ab. Set the table up at B, and turn it till the line bd on the paper coincides with BA on the ground, as in the Method of Progression. Then sight to C, and draw an indefinite line by the ruler. Set up at C, and turn the line last drawn so as to point to B. Fix a needle at a on the table, place the alidade against the needle and turn it till it sights to A. Then the point in which the edge of the ruler cuts the line drawn from B will be the point c on the table. Next sight to D, and draw an indefinite line. Set up at D, and make the line last drawn point to C. Then fix the needle at a or b, and by the alidade, as at the last station, get a new line back from either of them, to cut the last drawn line at a point which will be d. So proceed as far as desired. (4g6) To orient the taMle.' The operation of orientation con. sists in placing the table at any point so tlat its lines shall have the same directions as when it was at previous stations in the same survey. The French phrase, To orient one's self, meaning to determine one's position, asually with respect to the four quarters of the heavens, of which the Orient is she leading one, well deserves naturalization in our language. PART vIIm.] PLNE I IABLE SURVEYrIT. 309 With a compass, this is very easily effected by turning the table till the needle of the attached compass, or that of the Declinator, placed in a fixed position, points to the same degree as when at the previous station. Without a compass the table is oriented, when set at one end of a line previously determined, by sighting back on this line, as at C in the Method of Progression, Art. (453). To orient the table, when at a station unconnected with others. is more difficult. It may be Fig. 311. effected thus. Let ab on the ta- a ble represent a line AB on the -- ground. Set up at A, make ab. _- coincide with AB, and draw a line from a directed towards a steeple, or other conspicuous ob- ject, as S. Do the same at B. Draw a line cd, parallel to ab, and intercepted between aS, and 5S. Divide ab and cd into the same number of equal parts. The table is then prepared. Now let there be a station, P, p on the table, at which the table is to be oriented. Set the table, so that p is over P, apply the edge of the ruler to p, and turn it till this edge cuts cd in the division corresponding to that in which it cuts ab. Then turn the table till the sights point to S, and the table will be oriented. (457) To find oes Mlace on tie gr oun. This problem may be otherwise expressed as Interpolating a point in a plat. It is most easily performed by reversing the Method of Intersection. Set up the table over the station, Fig. 312. 0 in the figure, whose place on the plat already on the table is desired, and orient it, by one of A< the means described in the last \ article. Make the edge of the \ I, ruler pass through some point, a / on the table, and turn it till the sights point to the corresponding o station, A on the ground. Draw a line by the ruler. The' desired 310 PLANE TABLE SURTEYING [PART VIXI point is somewhere in this line. Make the ruler pass through another point, b 9n the table, and make the sights point to B on the ground Draw a second line, and its intersection with the first will be the point desired. Using C in the same way would give a third line to prove the work. This operation may be used as a new method of surveying with the plane-table, since any number of points can have their places fixed in the same manner. This problem may also be executed on the principle of Trilinear Surveying. Three points being given on the table, lay on it a piece of transparent paper, fix a needle any where on this, and with the alidade sight and draw lines towards each of these three points on the ground. Then use this paper to find the desired point, precisely as directed in the last sentence of Art. (398), page 277. (458) Enaccessible distanceSo Many of the problems in Part VII. can be at once solved on the ground by the plane-table, since it is at the same time a Goniometer and a. Protractor. Thus, the Problem of Art. (435) may be solved as follows, on the principle of the construction in the last paragraph of that article. Set the table at C. Mark on it a point, c', to represent C, placing c' vertically over C. Sight to A, B and D, and draw corresponding lines from c'. Set up at D, mark any point on the line drawn from c' towards D, and call it d'. Let d' be exactly over D, and direct d'c' toward C. Then sight to A and B, and draw corresponding lines, and their intersections with the lines before drawn towards A and B will fix points a' and b'. Then on the line joining a and b, given on the paper to represent A and B, ab being equal to AB on any scale, construct a figure, abed, similar to a'b''dc', and the line ed thus determined will represent CD on the same scale as AB PART IX. SURVEYING WITHOUT INSTRUMENTS. (459) THE Principles which were established in Part I, and subse. quently applied to surveying with various instruments, may also be employed, with tolerable correctness, for determining and representing the relative positions of larger or smaller portions of the earth's surface without any Instruments but such as can be extemporized. The prominent objects on the ground, such as houses, trees, the summits of hills, the bends of rivers, the crossings of roads, &c., are regarded as " points" to be "determined." Distances and angles are consequently required. Approximate methods of obtaining these will therefore be first given. (4C0) Bistances Dy pacngg. Quite an accurate measurement of a line of ground may be made by walking over it at a uniform pace, and counting the steps taken. But. the art of walking in a straight line must frst be acquired. To do this, fix the eye on two objects in the desired line, such as two trees, or bushes, or stones, or tufts of grass. Walk forward, keeping the nearest of these objects steadily covering the other. Before getting up to the nearest object, choose a new one in line farther ahead, and then proceed as before, and so on. It is better not to attempt to make each of the paces three feet, but to take steps of the natural length, and to ascertain the value of each by walking over a known distance, and dividing it by the number of paces required to traverse it. Every person should thus determine the usual length of his own steps, repeating the experiment sufficiently often. The French' Geographical Engineers" accustom themselves to take regular 312 STlRVEYING WITHOUT I'STERUIENTS [PART IX. steps of eight-tenths of a metre, equal to two feet seven and a half inches. The English military pace is two feet and six inches. This is regarded as a usual average. 108 such paces per minute give 3.07 English miles per hour. Quick pacing of 120 such paces per minute gives 3.41 miles per hour. Slow paces, of three feet each and 60 per minute, give 2.04 miles per hour.* An instrument, called a Pedometer, has been contrived, which counts the steps taken by one wearing it, without any attention on his part. It is attached to the body, and a cord, passing from it to the foot, at each step moves a toothed wheel one division, and some intermediate wheelwork records the whole number upon a dial. (461t) Distances by visual angles. Prepare a scale, by marking off on a pencil what length of it, when it is held off at arm's length, a man's height appears to cover at different distances (previously measured with accuracy) of 100, 500, 1000 feet, &c. To apply this, when a man is seen at any unknown distance, hold up the pencil at arm's length, making the top of it come in the line from the eye to his head, and placing the thumb nail in the line from Fig. 313. the eye to his feet, as in Fig. 313. The pencil having been previously graduated by the method above explained, the portion of it now intercepted between these two lines will indicate the correspending distance. If no previous scale have been prepared, and the distance of a man be required, taKe a foot-rule, or any measure minutely divided, hold it off at arm's length as before, and see how much a man's height covers. Then knowing the distance from the eye to the rule, a statement by the Rule of Three (on the principle of similar triangles) will give the distance required. Suppose a man's height, of 70 inches, covers 1 inch of.the rule. He is then 70 times as far * A horse, on a walk, averages 330 feet per minute, on a trot 6(50, and on a coin mon gallop 1040. For longer times, the difference in horses is more apparent. PART ix.] U RVEYIG WITIOUVT INSTRiT rENTSe 13 from the eye as the rule; and if its distance be 2 feet, that of the man is 140 feet. Instead of a man's height, that of an ordinary house, of an apple-tree, the length of a fence-rail, &c., may be be taken as the standard of comparison. To keep the arm immovable, tie a string of known length to the pencil, and hold between the teeth a knot tied at the other end of the string. (462) Distances by visibility. The degree of visibility of various well-known objects will indicate approximately how far distant they are. Thus, by ordinary eyes, the windows of a large house can be counted at a distance of about 13000 feet, or 2A miles; men and horses will be perceived as points at about half that distance, or 11 miles; a horse can be clearly distinguished at about 4000 feet; the movements of men at 2600 feet, or half a mile; and the head of a man, occasionally, at 2300 feet, and very plainly at 1300 feet, or a quarter of a mile. The Arabs of Algeria define a mile as " the distance at which you can no longer distinguish a man from a woman." These distances of visibility will of course vary somewhat with the state of the atmosphere, and still more with individual acuteness of sight, but each person should make a corre sponding scale for himself. (463) Distances by souni. Sound passes through the air math a moderate and known velocity; light passes almost instantaneously. If, then, two distant points be visible from each other, and a gun be fired at night from one of them, an observer at the other, noting by a stop-watch the time at which the flash is seen, and then that at which the report is heard, can tell by the intervening number of seconds how far apart the points are, knowing how far sound travels in a second. Sound moves about 1090 feet per second in dry air, with the temperature at the freezing point, 320 Fahrenheit. For higher or lower temperatures add or subtract 1 - foot for each degree of Fahrenheit. If a wind blows with or against the movement of the sound, its velocity must be added or subtracted. If it blows obliquely, the correction will evidently equal its velocity multiplied by the cosine of the angle which the direction of the wind makes 314 STREYING WiITHUT INSTRUIIENTS [PAiT Ix. with the direction of the sound.* If the gun be fired at each end of the base in turn, and the means of the times taken, the effect of the wind will be eliminated. If a watch is not at hand, suspend a pebble to a string (such as a thread crawn from a handkerchief) and count its vibrations. If it be 39j inches long, it will vibrate in one second; if 93 inches long, in half a second, &c. If its length is unknown at the time, still count its vibrations; measure it subsequently; and then will //length of string the time of its vibration, in seconds, = \/(ln of strin (41a) Angles, Right angles are those most frequently required in this kind of survey, and they can be estimated by the eye with much accuracy. If other angles are desired, they will be determined by measuring equal distances along the lines which make the angle, and then the line, or chord, joining the ends of these distan ces, thus forming chain angles, explained in Art. (t10). (465) iM]ethods of operation. The " First Method" of deter mining the position of a point, Art. (5), is the one most generally applicable. Some line, as AB in Fig. 1, is paced, or otherwise measured, and then the lines AS and BS; the point S is thus determined. The " Second Method," Art. (6), is also much employed, the right angles being obtained by eye, or by the easy methods given in Part II, ChapterV, Arts. (140), &c. It is used for offsets, as in Part II, Chapter III, Arts. (114), &c. The "' Third Method," Art. (7), may also be used, the angles being determined as in Art. (64G). The " Fourth Method," Art. (8), may also be employed, the angles being similarly determined. The " Fifth Method," Art. (10), wouid seldom be used, unless by making an extempore plane-table, and proceeding as directed in the last paragraph of Art. (457). * A gentle, pleasant wind has a velocity of 10 feet per second; a brisk gale 20 feet per second; a very brisk gale 30 feet; a high wind 50 feet; a very high wind 70 feet; a storm or tempest 80 feet; a great storm 100 feet; a hurricane 120 feet; and a violent hurricane, that tears up trees, &c., 150 feet per second PART Ix.] SURVEYING WVITHOUT imSTRtIENTS. 315 The method referred to in Art. (ll) may also be employed. When a sketch has made some progress, new points may bo. fixed on it by their being in line with others already determined. All these methods of operation are shown in the following figure AB is a line paced, or otherwise measured approximately. Fig. 314. /'___-,..', —- / —---,.''.I, f/ / \ G The hill C is determined by the first method. The river on the, other side of AB is determined by offsets according to the Second Method. The house D is determined by the Third Method, EBF being a chain angle. The house G is determined by the Fourth Method, chain angles being measured at B and H, a point in AB prolonged. The pond K is determined, as in Art. (11), by the intersection of the alinements CD and GH prolonged. The bend of the river at L is determined by its distance from II in the line of AH prolonged. A new base line, HM, is fixed by a chain angle at IH, and employed like the former one so as to fix the hill at N, &c. All these methods may thus be used collectively and successively. The necessary lines may always be ranged with rods, as directed in Art. (169), and very many of the instrumental methods already explained, may be practiced with extempore contrivances. The use of the Plane-table is an admirable preparation for this style of surveying or sketching, which is most frequently employed by Military Engineers, though they generally use a prismatic Compass, or pocket Sextant, and a sketching case, which may serve as a Plane-table, PART X. MA PPING. CHAPTER 1. COPYING PLATS, (466) THE Plat of a survey necessarily has many lines of construe tion drawn upon it, which are not needed in the finished map These lines, and the marks of instruments, so disfigure the papel that a fair copy of the plat is usually made before the map is finished. The various methods of copying plats, &c., whether on the same scale, or reduced or enlarged, will therefore now be described. (I37) Stretching the paper. If the map is to be colored, the paper must first be wetted and stretched, or the application of the wet colors will cause its surface to swell or blister and become uneven. Therefore, with a soft sponge and clean water wet the back of the paper, working from the centre outward in all directions. The' water-mark" reads correctly only when looked at from the front side, which it thus distinguishes. When the paper is thoroughly wet and thus greatly expanded, glue its edges to the drawing board, for half an inch in width, turning them up against a ruler, passing the glue along them, and then turning them down and pressing them with the ruler. Some prefer gluing down opposite edges in succession, and others adjoining edges. The paper must be mode rately stretched smooth during the process. Hot glue is best. Paste or gum may be used, if the paper be kept wet by a damp cloth, so that the edges may dry first. " Mouth-glue " may be used CHAP. I.] Copying Plats. 317 by rubbing it (moistened in the mouth or in boiling water) along the turned up edges, and then rubbing them dry by an ivory folder, a piece of dry paper being interposed. As this is a slower process, the middle of each side should first be fastened down, then the four angles, and lastly the intermediate portions. When the paper becomes dry, the creases and puckerings will have disappeared, and it will be as smooth and tight as a drum-head. (418) Copying by tracing. Fix a large pane of clear glass in a frame, so that it can be supported at any angle before a window. or, at night, in front of a lamp. Place the plat to be copied or this glass, and the clean paper upon it. Connect them by pins, &c. Trace all the desired lines of the original with a sharp pencil, as lightly as they can be easily seen. Take care that the paper does not slip. If the plat is larger than the glass, copy its parts successively, being very careful to fix each part in its true relative position. Ink the lines with India ink, making them very fine and pale, if the map is to be afterwards colored. (469) Copying on tRacing paper, A thin transparent paper is prepared expressly for the purpose of making copies of maps and drawings, but it is too delicate for much handling. It may be prepared by soaking tissue paper in a mixture of turpentine and Canada balsam or balsam of fir (two parts of the former to one of the latter), and drying very slowly. Cold drawn linseed oil will answer tolerably, the sheets being hung up for some weeks to dry. Linen is also similarly prepared, and sold under the name of " Vellum tracing paper." It is less transparent than the tracing paper, but is very strong and durable. Both of these are used rather for preserving duplicates than for finished maps. (470) Copying by raansfer paper. This is thin paper, one side of which is rubbed with blacklead, &c., smoothly spread by cotton. It is laid on the clean paper, the blackened side downward, and the plat is placed upon it. All the lines of the plat are then gone over with moderate pressure by a blunt point, such as the eye-end of a small needle. A faint tracing of these lines will then be found 818 MAPPINGO [PART X. on the clean paper, and can be inked at leisure. If the originai cannot be thus treated, it may first be copied on tracing paper. and this copy be thus transferred. If the transfer paper be prepared by rubbing it with lampblack ground up with hard soap, its lines will be ineffaceable. It is then called " Camp-paper." (471) Copying by punctares, Fix the clean paper an a drawing board and the plat over it. Prepare a fine needle with a sealing —wax head. Hold it very truly perpendicular to the board, and prick through every angle of the plats and every corner and intersection of its other lines, such as houses, fences, &c., or at least the two ends of every line. For circles, the centre and one point of the circumference are sufficient. For irregular curves, such as rivers, &c., enough points must be pricked to indicate all their sinuosities. Work with system, finishing up one strip at a time, so as not to omit any necessary points nor to prick through any twice, though the latter is safer. Whben completed, remove the plat. The copy will present a wilderness of fine points. Select those which determine the leading lines, and then the rest will be easily recognized. A beginner should first pencil the lines lightly, and then ink them. An experienced draftsman will omit the pencilling. Two or three copies may be thus pricked through at once. The holes in the original plat may be made nearly invisible by rubbing them on the back of the sheet with a paper-folder, or the thumb nail. (472) Copyin, by intersections0 Draw a line on the clean paper equal in length to some important line of the original. Two starting points are thus obtained. Take in the dividers the distance from one end of the line on the original to a third point.. From the corresponding end on the copy, describe an arc with this distance for radius and about where the point will come. Take the distance on the -original from the other end of the line to the point, and describe a corresponding arc on the copy to intersect the former arc in a point which will be that desired. The principle of the operation is that of our "First Method," Art. (5). Two pairs of dividers may be used as explained in Art. (90). " Tri CHAP. T.] Copying Plats, 319 angular compasses,' having three legs, are used by fixing two of their legs on the two given points of the original, and the third leg on the point to be copied, and then transferring them to the copy. All the points of the original can thus be accurately reproduced. The operation is however very slow. Only the chief points of a plat may be thus transferred, and the details filled in by the following method. (473) Copying by squarest On the original plat draw a series of parallel and equidistant lines. The T square does this most readily. Draw a similar series at right angles to these. The plat will then be covered with squares, as in Fig. 38, page 48. On the clean paper draw a similar series of squares. The important points may now be fixed as in the last article, and the rest copied by eye, all the points in each square of the original being properly placed in the corresponding square of the copy, noticing whether they are near the top or bottom of each square, on its right or left side, &c. This method is rapid, and in skilful hands quite accu. rate. Instead of drawing lines on the original, a sheet of transparent paper contaliinig them may be placed over it; or an open frame with threads stretched across it at equal distances and at right angles. This method supplies a transition to the Reduction and Enlcargement of plats in any desired ratio; under which head Co2yingby the Pantagraph and Camera Lucida will be noticed. (474) lefldCTB ly sqlaares, Begin, as in the preceding article, by drawing squares on the original, or placing them over it. Then on the clean paper draw a similar set of squares, but with their sides one-half, one third, &c., (according to the desired reduction), of those of the original plat. Then proceed as before to copy into each small square all the points and lines found in the large square of the plat in their true positions relative to the sides and corners of the square, observing to reduce each distance, by eye or as directed in the following article, in the given ratio. 820 AIPPINGo [PART X. (47,5) Reducing by pro'lortional seales. Many graphical methocs of finding the proportionate length on the copy, of any line of the original, may be used. The "Angle of reduction" is con structed thus. Draw any line AB. With it for radius and A Fig. 315. for centre, describe an indefinite \\ \\ are. With B for centre and a \\ radius equal to one-half, one-third, \ \ \ \ \ \ \ \ \\ &c., of AB according to the de- ^< \\ \ \'l\ \I \ \\ \1 sired reduction describe another A D B arc intersecting the former arc in C. Join AC. From A as centre describe a series of arcs. Now to reduce any distance, take it in the dividers, and set it off from A on AB, as to D. Then the distance from D to E, the other end of the arc passing through D, will be the proportionate length to be set off on the copy, in the manner directed in Art. (472). The Sector, or " Compass of proportion," described in Art. (52), presents such an "Angle of reduction," always ready to be used in this manner. The "; Angle of reduction may be simplified Fig. 316. thus. Draw a line, AB, parallel to one side C of the drawing board, and another, 3C, at right angles to it, and one-half, &c., of it, as desired. Join AC.. Then let AD be the distance re- quired to be reduced. Apply a T square so as to pass through D. It will meet AC in some point E, and DE will be the reduced A length required. Another arrangement for the same object is shown in Fig. 317, Draw two lines, AB, AC, at any angle, and de- Fig. 317. scribe a series of arcs from their intersection, A, as in the figure. Suppose the reduced scale i, to be half the original scale. Divide the outermost arc into three equal parts, and draw a line from A to one of the points of division, as D. Then: each arc will be divided into parts, one of which is twice the other. Take any distance on the original scale, and find by trial which of the arcs on CHAP. I.] Copying Platso 321 the right hand side of the figure it corresponds to. The other part of that arc will be half of it, as desired. " Proportional compasses," being properly set, reduce lines in any desired ratio. A simple form of them, known as "Wholes and halves;" is often useful. It consists of two slender bars, pointed at each end, and united by a pivot which is twice as far from one pair of the points as from the other pair. The long ends being set to any distance, the short ends will give precisely half that distance. (476) Reducing by a pantagraph. This instrument consists of two long and two short rulers, connected so as to form a parallelo gram, and capable of being so adjusted that when a tracing point attached to it is moved over the lines of a map, &c., a pencil attached to another part of it will mark on paper a precise copy, reduced on any scale desired. It is made in various forms. It is troublesome to use, though rapid in its work. (4T7) Redlaing by a camera lucida. This is used in the Coast Survey Office. It cannot reduce smaller than one-fourth, without losing distinctness, and is very trying to the eyes. Squares drawn on the original are brought to apparently coincide with squares on the reduction, and the details'are then filled in with the pencil, as seen through the prism of the instrument. (478) Enlarging plats. Plats may be enlarged by the principal methods which have been given for reducing them, but this should be done as seldom as possible, since every inaccuracy in the original becomes magnified in the copy. It is better to make a new plat from the original data. 21 Z 21 2 M3APPING. [PART L CHAPTER IL CONVENTIONAL SIGNS. (4t7) Various conventional signs or marks have been adopted, more or less generally, to represent on maps the inequalities of the surface of the ground, its different kinds of culture or natural pro. ducts, and the objects upon it, so as not to encumber and disfigure it with much writing or many descriptive legends. This is the purpose of what is called Topographieal Maapping. (480) The relief of groumd. The inequalities of the surface of the earth, its elevations and depressions, its hills and hollows, constitute its " Relief." The representation of this is sometimes called " Hill drawing." Its difficulty arises from our being accustomed to see hills sideways, or " in elevation," while they must be represented as they would be seen from above, or " in plan." Various modes of thus drawing them are used; their positions being laid down in pencil as previously sketched by eye or measured. If light be supposed to fall vertically, the slopes of the ground will receive less light in proportion to their steepness. The relief of ground will be indicated on this principle by making the steep slopes very dark, the gentler inclinations less so, and leaving the level surfaces white. The shades may be produced by tints of India ink applied with a brush, their edges, at the top and bottom of a hill or ridge, being softened off with a clean brush. If light be supposed to fall obliquely, the slopes facing it will be light, and those turned from it dark. This mode is effective, but not precise. In it the light is usually supposed to come from the upper left hand corner of the map. Horizontal contour lines are however the best convention for this purpose. Imagine a hill to be sliced off by a number of equi. distant horizontal planes, and their intersections with it to be drawn as they would be seen from above, or horizontally projected on the CHAP II.] Cos ventiontal Sgns 323 map. These are C' Contour lines." They are the same lines as would be formed by water surrounding the hill, and rising one foot at a time (or any other height) till it reached the top of the hill. The edge of the water, or its shore, at each successive rise, would be one of these horizontal contour lines.'It is plain that their nearness or distance on the map would indicate the steepness or gentleness of the slopes. A right cone would thus be repreFig. 318. Fig. 319. Fig. 320. sented by a series of concentric circles, as in Fig. 318; an oblique cone by circles not concentric, but nearer to each other on the steep side than on the other, as in Fig. 319; and a half-egg, somewhat as in Fig. 320. Vertieal sections, perpendicular to these contour lines, are usually combined with them. They are the "' Lines of greatest slope," and may be supposed to represent water running down the sides of the hill. They are also made thicker and nearer together on the steeper slopes, to produce the effect required by the convention of vertical light Fi. 321. already referred to. \ II//h!i The marginal figure shews an elongated. half-egg, or oval h ill, thus represented. The spaces betweenw, the rows of vertical "Hatchings" indicate the contour lines, which are not actually dranwn. The beauty of the graphical execution of this work depends on the uniformity of the strokes representing uniform slopes, on their perfectly regular gradation in thickness and nearness for varying slopes, and on their being made precisely at right angles to the contour lines between which they are situated. 824 MA3PPING [PART x The methods of determining the contour lines are applications of Levelling, and will therefore be postponed, together with the farther details of " Hill-drawing," to the volume treating of that subject, which is announced in the Preface. (481) Signs for natural surface Sand is represented by fine dots made with the point of the pen; gravel by coarser dots. Rocks are drawn in their proper places in irregular angular forms, imitating their true appearance as seen from above. The nature of the rocks, or the aeology of the country, may be shown by applying the proper colors, as agreed on by geologists, to the back of the map, so that they may be seen by holding it up against the light, while they will thus not confuse the usual details. (482) Sgns' for vegetation. TYoods are represented by scolloped circles, irregularly disposed, Fig. 392. imitating trees seen "in plan," and A,j Y X i closer or farther apart according to, 1 1 f / the thickness of the forest. It is..' c usual to shade their lower and right hand sides and to represent their' 4 shadows, as in the figure, though, in strictness, thiis is nconsistent with the hypothesis of vertical light, adopted for " hill-drawing." For pine and similar forests, the signs may have a star-like form, as on the right hand side of the figure. Trees are sometimes drawn " in elevations" or sideways, as usually seen. This makes them more easily recognized, but is in utter violation of the principles of mapping in horizontal projection, though it may be defended as a pure convention. Orchards are represented by trees arranged in rows. Bushes may be drawn like trees, but smaller. Grass-land is drawn with irregularly Fig. 323. scattered groups of short lines, as in the',L figure, the lines being arranged in odd l,......., V. numbers, and so that the top of each group is,,-,,f^.,,. VV\ convex and its bottom horizontal or parallel vto the base of the drawing. Jfeadows are l-,,. ^ sometimes represented by pairs of diverging lines, (as on the right CHAP. II.] Conventional Signs. 325 of the figure), which may be regarded as tall blades of grass. Uncultivated land is indicated by appropriately intermingling the signs for grass land, bushes, sand and rocks. Caltivated land is shown by parallel rows of broken and dotted Fig. 324. lines, as in the figure, representing furrows. 1 1i il! Crops are so temporary that signs for them are, I I I i unnecessary, though often used. They areusu- u Ii I ll I 1 ji ally imitative, as for cotton, sugar, tobacco, rice, vines, hops, &c. Gardens are drawn with circular and other beds and walks. (483) Signs for water. The Sea-coast is represented by drawing a line parallel to the shore, following all its windings and indentations, and as close to it as possible, then another parallel line a little more distant, then a third still more distant, and so on. Examples are seen in figures 287, &c. If these lines are drawn from the low tide mark, a similar set may be drawn between that and the high tide mark, and dots, for sand, be made over the included space. Rivers have each shore treated like the sea shore, as in the figures of Part VII.* Brooks would be shown by only two lines, or one, according to their magnitude. Ponds mzay be drawn like sea shores, or represented by _ Fig 325. parallel horizontal lines ruled across them. "_-..=..: J3arshes and Swamps are represented by an — _ —., - _ _ irregular intermingling of the preceding _ -.. -ii, sign with that for grass and bushes, as in the.,. _~figure...(484) Colored Topography. The conventional signs which have been described, as made with the pen, require much time and labor. Colors are generally used by the French as substitutes for them, and combine the advantages of great rapidity and effectiveness. Only three colors (besides India ink) are required; viz. Gamboge (yellow), Indigo (blue), and Lake (pink). Sepia, Burnt Sienna, Yellow ochre, Red lead, and Vermillion, are also sometimes used. The last three are difficult to work with. Tc Those in Part II, Chapter V, have the lines too close together in the middle. 826 MAPPING, [PART x use these paints, moisten the end of a cake and rub it up with a drop of water, afterwards diluting this to the proper tint, which should always be light and delicate. To cover any surface with a uniform flat tint, use a large camel's hair or sable brush, keep it always moderately full, incline the board towards you, previously moisten the paper with clean water if the outline is very irregular, begin at the top of the surface, apply a tint across the upper part, and continue it downwards, never lettzng tAe edge dry. This last is the secret of a smooth tint. It requires rapidity in returning to the beginning of a tint to continue it, and dexterity in following the outline. Marbling, or variegation, is produced by having a brush at each end of a stick, one for each color, and applying first one, and then' the other beside it before it dries, so that they may blend but not mix, and produce an irregularly clouded appearance. Scratched parts of the paper may be painted over by first applying strong alum water to the place. The conventions for colored Topography, adopted by the French Military Engineers, are as follows. WooDS, yellow; using gamboge and a very little indigo. GRASS-LAND, green; made of gamboge and indigo. CULTIVATED LAND, brown; lake, gamboge, and a little India ink. "Burnt Sienna" will answer. Adjoining fields should be slightly varied in tint. Sometimes furrows are indicated by strips of various colors. GARDENS are represented by small rectangular patches of brighter green and brown. UNCULTIVATED LAND, marbled green and light brown. BRUSH, BRAMBLES, &C., marbled green and yellow. HEATH, FURZE, &c., marbled green and pink. VINEYARDS, purple; lake and indigo. SANDS, a light brown; gamboge and lake. " Yellow ochre" will do. LAKES and RIVERS, light blue', with a darker tint on their upper and left hand sides. SEAS, dark blue, with a little yellow added. MIARSHES, the blue of water, with spots of grass green, the touches all' lying horizontally. ROADS, brownz; between the tints for sand and cultivated ground, with more India ink. HILLS, greenish brown; gamboge, indigo, lake and India ink, instead of the pure India ink, directed in Art. (480). WOODs may be finished up by drawing the trees as in Art. (482) and coloring them green, with touches of gamboge towards the light (the upper and left hand side) and of indigo on the opposite side. EIAP. II.] Conventional Signs~ 327 (485) Signs for detached objects. Too great a number of these will cause confusion. A few leading ones will be given, the meanings of which are apparent. Figs. Figs. Court house, 326. Wilnd mill, X334. Post office, 327. Steam mill, 335. Tavern, 328. Furnace, p 336. Blacksmith's shop, ^ 329. Woollen factory, 33. Guide board, t 330. Cotton factory, 1 338. Quarry, 331. Glass works, A 339 Grist mill, 0 332. Church, d 340. Saw mill, 333. Grave yard,' 4 341. An ordinary house is drawn in its true position and size, and the ridge of its roof shown if the scale of the map is large enough. On a very small scale, a small shaded rectangle represents it. If colors are used, buildings of masonry are tinted a deep crimson, (with lake), and those of wood with India ink. Their lower and right hand sides are drawn with heavier lines. Fences' of stone or wood, and hedges, may be drawn in imitation of the.realities; and, if desired, colored appropriately. Mines may be represented by the signs of the planets which were anciently associated with the various metals. The signs here given represent respectively, Gold, Silver, Iron, Copper, Tin, Lead, Quicksilver. A large black circle,, may be used for Coal. Boundary lines, of private properties, of townships, of counties, and of states, may be indicated by lines formed of various combinations of short lines, dots and crosses, as below.' * Very minute directions for the execution of the details described in this chap ter, are given in Lieut. R. S. Smith's " Topographical Drawing." Wiley, N. Y. 228 MAPPlING rPART X CHAPTER III. FINISHING THE mlAP (480) Orientation, The map is usually so drawn that the top of the paper may represent the North. A Meridian line should also be drawn, both True and Magnetic, as in Fig. 199, page 189. The number of degrees and minutes in the Variation, if known, should also be placed between the two North points. Sometimes a compass-star is drawn and made very ornamental. (487) Lettering. The style in which this is done very much affects the general appearance of the map. The young surveyor should give it much attention and careful practice. It must all be in imitation of the best printed models. No writing, however beautiful, is admissible. The usual letters are the ordinary ROMAN CAPITALS, Small Roman, ITALIC CAPITALS, Small Italic, and GO T HIC O R EG Y PT A N. This last, when well done, is very effective. For the Titles of maps, various fancy letters may be used. For very large letters, those formed only of the shades of the letters regarded as blocks (the body being rubbed out after being pencilled as a guide to the placing of the shades) are most easily made to look well. The simplest lettering is generally the best. The sizes of the names of places, &c., should be proportional to their importance. Elaborate tables for various scales have been published. It is better to make the letters too small than too large. They should not be crowded. Pencil lines should always be ruled as guides. The lettering should be in lines parallel to the bottom of the map, except the names of rivers, roads, &c., whose general course should be followed. (488) Borderse The Border may be a single heavy line, enclosing the map in a rectangle, or such a line may be relieved by a finer line drawn parallel and near to it. Time should not be wasted in ornamenting the border. The simplest is the best. cRIAP II.] Finiishing the Mapo 329 (489) Joining paperi If the map is larger than the sheets of paper at hand, they should be joined with a feather-edge, by pro" ceeding thus. Cut, with a knife guided by a ruler, about onethird through the thickness of the paper, and tear off on the under side, a strip of the remaining thickness, so as to leave a thin sharp edge. Treat the other sheet in the same way on the other side of it. When these two feather edges are then put together, (with paste, glue or varnish), they will make a neat and strong joint. The sheet which rests upon the other must be on the right hand side, if the sheets are joined lengthways, or below if they are joined in that direction, so that the thickness of the edge may not cast a shadow, when properly placed as to the light. The sheets must be joined before lines are drawn across them, or the lines will become distorted. Drawing paper is now made in rolls of great length, so as to render this operation unnecessary. (490)' Mounting maps. A map is sometimes required to be mounted, i. e. backed with canvas or muslin. To do this, wet the muslin and stretch it strongly on a board by tacks driven very near together. Cover it with strong paste, beating this in with a brush to fill up the pores of the muslin. Then spread paste over the back of the paper, and when it has soaked into it, apply it to the muslin, inclining the board, and pasting first a strip, about two inches wide, along the upper side of the paper, pressing it down with clean linen in order to drive out all air bubbles. Press down another strip in like manner, and so proceed till all is pasted. Let it dry very gradually and thoroughly before cutting the muslin from the board. Maps may be varnished with picture varnish; or by applying four or five coats of isinglass size, letting each dry well before applying the next, and giving a full flowing coat of Canada balsam diluted with the best oil of turpentine. PART XL LAYING OUT, PAINGING OFF, AND DIVIDING UP LANDS. CHAPTER I. LAYING OUJT LAND. (491) Its nature. This operation is precisely the reverse of those of Surveying properly so called. The latter measures certain lines as they are; the former marks them out in the ground where they are required to be, in order to satisfy certain conditions. The same instruments, however, are used as in Surveying. Perpendiculars and parallels are the lines most often employed. The Perpendiculars may be set out either with the chain alone, Arts. (140) to (159); still more easily with the Cross-staff, Art. (104), or the Optical-square, Art. (107); and most precisely with a Transit or Theodolite, Arts. (402) to (406). Parallels may also be set out with the chain alone, Arts. (160) to (166); or with Transit, &c., Arts. (407) and (408). The ranging out of lines by rods is described in Arts. (169) and (1178), and with an Angular instrument, in Arts. (376), (409) and (415). (492) To lay out squares, Reduce the desired content to square chains, and extract its square root. This will be the length of the required side, which is to be set out by one of the methods indicated in the preceding article. An Acre, laid out in the form of a square, is frequently desired by farmers. Its side must be made 3161 links of a Gunter's * The Demonstrations of the Problems in this part, when required, will be found in Appendix B. CHAP. I.] Laying out Lansd 331 chain; or 208-1O - feet; or 69V-o yards. It is often taken at 70 paces. The number of plants, hills of corn, loads of manure, &c., which an acre will contain at any uniform distance apart, can be at once found by dividing 209 by this distance in feet, and multiplying the quotient by itself; or by dividing 43560 by the square of the distance in feet. Thus, at 3 feet apart, an acre would contain 4840 plants, &c.; at 10 feet apart, 436; at a rod apart, 160; and so on. If the distances apart be unequal, divide 43560 by the product of these distances in feet; thus, if the plants were in rows 6 feet apart, and the plants in the rows were 3 feet apart, 2420 of them would grow on one acre. (493) To lay out rcetangles, The content and length being given, both as measured by the same unit, divide the former by the latter, and the quotient will be the required breadth. Thus, 1 acre or 10 square chains, if 5 chains long, must be 2 chains wide. The content being given and the length to be a certain number of times the breadth. Divide the content in square chains, &c., by the ratio of the length to the breadth, and the square root of the quotient will be the shorter side desired, whence the longer side is also known. Thus, let it be required to lay out 30 acres in the form of a rectangle 3 times as long as broad. 30 acres = 300 square chains. The desired rectangle will contain 3 squares, each of 100 sq. chs., having sides of 10 chs. The rectangle will therefore be 10 chs. wide and 30 long. An Acre laid out in a rectangle twice as long as broad, will be 224 links by 448 links, narly; or 1471 feet by 295 feet; or 491 yards by 985 yards. 50 paces by 100 is often used as an approximation, easy to be remembered. The content being given, and the difference between the length and breadth. Let c represent this content, and d this difference, Then the longer side d + 1 V(d2 + 4c). Example. Let the content be 6.4 acres, and the difference 12 chains. Then the sides of the rectangle will be respectivetly 16 chains and 4 chains. AV32 LALYING OUT FAND DIVIDIN UP LANDO. [PART xi The content being given, and the sum of the length and breadth. Let c represent this content, and s this sum. Then the longer side = s + ~ (s2 -4 e). Examplle. Let the content be 6.4 acres, and the sum 20 chains. The above formula gives the sides of the rectangle 16 chains and 4 chains as before. (441) To lay out triangles, The content and the base being given, divide the former by half the latter to get the height. At any point of the base erect a perpendicular of the length thus obtained, and it will be the vertex of the required triangle. The content being given and the base having to be m times the height, the height will equal the square root of the quotient obtained by dividing twice the given area by mn. The content being given and the triangle to be equilateral, take the square root of the content and multiply it by 1.520. The product will be the length of the side required. This rule makes the sides of an equilateral triangle containing one acre to be 480. links. A quarter of an acre laid out in the same form would have each side 240 links long. An equilateral triangle is very easily set out on the ground, as directed in Art. (90), under'" Platting," using a rope or chain for compasses. (495) The content and base being given, and one side having to make a given angle, as B, with the base Fig. 342. ABthe length of the sideBC 2 x ABC AB. sin. B Example. Eighty acres are to be laid out in the form of a triangle, on a base, AB, of sixty chains, bearing N. 80~ W. a the bearing of the side BC being N. 70~ E. Here the angle B is found from the Bearings (by Art. (243), reversing one of them) to be 30~. Ience BC = 53.23. The figure is on a scale of 50 chains to 1 inch -= 1:39600. Any right-line figure may be laid out by analogous methods. (496) To lay sot crc'les. Multiply the given content by 7, divide the product by 22, and take the square root of the quotient. C01AP. i.] Laying out Lande 33b This will give the radius, with which the circle can be described on the ground with a rope or chain. A circle containing one acre has a radius of 1781 links. A circle containing a quarter of an acre will have a radius of 89 links. (497) Town liotse House lots in cities are usually laid off as rectangles of 25 feet front and 100 feet depth, variously combined in blocks. Part of New-York is laid out in blocks 200 feet by 800, each containing 64 lots, and separated by streets, 60 feet wide, running along their long sides, and avenues, 100 feet wide, on their short sides. The eight lots on each short side of the block, front on the avenues, and the remaining forty-eight lots front on the streets. Such a block covers almost precisely 3- acres, and 171 such lots about make an acre. But, allowing for the streets, land laid out into lots, 25 by 100, arranged as above, would contain only 11.9, or not quite 12 lots per acre. Lots in small towns and villages are laid out of greater size and less uniformity. 50 feet by 100 is a frequent size for new villages, the blocks being 200 feet by 500, each therefore containing 20 lots. (498) Lanid sold or taxes, A case occurring in the State of New-York will serve as an application of the modes of laying out squares and rectangles. and Fig. 343. on which taxes are unpaid is ~..~ C sold at auction to the lowest bidder; i. e. to him who will accept the smallest portion of it in return for paying the taxes on the whole. The lot in question was originally the east half of the square lot ABCD, containing 500 acres. At a sale for taxes in 1830, 70 acres were bid off, and this area was D set off to the purchaser in a square lot, from the north-east corner. Required the side of the square in links. Again, in 1834, 29 acres more were thus sold, to be set off in a strip of equal width 384 LAYING OUT AND DIVlDING'UP LAND [PART xi around the square previously sold. Required the width of this strip. Once more, in 1839, 42 acres more were sold, to be set off around the preceding piece. Required the dimensions of this third portion. The answer can be proved by calculating if the dimensions of the remaining rectangle will give the content which it should have, viz. 250 - (70 + 29 + 42) = 109 Acres. The figure is on a scale of 40 chains to 1 inch = 1: 31680. (49) New coM ntries. The operations of laying out land for the purposes of settlers, are required on a large scale in new countries, in combination with their survey. There is great difficulty in uniting the necessary precision, rapidity and cheapness. "Triangular Surveying" will ensure the first of these qualities, but is deficient in the last two, and leaves the laying out of lots to be subsequently executed. " Compass Surveying" possesses the last two qualities, but not the first. The United States system for surveying and laying out the Public Lands admirably combines an accurate determination of standard lines (Meridians and Parallels) with a cheap and rapid subdivision by compass. The subject is so important and extensive that it will be explained by itself in Part XII. CHAPTER II. PAIRTIWI OFF LAND, (500) It is often required to part off from a field, or from an indefinite space, a certain number of acres by a fence or other boundary line, which is also required to run in a particular direction, to start from a certain point, or to fulfil some other condition. The various cases most likely to occur will be here arranged according to these conditions. Both graphical and numerical me~ thods will generally be given.* * The given lines will be represented by fine full lines; the lines of construction by broken lines, and the lines of the result by heavy full lines. CHP. I.] Parting off Land. 335 The given content is always supposed to be reduced to square chains and decimal parts, and the lines to be in chains and decimals. A. BY A LINE PARALLEL TO A SIDE. (501) To part off a rectangle, If the sides of the field adjacent to the given side make right angles with it, the figure parted off by a parallel to the given side will be a rectangle, and its breadth will equal the required content divided by that side, as in Art. (493). If the field be bounded by a curved or zigzag line outside of the given side, find the content between these irregular lines and the given straight side, by the method of offsets, subtract it from the content required to be parted off, and proceed with the remainder as above. The same directions apply to the subsequent problems. (502) To part off a parallelograma If the sides adjacent to the given side be parallel, the Fig. 344. figure parted off will be a parallel- D ogram, and its perpendicular width,^ CE, will be obtained as above. / The length of one of the parallel A B CE ABDa sides, as AC -. - -AB sm. A AB. sino A (503) To part off a trapezoid. When the sides of the field adjacent to the given side are not parallel, the figure parted off will be a trapezoid. When the field or figure is given on the ground, or on a plat begin as if the sides were parallel, Fig. 345. dividing the given content by the base AB. The quotient will be c - an approximate breadth, CE, or DF; too small if the sides con- verge, as in the figure, and vice / \ X versa. Measure CD. Calculate r the content of ABDC. Divide the difference of it and the required 336 LAYING IOT AND DIVIDING UP LANDe [PART xI. content by CD. Set off the quotient perpendicular to CD, (in this figure, outside of it,) and it will give a new line, GH, a still nearer approximation to that desired. The operation may be repeated, if found necessary. (504) When the field is given by Bearings, de- Fig. 346. duce from them, as in Art. (243), the angles at A B and B. The required sides will then be given by these formulas: CD-I/A A B2 2 x ABCD. sin. (A + B)) sin. B. B AD (AB CD) (A +B) sin. (A+B) sin. A ( -) ssin. (A B) A - When the sides AD and BC diverge, instead of converging, as in the figure, the negative term, in the expression for CD, becomes positive; and in the expressions for both AD and BC, the first factor becomes (CD- AB). The perpendicular breadth of the trapezoid =AD. sin. A; or -- BC. sin. B. Example. Let AB run North, six chains; AD, N. 80" E.; BC, S. 60~ E. Let it be required to part off one acre by a fence parallel to AB. Here AB - 6.00, ABCD = 10 square chains, A = 80, B =- 60. Ans. CD = 4.57, AD =1.92, BC = 2.18, and the breadth = 1.89. The figure is on a scale of 4 chains to 1 inch = 1:3168. B. BY A LINE PERPENDICULAR TO A SIDE. (505) To part off a triangle. Let FG be the required line. When the field is given on the Fig. 347. ground, or on a plat, at any point, as G C D, of the given side AB, set out a. "; guess line," DE, perpendicular to AB, and calculate the content of B D AF DEB. Then the required distance BF, from the angular point to the foot of the desired perpendicular, = BD ~(BFG B " a - RE~ CHAP. ii.] Parting off Lando 337 Example. Let BD= 30 chains; ED =-12 chains; and the desired area = 24.8 acres. Then BF = 35.22 chains. The scale of the figure is 30 chains to 1 inch =1: 23760. (580) When the field is given by Bearings, Fig. 34t find the angle B from the Bearings; then is BE 2 x BFG) V ^ tang. -B / Example. Let BA bear S. 75~ E., and BC N. 60~ E., and let five acres be required to be A parted off from the field by a perpendicular to BA. Here the angle B - 45~, and BF = 10.00 chains. The scale of the figure is 20 chains to 1 inch = 1:15840. (507) To part off a quadrilateral Produce the converging sides to meet at B. Calculate the Fig. 369. content of the triangle 1HKB, whe- C ther on the ground or plat, or from Bearings. Add it to the content -- of the quadrilateral required to be B ^ parted off, and it will give that of the triangle FGB, and the me. thod of the preceding case can then be applied. (50S) To part of any figure. If the field be very irregularly shaped, find by trial any line which will part off a little less than the required area. This trial line will represent HK in the preceding figure, and the problem is reduced to parting off, according to the required condition: a quadrilateral, comprised between the trial line, two sides of the field, and the required line, and containing the difference between the required content and that parted off by the trial-line. C. BY A LINE RUNNING IN ANY GIVEN DIRECTION. (509) To part of a triangle. By construction, on the ground or the plat, proceed nearly as in Art. (595), setting out a line in the required direction, calculating the triangle thus formed, and obtaining BF by the same formula as in that Article. 22 338 LAYING OUT AND DIVIDING ULP LiSD. [PART XL (510) If the field be given by Bearings, find Fig. 350. from them the angles CBA and GFB; then is G ~BF=J(2 x BFG sin (B + F)) sin.[ B. sin. F t Example. Let BA bear S. 30~ E.; BC, N. 80~ E.; and a fence be required to run, from \ some point in BA, a due North course, and to part off one acre. Required the distance from B to the point F, whence it must start. Ans. The angle B = 70, and F — 30~ Then BF= 6.47. The scale of Fig. 350 is 6 chains to 1 inch 1: 4752. (511) To part off a quadrilateral. Let it be required to part off, by a line running in a Fig. 351. given direction, a quadrila- "7 teral from a field in which " are given the side AB, and lp \ the directions of the two B 3 \ other sides running from A and from B.\ \ On the ground or plat \ produce the two converging sides to meet at some point / E. Calculate the content A of the triangle ABE. Measure the side AE. From ABE subtract the area to be cut off, and the remainder will be the content of the triangle CDE. From A set out a line AF parallel to the given direction. Find the content of ABF. Take it from ABE, and tAus obtain AFE. Then this formula, ED = AE /AE' will fix V YXAE' the point D, since AD = AE- ED. (5IS) When the field and the dividing line are given by Bear ings, produce the sides as in the last article. Find all the angles from the Bearings. Calculate the content of the triangle ABE, by the formula for one side and its including angles. Take the CHAP. i.] Parting off Lamnd 339 desired content from this to obtain CDE. Calculate the side s Asin. B 2 =X CDE. sin. DCE' AE=AB ~ s. ThenisAD=- AE- 2 si. E. _ si-. E-IExample. Let DA bear S. 201~ W.; AB, N. 51V W., 8.19; BC, N. 738 E.; and let it be required to part off two acres by a fence, DC, running N. 45~ W. Ans. ABE = 32.50 sq. chains; whence CDE- 12.50 sq. chs. Also, AE 8.37; and finally AD = 87 5.49- 2.88 chains. The scale of Fig. 351 is 5 chains to I inch 1: 3960. If the sum of the angles at A and B was more than two right angles, the point E would lie on the other side of AB. The necessary modifications are apparent. (513) To part off any figureo Proceed in a similar manner to that described in Art. (508), by getting a suitable trial-line, producing the sides it intersects, and then applying the method just given. D. BY A LINE STARTING FROMI A GIVEN POINT IN A SIDE. (514) To part of a triaangle Let it be required to cut off from a corner of a field a triangu- Fig. 352. lar space of given content, by a D line starting from a given point A Ur on one of the sides, A in the figure, \ the base, AB, of the desired tri- angle being thus given. If the field be given on the ground or on A a plat, divide the given content by half the base, and the quotient will be the height of the triangle. Set off this distance from any point of AB, perpendicular to it, as from A to C; trom C set out a parallel to AB, and its intersection with the second side, as at D, will be the vertex of the required triangle. Otherwise, divide the required content by half of the perpendicular distance from A to BD, and the quotient will be BD. * This original formula is very convenient for logarithmic computation. 340 LYING OUT AND DIVIDING IP LAND. [PART XI (515) If the field be given by the Bearings of two sides and the length of one of them, deduce the angle B (Fig. 352) from the 2 x ABD Bearings, as in Art. (243). Then is BD - 2x A AB. sin. B' If it is more convenient to fix the point D, by the Second Method, Art. (6), that of rectangular co-ordinates, we shall have BE = BD. cos. B; and ED = BD. sin. B. The Bearing of AD is obtained from the angle BAD; which is ED ED known, since E- AB - tang. BAD. Example. Eighty acres are to be set off from a corner of a field, the course AB being N. 800 W., sixty chains; and the Bearing of BD being N. 700 E. Ans. BD= 53.33; BE= 46.19; ED = 26.67; and the Bearing of AD, N. 17i 23' W. The scale of Fig. 352 is 40 chains to 1 inch = 1: 31680. 2 ABD If the field were right angled at B, of course BD ~ — AB (516) To part oa a quadrilateral. Imagine the two converge ing sides of the field produced to meet, as in Art. (511). Calculate the content of the triangle thus formed, and the question will then be reduced to the one explained in the last two articles. (517) To part off any fgure. Proceed as directed in Art.(513). Otherwise, proceed as follows.'The field being given on the ground or on a plat, find on which side of it the required line will end, by drawing or running 66 guess lines" from the given point to various angles, and roughly measure ing the content thus parted off. Fig. 353. If, as in the figure, A being the given point, the guess line AD C parts off less than the required con- / tent, and AE parts off more, then I, the desired division line AZ will / / end in the side DE. Subtract the I - -- area parted off by AD from the required content, and the difference will be the content of the triu CHAP. 11.] Parting off Land. 341 angle ADZ. Divide this by half the perpehdicular let fall from the given point A to the side DE, and the quotient will be the base, or distance from D to Z. Or, find the content of ADE and make this proportion; ADE: ADZ:: DE: DZ. (518) The field being given by Bearings and distances, find as before, by approximate trials on the plat, or otherwise, which aide the desired line of division will terminate in, as DE in the last figure. Draw AD. Find the Latitude and Departure of this line, and thence its length and Bearing, as in Art. (440). Then calculate the area of the space this line parts off, ABCD in the figure, by the usual method, explained in Part III, Chapter VI. Subtract this area from that required to be cut off, and the remainder will be the area of the triangle ADZ. Then, as in Art. (515), D 2 ADZ AD. sin. AD-Z This problem may be executed without any other Table than that of Latitudes and Departures, thus. Find the Latitude and Departure of DA, as before, the area of the space ABCD, and thence the content of ADZ. Then find the Latitude and Departure of EA, and the content of ADE. Lastly, make this proportion: ADE: ADZ:: DE: DZ.* Example. In the field ABCDE, &c., part of which is shown in Fig. 353, (on a scale of 4 chains to 1 inch = 1 3168), one acre is to be parted off on the west side, by a line starting from the angle A. Required the distance from D to Z, the other end of this dividing line.t The only courses needed are these. AB, N. 530 W., 1.55, BC, N. 200 E., 2.00; CD, N. 538 E., 1.32; DE, S. 570 E., 5.79. A rough measurement will at once shew that ABCD is less than an acre, and that ABCDE is more; hence the desired line will fall * The problem may also be performed by making the side on which the divi sion line is to fall, a Meridian, and changing the Bearings as in Art. (244). The difference of the new Departures will be the Departure of the Division line. Its position can then be easily determined, by calculations resembling those in Part VII, Chapter IV, Arts. (443), &c. t If the whole field has been surveyed and balanced, the balanced Latitudes and Departures should be used. We will here suppose the survey to have proved perfectly correct. 342 LAYING OUT AND DIVIDINB UP LANbD [PART XI. on DE. The Latitudes and Departures of AB, BC and CD are then found. From them the course AD is found to be N. 80 E., 3.63. The content of ABCD will be 3.19 square chains. Sub tracting this from one acre, the remainder, 6.81 sq. chs., is the content of ADZ. AP 3.63 x sin. 650 = 3.29. Dividing ADZ by half of this, we obtain DZ = 4.14 chains. By the Second Method, the Latitude and Departure of DA, the area of ABCD, and of ADZ, being found as before, we next find the Latitude and Departure of EA, from those of AD and DE, and thence the area of ADE = 9.53. Lastly, we have'he proportion 9.53: 6.81: 5.79: DZ= 4.14, as before. E. BY A LINE PASSING THROUGH A GIVEN POINT WITHIN THE FIELD. (519) To part off a t'iangle. Let P be a point within a field through which it is required to' iB run a line so as to part off from Fig. 354. the field, a given area in the form of a triangle. When the field is given on the // I \ ground or on a plat, the division / can be made by construction // / thus. From P draw PE, paral-,/ p lel to the side BC. Divide the // given area by half of the perpen-, / /, 1 dicular distance from P to AC, / /\ \ and set off the quotient from C /' / 2/' z_ to G. Bisect GC in H. On — / \ G A I HE describe a semi-circle. On C it set off EK = EC. JoinKH. K. Set off HL - HK. The line LM, drawn from L through P, will be the division line required.' If HK be set off in the contrary direction, it will fix another line L'PM', meeting CB produced, and thus parting off another triangle of the required content. Example. Let it be required to part off 31.175 acres by a fence passing through a point P, the distance PD of P from the * As some lines in the figure are not used in the construction, though needed for the Demonstration, the student should draw it himself to a large scale. CHAP. ii.] Parting off Land. 343 side BC, measured parallel to AC, being 6 chains, and DOC 18 chains. The angle at C is fixed by a "tie-line" AB= 48.00 BC being 42.00, and CA being 30.00. Ans. CL= 27.31 chains, or CL'/ 7.69 chains. The figure is on a scale of 20 chains to 1 inch = 15840. (520) If the angle of the field and the position of the point P are Fig. 55. given by Bearings or angles, proceed thus. Find the perpendicular distances, PQ and PR, from the given I point to the sides, by the formulas PQ==PC. sin. PCQ; and PR= \ PC. sin. PCR. Let PQ = q, PR ___ -=, and the required content= c. = a 1' c Then CL i /! ) Then V sin. LCM.Example. Let the angle LCM-== 820. Let it be required to part off the same area as in the preceding example. Let PC = 19.75, PCQ=170 301' PCR- 640 29'. Required CL. Ans. PQ = 5.94, PR = 17.82, and therefore, by the formula, CL= 27.31, or CL' —= 7.69; corresponding to the graphical solution. The figure is on the same scale. If the given point were without the field, the division line could be determined in a similar manner. (521) To part off a quadrilateral, Conceive the two sides of the field which the division line will intersect, Fig. 356. DA and CB, produced till they meet at a C point G, not shown in the figure. Calculate the triangle thus formed outside of the field. Its area increased by the required area, will be that of the triangle EFG. Then the problem is identical with that in the last article. The following example is that given in Gummere's Surveying. The figure A E represents it on a scale of 20 chains to I inch = 1 15840. 844 LAYING OUT AND DIVIDING UP LAND, [PART XI Example. A field is bounded thus: N. 140 W., 15.20; N. 701~ E., 20.43; S. 6~ E., 22.79; N. 86- W.V., 18.00. A spring within it bears from the second corner S. 75~ E., 7.90. It is required to cut off 10 acres from the West side of the field by a straight fence through the spring. How far will it be from the first corner to the point at which the division fence meets the fourth side? Ans. 4.6357 chains. (522) To part off any figure. Let it be required to part off from a field a certain area by Fig. 357. a line passing through a given /e point P within the field. Run / a guess-line AB through P.. —--- Calculate the area which it C parts off. Call the difference between it and the required area -- d. Let CD be the desired line of division, and let P represent the angle, APC or BPD, which it makes with the given line. Obtain the angles PAC = A, and PBD = B, either by measurement, or by deduction from Bearings. Measure PA and PB. Then the desired angle P will be given by the following formula. ot. P (cot. A + cot. B AP2 BP2~ AP2. cot. B BP2. cot. A - ~ -cot. A. cot. B + 2d t (cot. A + cot B- AP2 B P2) ] If the guess line be run so as to be perpendicular to one of the sides of the field, at A, for example, the preceding expression reduces to the following simpler form. Cot. P - -- (cot. B- Al) BP / AP2 t.Bc ot., B- AP - BPa2 2 2 d + c B 2 ) CAP. Ii.] Partiig off Lando 345 Excample. It was required to cut off from a field twelve acres by a line passing through a spring, P. A guess-line, AB, was run making an angle with one side of the field, at A, of 5550 and with the opposite side, at B, of 810. The area thus cut off was found to be 13.10 acres. From the spring to A was 9.80 chains, and to B 38.30 chains. Required the angle which the required line, CD, must make with the guess line, AB, at P. Ans. 200 45'; or - 886 25'. The heavy broken line, C('D' shows the latter. The scale of the figure is 10 chains to 1 inch = 1: 7920. If the given point were outside of the field, the calculations would be similar F. BY THE SHORTEST POSSIBLE LINE. (523) To part off a triangle. Let it be required to part off a triangular space, BDE, of given content, from the Fig.. 358. corner of a field, ABC, by the shortest possible line, DE. From B set off BD and BE each equal td /| /(2 BDE). The line DE thus obtained will be / V sin. B 7 perpendicular to the line, BF, which bisects the angle B. The length of DE = (2. DBE. sin. B) cos. ~ B /A C.Example. Let it be required to part off 1.3 acre from the corner of a field, the angle, B, being 300. Ans. BD = BE 7.21; and DE = 3.73 The scale of the figure is 10 chains to 1 inch ==1: 7920. G. LAND OF VARIABLE VALUE. (521) Let the figure represent a field in which Fig. 359. the land is of two qualities and values, divided by B C the 4" quality line EF. It is required to part off from it a quantity of land worth a certain sum, by E a straight fence parallel to AB. Multiply the value per acre of each part by its length (in chains) on the line AB, add the products, multiply the value to be set off by 10, divide A 346 LAYING OUT AND DIVIDING IP LAND. [PART XI. by the above sum, and the quotient will be the desired breadth, BC or AD, in chains. Example. Let the land on one side of EF be worth $200 pel acre, and on the other sile $100. Let the length of the former, BE, be 10 chains, and EA be 30 chains. It is required to part off a quantity of land worth $7500. Ans. The width of the desired strip will be 15 chains. The scale of the figure is 40 chains to 1 inch -- 1: 31680. If the' quality line" be not perpendicular to AB, it may be made so by " giving and taking," as in Art. (124), or as in the article following this one. The same method may be applied to land of any number of different qualities; and a combination of this method with the preceding problems will solve any case which may occur. H. STRAIGHTENING CROOKED FENCES. (525) It is often required to substitute a straight fence for a crooked one, sq that the former shall part off precisely the same quantity of land as did the latter. This can be done on a plat by the method given in Art. (83), by which the irregular figure Fig. 360. 2 1...2...3...4...5 is reduced to the equivalent triangle 1...5...3', and the straight line 5...3' therefore parts off the same quantity of land on either side as did the crooked one. The distance from 1 to 3', as found on the plat, can then be set out on the ground and the straight fence be then ranged from 3' to 5 The work may be done on the ground more accurately by running a guess line, AC, Fig. 361, across the bends of the fence which crooks from A to B, measuring offsets to the bends on each side of the guess line, and calculating their content. If the sums of these areas on each side of AC chanced to be equal, that would be the line desired; but if, as in the figure, it passes too far on one CHAP. xII] Dividing up Land.e 47 Fig. 361. — ( -....... side, divide the difference of the areas by half of AC, and set it off at right angles to AC, from A to D. DC will then be a line parting off the same quantity of land as did the crooked fence. If the fence at A was not perpendicular to AC, but oblique, as AE, then from D run a parallel to AC, meeting the fence at E, and EC will be the required line. CHAPTER III. DIIDITNG UP LAND. (526) MOST of the problems for " Dividing up" land may be brought under the cases in the preceding chapter, by regarding one of the portions into which the figure is to be divided, as an area to be " Parted off" from it. Many of them, however, can be most neatly executed by considering them as independent pro* blems, and this will be here done. They will be arranged, firstly, according to the simplicity of the figure to be divided up, and then sub-arranged, as in the leading arrangement of Chapter II, according to the manner of the division. DIVISION OF TRIANGLES. (527) By lines parallel to a sed Sup- Fig. 362. pose that the triangle ABC is to be divided into two equivalent parts by a line parallel to AC. The desired point, D, from which this line is to start, will be obtained by measuring BD = A1B, So, too, E is fixed by BE = BC /. 348 LYINtY OUT TAND IVIDNG UP LAND. [PART XL Generally, to divide the triangle into two parts, BDE and ACED which shall have to each other a ratio =-: n, we have BD= AB / Vn q- n This may be constructed thus. Describe a Fig. 363. semicircle on AB as a diameter. From B set ~~ -- B off BF= "-"- BA. At F erect a perpendi-' \ cular meeting the semicircle at G. Set off BG \ from B to D. D is the starting point of the divi- A c sion line required. In the figure, the two parts are as 2 to 8, and BF is therefore =2 BA. To divide the triangle ABC into five Fig. 364. equivalent parts, we should have, similarly, BD=AB V/; BD'=AB V —; BD" =AB V/; BD"'=AB V D E The same method will divide the trian- gle into any desired number of parts hav- 0" ing any ratios to each others A (528) By Unes perpendIiular to a sdle, Suppose that ABC is to be divided into two parts having Fi. 365. a ratio = m: n, by a line perpendicular to AC. Let EF be the dividing line whose position is required. Let BD be a perpendicular let fall from B to A E,. AC. ThenisAE = AC x AD X - In this figure, AFE EFBC:: m: n:: 1: 2. If the triangle had to be divided into two equivalent parts, the above expression would become AE = V/( AC x AD). (529) By lines rnuning in any gaven dlrection, Let a triangle, ABC, be given to be divided into two parts, having a ratio = m: n, by a line making a given angle with a side. Part off, as in Art (509) or (510), Fig. 350, an area BFG = +. ABC. + 7?t 1 CHAP. III.] Dividming up Landa 349 (539) By lines starting frio ag angie. Divide the side oppoa site to the given angle into the required num- Fig. 366. ber of parts, and draw lines from the angle' to the points of division. In the figure the triangle is represented as being thus divided into two equivalent parts. A^ If the triangle were required to be divided into two pats, having to each other a ratio in: n, we should have AD = AC m- n and DC = AC + n If" the triangle had to be divided into three Fig. 367. parts which should be to each other:: n: n:' we should have AD= AC Frm DE?m + n +- p - AC- ~,and EC = AC - D C m + n - p n + n + 7' Suppose that a triangular field ABC, had to be divided among five men, two of them to have a quarter each, and three of them each a sixth. Divide AC into two equal parts, one of these again into two equal parts, and the other one into three equal parts. Run the lines from the four points thus obtained to the angle B. (53I) ly Iaies starting from a point in a side, Suppose that the triangle ABC is to be divided into two Fig. 368 equivalent parts by a line starting from a point D in the side AC. Take a point E in the middle of AC. Join BD, and from E draw a\ parallel to it, meeting AB in F. DF will be A -~ Fc the dividing line required. The point F will be most easily obtained on the ground by the proportion AD::: AB: A F The altitude of AFD of course equals A ABC AL). If the triangle is to be divided into two parts having any other ratio to each other, divide AC in that ratio, and then proceed as Ab x AC mn before. Let this ratio = m: n, then AF = —. AD mn + n 8508 LANiLM OUT AND DIVIDING TP LAND. [PART x (. 2) Next suppose that the trian- Fig. 369. gle ABC is to be divided into three equivalent parts, meeting at D. The' altitudes, EF and GH, of the parts kDE and DCG, will be obtained by A..~~ D 1~ lividing I ABC, by half of the respective bases AD and DC. If one of -these quotients gives an altitude greater than that of the triangle ABG, it will shew that the two lines DE and DG would both cut the same side, as in Fig. 370, in Fig. 370. which EF is obtained as above, and GH = I ABGC - AD. In practice it is more convenient to de- termine the points F and G, by these \ proportions; F I KH D BK: AK: EF: AF; and BK: AK:: GH: AH. The division of a triangle into a greater number of parts, having any ratios, may be effected in a similar manner. (533) This problem admits of a more elegant solution, analogous to that given for the division into two Fig. 371." parts, graphically. Divide AC into \G three equal parts at L and M. Join BD, and from L and M draw paral- lels to it, meeting AB and BC in E ^ i and G. Draw ED and GD, which will be the desired lines of division. The figure is the same triangle as Fig. 369. The points E and G can be obtained on the ground by measuring AD and AB, and making the proportion AD: AB:: ~ AC: AE. The point G is similarly obtained. The same method will divide a triangle into a greater number of parts. (534) To divide a triangle into four equivalent triangles by lines terminating in the sides, is very Fig. 372. easy. From D, the middle point of AB, draw DE parallel to AC, and from F, the middle of AC, draw FD and FE. The problem is now solved. ^ F C CHAP. iii.] Dividing up Land. 351 (535) By lnes passing through a point wit iin the triangle. Let D be a given point (such as a well, Fig. 273. &c.) within a triangular field ABC, from which fences are to run so as to divide the triangle into two equivalent parts. Join AD. Take E in the middle of BC, A - ~ c and from it draw a parallel to DA, meeting AC in F. EDF is the fence required. (536) If it be required to di- l B vide a triangle into two equiva- Fig. 374. lent parts by a straight line pass- \ ing through a point within it, pro- / \ ceed thus. Let P be the given // point. From P draw PD paral- // lel to AC, and PE parallel to BC. //' Bisect AC at F. Join FB. From,/ 1 B draw BG parallel to DF. Then / / " bisect GC in H. On HE de- // scribe a semicircle. On it set off / I EK = EC. Join KH. Set off / / / \/ HL = IK. The line LM drawn. ~ c G'-A L 3 71 C from L, through P, will be the \'/ division line required. This figure is the same as that of Art. (519). The triangle ABC contains 62.35 acres, and the distance CL = 27.31 chains, as in the example in that article. (537) Next suppose that the trian- Fig. 375, gle ABC is to be divided into three equivalent parts by lines starting from / a point D, within the triangle, given by the rectangular co-ordinates AE and / and ED. Let ED be one of the lines K E of division, and F and G the other points required. The point F will be determined if AH is known; AH and HF being its rectangular co-ordinates. From B let fall the perpendicular BK on AC. 352 LAYING OUT AND DIVIDIMNG UP LANDS [PANR xi. mi ATT AK ( ABC - AE x ED) v ^. ^ Then is AH AK ABC - AE x3 ED) The position of the AE x BK- ED x AK' other point, G, is determined in a similar manner. (538) Let DB, instead of DE, Fig. 376 be one of the required lines of division. Divide ~ ABC by half \, of the perpendicular DHI, let fall from D to AB, and the quotient will be the distance BF. To find / G, if, as in this figure, the trian- At - I~ gle BDC (= BC x IDK) is less than ABC, divide the excess of the latter (which will be CDG) by I DE, and the quotient will be CG. Exazmple. Let AB = 30.00; BC = 45.00; CA - 50.00. Let the perpendiculars from D to the sides be these; DE - 10.00; DH -= 20.00; DKi = 5.17'. The content of the triangle ABC will be 666.6 square chains. Each of the small triangles must therefore contain 222.2 sq. chs., BD being one division line. We shall therefore have BF = 222.2 - IDH= 22.2 chains. BDC = 45 x } x 5.17-1 = 116.4 sq. chs., not enough for a second portion, but leaving 105.8 sq. chs. for CDG; whence CG = 21.16 chs. To prove the work, calculate the content of the remaining portion, GDFA. We shall find DGA = 144.2 sq. chs., and ADF = 78.0 sq. chs., making together 222.2 sq. chs., as required. The scale of Fig. 376 is 30 chains to 1 inch = 1: 23760. (539) The preceding case may Fig. 377. be also solved graphically, thus. Take CL= A AC. Join DL, and from B draw BG parallel to DL. Join DG. It will be a second line of division. Then take a point, M, in the middle of BG, and from A it draw a line, MF, parallel to DA. DF will be the third line of division. This method is neater on paper than the preceding; but less convenient on the ground. CHAP. Ilm.j Dividing up Lanao 353 (540) Let it be required to divide Fig. 378. the triangle ABC into three equiva- lent triangles, by lines drawn from the three angular points to some tu/known point within the triangle. This point is now to be found. On any A " side, as AB. take AD = - AB. From D draw DE parallel to AC. The middle, F, of DE, is the point required. If the three small triangles are not to be equivalent, but are to have to each other the ratios:: m: n:p, Fig. 379. divide a side, AB, into parts having these ratios, and through each point of division, D, B, draw a parallel to the side nearest to it. The intersection of these parallels, in F, is the cA point required. In the figure the parts ACF, ABF, BCF, are as 2: 3:4. (til) Let it be required to find Fig. 380. the position of a point, D, situated within a given triangle, ABC, and equally distant from the points A, B, C; and to determine the ratios to each other of the three triangles into A which the given triangle is divided. By construction, find the centre of the circle passing through A, B, C. This will be the required point. By calculation, the distance DA = DB = AB X BC x C 4 x area ABC The three small triangles will be to each other as the sines of their angles at D; i.e. ADB: ADC: BDC:: sin. ADB: sin. ADC: sin. BDC. These angles are readily found, since the sine of half of each of them equals the opposite side divided by twice one of the equal distances. 23 854 LAYiNG OUT AND DIVIDING UP LANDI [PART xI (542) By the shortest possible line. Let it be Fig. 381. required to divide the triangle ABC by the short- est possible line, DE, into two parts, which shall be to each other:::,m n; or DBE: ABC:: m/: m + n. D a, and A is an acute angle. C =180~ — (A - B).' c= sn.C i a smin.A CASE 3. —Given two sides and their included angle. Applying Theorem II. (obtaining the sum of the angles opposite the given sides by subtracting the given included angle from 180~), we obtain the difference of the unknown angles. Adding this to their sum we obtain the greater angle, and subtracting it from their sum we get the less. Then Theorem I. will give the remaining side. Calling the given sides a and 6,, and the included angle C, we have A+-B=-180~-C. Then tan. j (A -B) =tan. (A+ B). a —b sin. C (A+ B) - A A (A —B) =A. (A+B) —(A-B)= B. c=a i A sin. A. In the first equation cot. A C may be used in the place of tan. A (A + B). CASE 4.-Given the three sides. Let s represent half the sum of the three sides ~i (a + b + c). Then any angle, as A, may be obtained from either of the fol. lowing formulas, founded on Theorem III.: sin., k A / i(s - b) (s —c) sin. A - -- b] cos. A = The third should not be used when A is nearly 180~; nor the fourth whe A is nearly 90~; nor the fifth when A is very small, The third is the most convenient when all the angles are required. APPENDIX B. DEMONSTRATIONS OF PROBLEMS, ETC. MANY of the problems, &c., contained in the preceding pages, require Demonstrations. These will be given here, and will be designated by the same numbers as those of the Articles to which they refer, As many of these Demonstrations involve the beautiful Theory of Transversals, &c., which has not yet found its way into our Geometries, a condensed summary of its principal Theorems will first be given. TRANSVERSALS. THEOREM I.-If a straight line be drawn so as to cut any two sides of a triangle, and the third side prolonged, thus dividing them into six parts (the prolonged side and its prolongation being two of the parts), then will the product of any three of those parts, whose extremities are not contiguous, equal the product of the other three parts. That is, in Fig. 403, ABC being the triangle, and Fig. 403. DF the Transversal, BEXAD X F=EA X DC XBF. A To prove this, from B draw BG, parallel to CA. From the similar triangles BEG and AED, we have BG: BE:: AD: AE. From the similar triangles BFG and CFD, we have CD: CF:: BG: BF. Multiplying these proportions together, we have F _\ BGXCD: BEXCF:: ADXBG: AEXBF. Multi- B plying extremes and means, and suppressing the common factor BG, we have BExADXCF EA XDCxBF. These six parts are sometimes said to be in involution. If the Transversal passes entirely out- Fig. 404. side of the triangle, and cuts the prolonga- A tions of all three sides, as in Fig. 404, the theorem still holds good. The same demonstration applies without any change.* TEaEORE II.-Conversely: If three points be taken on two sides of a triangle, and on the third side prolonged, or on the prolon- / gations of the three sides, dividing them into six parts, such that the product of E three non-consecutive parts equals the product of the other three parts; then will these three points lie in the sasme str..h l'ne. This Theorem is proved by a Reductio ad absurdum. * This Theorem may be extended to polygons. 388 TRANSVERSALS. [APPr., THEOREM III.-If from the summits of a trzangle, lines Fig 405. be drawn, to a point situated either. within or without the A triangle, and prolonged to meet the sides of the triangle, or their prolongations, thus dividing them into six parts; then will the product of any three non-consecutive parts be 3 equal to the product of the other three parts. That is, in Fig. 405, or Fig. 406, F AE X.BF X CD =EB X FO X DA. rig. 406. For, the triangle ABF being cut by the transversal EC, gives the relation (Theorem I.), / AE X BC X FP EB X FC X PA. The triangle ACF, being cut by the transversal DB, gives / DC X FB X PA=AD X CB X FP. Multiplying these equations together, E and suppressing the common factors PA, CB, and FP, we have AE X BF X CD =EB X FC X DA. THEOREM IV.-Conversely: If three points are situated on the three sides of a triangle, or on their prolongations (either one, or three, of these points being on the sides), so that they divide these lines in such a way that the product of any three non-con secutive parts equals the product of the other three parts, then will lines drawn from these points to the opposite angles meet in the same point. This Theorem can be demonstrated by a Reductio ad absurdum. COROLLARIES OF THE PRECEDING THEOREMS. CR. 1. —The MEDIANS of a triangle (i. e., the lines drawn from its summits to the middles of the opposite sides) meet in the samepoint. For, supposing, in Fig. 405, the points D, E, and F to be the middles of the sides, the products of the non-consecutive parts will be equal, i. e., AE X BF X CD= DA XEB X FC; since AE = EB, BF = FC, CD = DA. Then Theorem IV. applies. COR. 2.-The BISSECTRICES of a triangle (i. e., the lines bisecting its angles),eeet in the same point. For, in Fig. 405, supposing the lines AF, BD, CE to be Bissectrices, we have (Legendre IV. 17): BF: FC: AB: AC, BF X A =F X AB CD: DA:: B: BA, whence CD X BA=DA X B, AE: EB:: CA: CB, AE X CB = EB X CA. Multiplying these equations together, and omitting the common factors, we have BF X CD X AE =- F X DA X EB. Then Theorem IV. applies, APP. B.] TRANSVERSALS, 389 CoR. 3.-The ALTITUDES of a triangle (i. e., the lines arawn from its summits perpendicular to the opposite sides) meet in the same point. For, in Fig. 405, supposing the lines AF, BD, and CE, to be Altitudes, we have three pairs of similar triangles, BCD -and FCA, CAE and DAB, ABF and EBC, by comparing which we obtain relations from which it is easy to deduce BF XCD XAE EB XFC XDA; and then Theorem IV. again applies. COR. 4.-If, in Fig. 405, or Fig. 406, the point F be taken in the middle of BC, then will the line ED be parallel to B3C. For; since BF = FC, the equation of Theorem III. reduces to AE X CD=EB XDA; wrience AE. EB: AD: DC' consequently ED is parallel to BC. Con. 5.-Conversely: If El be parallel to BC, then is BF = FC. For, since AE: EB:: AD: D, we have AE X DC = EB X AD; whence, in the equation of Theorem III., we must have BF = FC. Con. 6.-From the preceding Corollary, we derive the following: If two sides of a triangle are divided proportionally, Fig. 407. starting from the same summit, as A, and lines are drawn A from the extremities of the third side to the points of divisiosn, the intersections of the corresponding lines will all lie in the same straight line joining the summit A, and the middle of the base. Con. 7.-A particular case of the preceding corollary is this: ---- B p C In any trapezoid, the straight line which joins the intersection of the diagonals and the point of meeting of the non-parallel sides produced, passes through the middle of the two parallel bases. CoRn 8.-If the three lines drawn through the corresponding summits of two triangles cut each other in the same point, then the three points in which the corresponding sides, produced if necessary, will meet, are situated in the same straight line. This corollary may be otherwise enunciated, thus: If two triangles have their summits situated, two and two, on three lines which meet in the same point, then, &c. This is proved by obtaining by Theorem I. three equations, which, being multiplied together, and the six common factors cancelled, give an equation to which Theorem II. applies. Triangles thus situated are called homologic; the common point of meeting of the lines passing through their summits is called the centre of homology; and the line on which the sides meet, the axis of homology. 890 HlARMONIC DIVISION. [APP. HARMONIC DIVISION. DEFINITIONS.-A straight line, AB, is said to Fig. 08. be harmonically divided at the points 0 and D, I --- - when these points determine two additive seg- A B ments, AC, BC, and two subtractive segments, AD, BD, proportional to one another; so that AC: BC:: AD: BD. It will be seen that AC must be more than BC, since AD is more than BD.* This relation may be otherwise expressed, thus: the product of the whole line by the middle part equals the product of:the extreme parts. Reciprocally, the line DC is harmonically divided at the pointsiB and A; since the preceding proportion may be written DB: CB: DA: C A. The four points, A, B, C, D, are called harmonics. The points 0 and D are called harmonic conjugates. So are the points A and B. When a straight line, as AB, is divided harmonically, its half is a mean proportional between the distance from the middle of the line to the two points, C and D, which divide it harmonically. If, from any point, 0, lines be drawn so as to Fig. 409. divide a line harmonically, these lines are called 0 an harmonic pencil. The four lines which compose it, OA, OC, OB, OD, in the figure, are called its radii, and the pairs which pass through the conjugate points are called conjugate radii. / A C B D THEOREM V.-In any harmonic pencil, a line drawn parallel to any one of the radii, is divided by the three other radii into two equal parts. Let EF be the line, drawn parallel to Fig. 410. OA. Through B draw GH, also parallel 0 to OA. We have, GB: OA:: BD: AD; and BH: OA:: BC: AC. / X But, by hypothesis, AC: BC:: AD: BD.,/ \ / \ Hence, the first two proportions reduce to /A — GB = BH; and consequently, EK = KF. / The Reciprocal is also true; i. e., If four lines radiating from a point are such that a line drawn parallel to one of them is divided into two equal parts by the other three, the four lines form an haro monic pencil. * Three numbers, m, n, p, arranged in decreasing order of size, form an 7larmonic proportion, when the difference of the first and the second is to the difference of the second and the third, as the first is to the third. Such are the numbers 6, 4, and 3; or 6, 8, and 2; or 15, 12, and 10; &c. So, in Fig. 408, are the lines AD, AB, and AC, which thus give BD: CB:: AD: AC; of AO: CB:: AD: BD. The series of fractions, -, - -,, -, &c., is called an harmonic proyres. sion, because any consecutive three of its terms form an harmonic proportion. APP. B.] THE COMPLETE QUADRiILTERALo 391 THEOREM V. —If any transversal to an harmonic pencil be drawn, it will be divided harmonically. Let LM be the transversal. Through K, where LM intersects OB, draw EF parallel to OA. It is bisected at K by the preceding theorem; and the similar triangles, FMK and LMO, EKN and LNO, give the proportions LM: KM:: OL: FK, and LN: NK:: OL: EK; whence, since FK = EK, we have LN: NK:: LM: KM. COROLLARY. — The two sides of any aznle, together with the bissectrices of the angle and of its supplement, form an harmonic pencil. THEOREM VII.-If, from the summits of any Fig. 411. triangle, ABC, through any point, P, there be drawn the transversals AD, BE, CF, and the trans- versal ED be drawn to meet AB prolonged, in F', the points F and F' will divide the base AB har- /. mnoically. __-,\: A F B F' This may be otherwise expressed, thus: The line, CP, which joins the intersection of the diagonals of anq quadrilateral, ABDE, with the point of meeting, C, of two opposite sides prolonged, cuts the side AB in a point F, which is the harmonic conjugate of the point of meeting, F', of the other two sides, ED and AB, prolonged. For, by Theorem I., AF' X BD X CE =F'B X DC X EA; and by Theorem III., AF X BD X CE - FB X DC X EA; whence AF: FB:: AF': F'B. THE COMPLETE QUADRILATERAL. A Complete Quadrilateral is formed by Fig. 412. drawing any four straight lines, so that each E of them shall cut each of the other three, so. \ as to give six different points of intersection. / It is so called because in the figure thus formed are found three quadrilaterals; viz.,. \ in Fig. 412, ABCD, a common convex quadri- / - lateral; EAFC, a uni-concave quadrilateral; r, and EBAFD, a bi-concave quadrilateral, com- / posed of two opposite triangles. The complete quadrilateral, AEBCDF, has \ N three diagonals; viz., two interior, AC, BD; and one exterior, EF. THEOREM VIII.-In every COMPLETE QUADRILATERAL the middle points of its three diagonals lie inl the same straightdine. AEBCDF is the quadrilateral, and LMN the middle points of its three diago. nals. From A and D draw parallels to BC, and from B and 0 draw parallels to 392 THE COMPLETE QUJADRILATERAL. LAPP. B1 AD. The triangle EDO being cut by the transversal BF, we have (Theorem I.), DF X CB X EA = F X EB X DA. From the equality of parallels between parallels, we have CB = E'B', EA = CA', EB - DB', DA = E'A'. Hence, the above equation becomes DF X E'B' X CA'= CF X DB' X E'A'; therefore, by Theorem II., the points, F, B', A', lie in the same straight line. Now, since the diagonals of the parallelogram ECA'A bisect each other at N, and those of the parallelogram EBB'D at M, we have EN: NA':: EM: MB'. Then MN is parallel to FA'; and we have EN: NA':: EL: LF, or EL = LF, so that L is the middle of EF, and the same straight line passes through L, M, and N. THEOREM IX.-In every complete quadrilateral each of the three diagonals is divided harmonically by the two others. CEBADF is the complete quadrilateral. Fig. 413. The diagonal EF is divided harmonically at A G and H-by DB and AC produced; since AH, DE, and FB are three transversals drawn from the summits of the triangle AEF through the same point C; and there- fore, by Theorem VII., DBGI and ACH di- / vide EF harmonically. _ —~-. So too, in the triangle ABD, CB, CA, CD, E are the three transversals passing through C; and G and K therefore divide the diagonal BD harmonically. So too, in the triangle, ABC, DA, DB, DC are the transversals, and H and K the points which divide the diagonal AC harmonically. THEOREM X.-If from apoint, A, any num- Fig. 414. ber of lines be drawn, cutting the sides of an A angle POQ, the intersections of the diagonals of the quadrilaterals thus formed will all lie in the same straight line passing through the B summit of the angle. / By the preceding Theorem, the diagonal BC' of the complete quadrilateral, BAB'C'CO, C C' C" Q is divided harmonically at D and E. Hence, OA, OP, OD, and OQ, form an harmonic pencil. So do OA, OP, OD', and OQ. Therefore, the lines OD, OD' coincide. So for the other intersections. If the point A moves on OA, the line OD is not displaced. If, on the contrary, OA is displaced, OD turns around the point 0. Hence, the point A is said to be a pole with respect to the line OD, which is itself called the polar of the point A. Similarly, D is a pole of OA, which is the polar of D. OD is likewise the polar of any other point on the line OA; and this property is necessarily reciprocal for the two conjugate radii OA, OD, with respect to the lines OP, OQ, which are also conjugate radii. Hence; In every harmonic pencil, each of the radii is a polar with respect to each point of its conjugate; and each point of this latter line is a pole with respect to the former. D E E0N S T AT I N S.* PART II.; CHAPTER V. (1.40)9 (141) The equality of the triangles formed in these methous provea their correctness. (14), (X. 4) These methods depend on the principle of the square of the bypothenuse. (146) CAD is an angle inscribed in a semicircle. (146) Let fall a perpendicular from B to AC, meeting it at a point E, not marked in Fig. 91. Then (Legendre, IV. 12), AB = AC2 + BCA - 2 AC. CE; whence CE= C2 ACBC2 When AC = AB, this becomes CE - 2 A The similar triangles, BCE and DCA, give EC CB:: AC: CD; whence CB X AC BC2 2 AC2 CD= =CB x AC- - CE 2AC BC (147) Mark a point, G, in the middle of DF, and join GA. The triangle AGD will then be isosceles, since it is equal to the isosceles triangle ABC, having two sides and the included angle equal. Then AG = GD -= AB = GF. The triangle AGF is:hen also isosceles. Now the angle FAG = 4 AGD; and GAD FGA. Therefore FAG + GAD = FAD = i (AGD + FGA) = (1800) =90~. (149) See Part VII., Art. (403). (1][0) The proof follows from the equal triangles formed. (151) The proof is found in the first half of the proof of Art. (146). (153) ACP is an angle inscribed in a semicircle. (154) Draw from C a perpendicular to the given line, meeting it at a point E. AC2 As in the proof of Art. (146), changing the letters suitably, we have AE = 2 A 2 AB' The similar triangles AEC and ADP give AP AP AC2 AP X AC AC: AE:: AP: AD =A- X AE =- X A- 2 AB (155) Similar triangles prove this. (15.6) The equal triangles which are formed give BP = CF. Hence FP is parallel to BC, and consequently perpendicular to the given line DG. (157) The proof of this is found in the " Theory of Transversals," corollary 8. (153) The proof of this is the same as the last. (161) The lines are parallel because of the equal angles formed. * Additional lines to the figures in the text will sometimes be employed. The student should draw them on the figures, as directed. 394 PEIfO0N$STATiOH~$ I APP. D. (162) The equal triangles give equal angles, and therefore parallels. (163) AB is parallel to PF, since it cuts the sides of the triangle proportionally. (164) The proof is found in corollary 4 of " Transversals." (1]65) From the similar triangles, CAD and CEP, we have CE: CD:: CP: CA. From the similar triangles, CEF and CBD, we have CE: CD:: CF: CB. These two proportions give the following; CP: CA:: CF: CB. Therefore PF is parallel to AB. (166) Draw PE. The similar triangles PCE and ACTD give PE I CE.: AD CD. The similar triangles CEF and CDB give EF: CE:: DB: CD. These proportions produce PE: EF:: AD: DB. Hence PEF is similar to ADB, and PF is parallel to AB. (1872) The equality of the symmetrical triangles which are formed, proves this method. (I74) ABP is a transversal to the triangle CDE. Then, by Theorem 1. of "Transversals," CA X EB X DP = AE X BD X CP; whence we have CP: DI':: CA XEB: AE X BD. By "division," CP —DP: DP:: CA X EB -AE X BD: AE X BD. DC XAE X BD Hence, since CP - DP = CD, we obtain DP = B CA X EB —AE X BD' The other formulas are simplified by the common factors obtained by making AE = AC, or BE = BD. (17T) By Theorem VII. " Harmonic Division," in the quadrilateral ABED, the line CF cuts DE in a point, L, which is the harmonic conjugate of the point at which AB and DE, produced, would meet. So too, in the quadrilateral DEHK, this same line, CG, produced, cuts DE in a point, L, which is the harmonic conjugate of the point at which DE and KH, produced, would meet. Consequently, AB, DE, and KH must meet in the same point. Otherwise; this problem may be regarded as the converse of Theorem X. of " Transversals," BCA being the angle, and P the point from which the radiating lines are drawn. (1t6) EGCFDH is the "Complete Quadrilateral." Its three diagonals are FE, DC, and HG; and their middle points A, B, and P lie in the same straight line, by our Theorem VIII. (l~2) This instrument depends on the optical principle of the equality of the angles of incidence and reflection. (1~4) The first method given, Fig. 120, is another application of the Theory of Transversals. The second method in the article is proved by supposing the figure to be constructed, in which case we should have a triangle QZR, whose base, QR, and a parallel to it, BD, would be cut proportionally by the required line PSZ; PD X QP so that QR: BD: QP: BS _ Q (1~9) By "Transversals," Theorem I., we obtain, regarding CD as the transversal of the triangle ABE, CB X AF XED = AC X FE X I)B; and since ED = DB, this becomes CB X AF = AC X FE; whence the proportion CB: AC:: FE: AF. By "division," we have CB- AC: AC:: FE AF: AF. Observing that AC AABC we obtain AB - CB - AC =AB, we obtain AB (FE AF). AF APP. B.] For Part 1l,, Chapter V. 395 (190) Take CH = CB; and from B let fall a perpen- g. 124, bis. dicular, BK, to AO. Then, in the triangle CBH, we have A ~(Legendre IV. 12), - I CI + BH2 -- BBC' B[IV HK= -= [n1 2 CH 2B' [1]C since CH =BC. In the triangle ABHI, we have (Leg. IV. 13) \ D F AB = AH2 + BH2 + 2 AH. HK. Substituting for HK, its value from [1], we get AB'=AA + BH 1 + 21 But A AC - CI = AC -BC AB2A11 El-I2x~+ (,+ AC - BC AC.. AB = AH + I-\l (1 C- + )=AH 2+ BHI. -. [2] In the above expression for AB, BH is unknown. To find it, proceed thus. Take CF = CD. Then DF is parallel to BH; and we have CD: CB:: DF: BH; whence CBa BH2 DF2. -. [3] In this equation DF' is unknown; but by proceeding as at the beginning of this CE investigation, we get an equation analogous to [2], giving ED2 EFa - DF. C; CD whence DF' = (DE'- EF2).. Substituting this value of DF' in [3], we have BH2 - (DE'- EFa)CD C Substituting this value of BHI in [2], we have ACxYBC ACXBC AB'=AHI2+(DE'- EF2) = (AC -BC)'+[DE2- (CE-CD)2] X CDX CE CDCE (191) Since BCD is a right angle, AC is a mean proportional between AB and AD. (92') rhe proof follows from the similar triangles constructed. (193) The similar triangles give DE: AC:: DB: AB; whence, by "division," DE-AC: AC::DB-AB:AB; whence, since DB-AB= AD, we have AC XAD A=DE-AC' (194) From the similar triangles, we have DE: CA:: EB AB; whence DE —CA: CA:: EB -AB: AB; whence, since EB-AB- AE, we get AC X AE =DE — AC' (195) The triangles DEF and BAF, similar because of the parallelogram which EDXAF ACXAF is constructed, give FE: ED:: AF: AB = FE- - iE' ACXDC The triangles DEF and BCD give similarly FE: ED::DC: CB = FE Jb^ 696 DEIMI$STggT IONS APP. (196) The equality of the triangles formed proves this problem. (1EL9) The proof of this problem also depends on the equality of the trianglea constructed. The details of the proof require attention. (19~) EB is the transversal of the triangle ACD. Consequently, CBXAFXDE ABXFD X CE; or, since CB = AB+AC, (ABAC) XAFXDE=ABXFD X CE: ACXAFXDE whence AB ACxAFxDE FDXCE-AFXDE' Taking E, in the middle of CD, CE = DE, and those lines are cancelled. Taking F in the middle of AD, AF = FD, and those lines are cancelled. (199) The line BE is harmonically divided at the points H and A, from Theorem X,, ECFBGD being a "Complete QuadriIateral." Consequently, AE: EH:: AB:HB. Hence, by "division," AE- EH AE:: AB - HB: AB. We therefore have, AE x AH since AB - HB = AH, AB = A- EAH AE - EHI (200) For the same reasons as in the last article, CF is harmonically divided at H and D; and we have CH: HF:: CD: DF; whence CH —HF: CH:: CD - DF: CD. Hence, since CD- DF = CF, CD = CI I- F CH -HF' The other two expressions come from writing CF as CH + HF, and HF as CF - CH. (201) The equality of the triangles formed proves the equality of the corresponding sides KD and DE, &c. (202) The similar triangles (made so by the measurement of CE) give ACXDE CD: DE:: CA: AB = CD (203) The similar triangles (made so by the parallel) give CE: EA:: CD: AB CDXEA CDX (AC-CE) CLGE CE (204) The similar triangles DFH and BCD give HF: FD:: DC:BC = F GH The similar triangles FGH and ABC give FG: GH:: BC: AB =BC G. DF X CD X GII Substituting for BC, its above value, we have AB = FHX FXG FH X FGWhen CD = CE, DF = CD, whence the second formula. (205) The equality of the symmetrical triangles which are formed, proves the equality of A'B' to AB. (206) The proof of this is similar to the preceding. (29f) Because the two triangles ABC and ADE have a common angle at A, we have ADE: ABC:: AD X AE: AB X AC; whence the expression for ABC. (20~) From B let fall a perpendicular to AC, meeting it at a point B'. Call this perpendicular BB' = p. From D let fall a perpendicular to AC, meeting it at a point D'. Call this perpendicular DD' = q. [APP. B.] For Part V, 39 The quadrilateral ABCD = AC X ~ (p -+ q). 2. BCE The triangle BCE = CE X p p; whence p = CE The similar triangles EDD' and BEB' give p::: BE: DE, whence DE 2.BCE X DE q-P BBE= CEXBE' RBCE BCE XDE BEDE DE Then (p +)=B + -C 9E BCE X + BCE X CT CEX BE CEXBE CEXBE' BD ACXBD Lastly, ABCD =AC X BCE X C BE =BCE X EXBE DEMONSTRATIONS FOR PART V. (3~9) Let B=-the measured inclined length, = this length reduced to a horizontal plane, and A =the angle which the measured base makes with the horizon. Then b B. cos. and the excess of B over b, i e., B -b = B (1 - cos. A). Since 1- cos. A = 2 (sin. ~ A)2 [Trigonometry, Art. (9)], we have B - b = 2 B (sin. i A)2. Substituting for sin. i. A, its approximate equivalent, i A < sin. 1' [Trigonometry, Art. (5)], we obtain B1 —b = 2 B (i A X sin. 1')2 = (sin. 1'). A. B, = 0.00000004231 A2 B. By logarithms, log. (B -6) -= 2.626422 4- 2 log. A + log. B. The greater precision of this calculation than that of b = B. cos. A, arises from the slowness with which the cosines of very small angles increase or decrease in length. (3S6) The exterior angle LER = LCR + CLD. Also, LER -LDR + CRD..'.LCR+C-LD =LDR+- CRD, and LCR — LDRCRD-CLD. CD From the triangle CRD we get sin. CRD = sin. CDR X CD From the triangle CLD we get sin. CLD = sin. LDC X L. As the angles CRD and CLD are very small, these values of the sines may be called the values of the arcs which measure the angles, and we shall have CD CD LCR = LDR + sin. CDR X -- sin. LDC X C CR CL' The last two terms are expressed in parts of radius, and to have them in seconds,they must be divided by sin. i' [Trigonometry, Art. (5), Note], which gives the formula in the text. Otherwise, the correction being in parts of radius, may be brought into seconds by multiplying it by the length of the radius in seconds; i. e., 180~ X 60 X 60 3. X 6,X 6 = 206264".80625 [Trigonometry, Art. (2) ]. 3.14159, &cQ (391) The triangles AOB, BOC, COD, &c., give the following proportions [Trigonometry, Art. (12), Theorem I.]; AO: OB:: sin. (2): sin. (1); OB: OC:: sin. (4): sin. (3); OC: OD: sin. (6): sin. (5); and so on around the polygon. Multiplying together the corresponding terms of all the proportions, the sides will all be cancelled, and there will result 1: 1:: sin (2) X sin. (4) X sin. (6) X sin. (8) X sin. (10) X sin. (12) X sin. (14) sin (1) X sin. (3) X sin. (5) X sin. (7) X sin. (9) X sin. (11) X sin. (13). Hence the equality of the last two terms of the proportion. 398 DEMINSTRATIONS LAPPr., DEMONSTRATION FOR PART VI. (399) In the triangle ABS, we have AB. sin. BAS co sin. IT' sin. ASB: sin. BAS: AB: SB -_. sin. iBAS. sin. sin. ASB sin. S 1 In the triangle CBS, we have n. BSC: asi. BOS -B: SB - =BO. sin.BCS a. sin. V sin. BSC in S' [2] c. sin. U a. sin.. Hence, S - i; whence, c. sin. S'. sin. U -a. sin. S. sin. V=0. [31 sin. S sin. S' In the quadrilateral ABCS, we have BCS 360~ -ASB-BSC-ABC-BAS; or V= 360~ -S-S'- B-U. Let T= 360~ - - S'- 1B, and we have V = T-U. [43 Substituting this value of V, in equation [3], we get [Trig., Art. (8)], c. sin S' sin. U- a. sin. S (sin. T. cos. U- cos. T. sin. U) =0. Dividing by sin. U, we get c. sin. S'- a. sin. S (sin. T. o'. U- cos. T) =0. \ sin. U 1 Whence we have cos. U c. sin. S'+ a. sin. S. cos. T - - =cot. U = sin. U a. sin. S. sin. T Separating this expression into two parts, and cancelling, we get. sin. S' cos. T a. sin. S. sin. T + sin. T' Separating the second member into factors, we get cos. T / c.sin. S' cot. U = sin. T a. sin. S. cos. T + 1 cot. U cot. T ( i. sin + 1) a. sin. S. cos. T Having found U, equation [4] gives V; and either [1] or [2] gives SB; and SA and SC are then given by the familiar " Sine proportion" [Trig, Art. (12)]. APP. B.] For Part VII, 399 DEMONSTRATIONS FOR PART VII. CP (403) If APC be a right angle, C = cos. CAB [Trigonometry, Art. (4)]. (405) AC = PC. tan. APC; and CB = PC. tan. BPC [Trigonometry, Art. (4)} Hence AC: CB:: tan. APC: tan. BPC; and AC: AC + B:: tan. APC: tan. APC + tan. BPC. tan. APC Consequently, since AC + CB = AB, AC - AB. tn tan, APO-O tan, FPO, (414) The equal and supplementary angles formed prove the operation. (421) In Fig. 285, CA:EG:: AB: GB. By "division," CA-EG: EG:: AB —GB: GB. Hence, observing that AB —GB -AG, we shall have A GB (CA —EG) AG= E e EG (423) Art. (12), Theorem III., [Trigonometry, Appendix A,] gives cos. C= 2; or2 = a + b - 2 ab. cos. C. This becomes [Trig., Art (6)], K being the supplement of C, = a2 +- b2 + 2 ab o cos. K. The series [Trig. Art. (5)] for the length of a cosine, gives, taking only its first two terms, since K is very small, cos. K = 1- K2. Hence, c a2 +b+ 2 ab - ab K = (a + b)'-ab K2= (a + b) ( ab ); whence, c =(a + b) - b Developing the quantity under the radical sign by the binomial theorem, and neg. lecting the terms after the second, it becomes ab K2'1-A-~ -(, &c. Substituting for K minutes, K. sin. 1' [Trig., Art. (5)], and performing the multiplication by a + b, we obtain c = ^a + b - abK. Now (sin. 0.0000000423079; 2(a+ ) 2 ab Ka whence the formula in the text, c a + b - 0.000000042308 X a.-. b (430) In the triangle ABC, designate the angles as A, B, C; and the sides op. posite to them as a, b, c. Let CD = d. The triangle BCD gives [Trig., Art. (12), sin. BDDC sin. ADG Theorem I], a= d si. D The triangle ACD similarly gives b d -sin. ADC sin. CBD' sin. CAD' In the triangle ABC, we have [Trig., Art. (12), Theorem II.], tan.: (A-B): cot. A C:: a-b: a +b; a -- b whence tan. - (A — B) = a- - cot. A C. [1] b a a a a a b Let K be an auxiliary angle, such that b = a. tan. K; whence tan. K = -, 400 DEMONSTRATIONS [APP. B. Dividing the second member of equation [1], above and below, by a, and substitub 1 -- tan. K ting tan. K for b, we get tan. i (A-B) =1 -tan K cot. i C. a I -- tan. K Since tan. 45~ = 1, we may substitute it for 1 in the preceding equation, and tan. 450 -- tan. K we get tan. j (A —B)= 4cot. j C. tan. 450 ~ tan. IK From the expression for the tangent of the difference of two arcs [Trig., Art. (8)], the preceding fraction reduces to tan. (45~ - K); and the equation becomes tan, - (A- B) = tan. (45~ - K) cot, CG, [2] In the equation tan. K =-, substitute the values of b and a from the formulas a at the beginning of this investigation. This gives sin. ADC sin. BDC sin. ADO. sin. CBD tan.. K = d..- J. dsin. CAD' sin. CBD sin. CAD. sin. BDC' (A - B) is then obtained by equation [2]; (A + B) is the supplement of C therefore the angle A is known. a. sin. C d. sin. BDC. sin. ACB Then c =AB ^~Then c=AB sin. A sin. CBD. sin. CAB The use of the auxiliary angle K, avoids the calculation of the sides a and b. (4S4) In the figure on page 292, produce AD to some point F. The exterior angles, EBC=A+-P; ECD =A+Q; EDF=A+ R. The triangle ABE BE sin, A CE sin. A gives - = -. The triangle ACE gives - = i Dividing member a sin. P a + x sin. m BE a. sin. Q by member, we get = (a sin. CE (a + x) sin.P' BE sin. (A q- R) In the same way the triangles BED and CED give b s=i (R P); b +- x sin. (R P) CE sin. (A + R) BE (b + x) sin. (R Q) and - sin. (A Whence as before, - b sin. (R -Q) CE b. sin. (R — P) Equating these two values of the same ratio, we get a. sin. Q (b + x) sin. (R - Q) - -~~. ~~ ~ ~- =; and thence (a +- x) sin. P b sin. (R - P) ab. sin. Q. sin. (R - P) (a + x) (b + x) - ab + (a + b) x + qx sin. P. sin. (R - Q) To solve this equation of the 2d degree, with reference to x, make t 4 ab sin. Q (sin. R — P) tan?2 K =: I (a — b)2 sin. P (sin. R Q) Then the first member of the preceding equation = ~ (a - b)2 X tan.2 K and we get 2 + (a + b) x = 4 (a -b). tan. K - ab, and x = ( a + b)::: [ (a - b) tan. K ab + (a + b)2] = — (+ 6b) ~: /[4 (ab) _ ~.tan. + 4 (a -- )] =- (a + b) i (a -b) V (tan. K + 1). Cr1 asi+ b a - 1 Or, since / (tan.2 K + 1) = secant K o we have x 2 2. Cos. K ye ha e 2 ~ 2. cos. K' APP. iB. For Part XI. 401 DEMONSTRATIONS FOR PART XI. (49g) The content being given, and the length to be n times the breadth Breadth X i times breadth = content; whence, Breadth =/ (on ). Given the content = c, and the difference of the length and breadth — d; to find the length 1, and the breadth b. We have I X b = o; and I-b -=, From these two equations we get Z = - d + - / (d' -+- 4 c). Given the content = c, and the sum of the length and breadth = s; to find I and b, We have I X b = c; and I+ b = s; whence we get l =I s -+ - (s2 - 4 c). (4194) The first rule is a consequence of the area of a triangle being the product of its height by half its base. To get the second rule, call the height h; then the base = mh; and the area ih X mhk; whence h = 2 X aea) For the equilateral triangle, calling its side e, the formula for the area of a triangle [ (~ s ) (A s - a) (~ s b)( s-c)] reduces to i e2 v3. Hence e = 2- (/3 ) =1.5191 / area. (495) By Art. (65), Note, a. AB X BC X sin. B = content of ABC; whence, 2 X ABC ~AB. sin. B' (496) The area of a circle = radius X 22; whence radius=,( X area 22 (497) The blocks, including half of the streets and avenues around them, are 900 X 260 = 234000 square feet. This area gives 64 lots; then an acre, or 43560 feet, would give not quite 12 lots. (502) The parallelogram ABDO being double the triangle ABC, the proof for Art. (495), slightly modified, applies here. (504) Produce BC and AD to meet in E. Fig. 846, bis. By similar triangles, ABE: DCE:: AB2: DO2.C ABE -DCE: ABE:: AB2 -DC: AB' F Now ABE - DCE = ABCD; also, by Art. (65), Note, sA=A in. A. sin. B ABE = A 2. si. (+B)' A The above proportion therefore becomes ABCD: AB2. sin. A. sin. B ABT-CD: A13 2. sin. (A + B) - Multiplying extremes and means, cancelling, transposing, and extracting the square 2.ABCD.sin. (A+B)J root, we get CD i [A 2. A. sin. B+ ) 26 402 DEMiONSTRITIONS [APP. B When A -+ B > 180~, sin. (A + B) is negative, and therefore the fraction in which it occurs becomes positive. CF being drawn parallel to DA, we have sin. B sin. B sin. B AD =FC - FB. =FB.. -- (AB —CD) sin. BCF sin. (180 - B) sin. (1A - B)) sin. A BC = (AB - CD). s~n' sin. (A + B)' (505) Since similar triangles are as the squares of their homologous sides BPD)E: BEGa:: BD2 BrF2; whence BF =-BD (/D(Bi ). (506) BFG=-.BF X FG =.BF X BF.tan. B; whence, BF /(2 tan. B sin. B. sin. F (510) By Art. (65), Note, BFG = BF. 2i. in. F 2. sin. (B+-F).BF whence, BF 2. sin. (B-~-'~ ).F whence B y sin. B. sin. F!' (511) The final formula results from the proportion FAE CDE:: AE: ED2. (512) Since triangles which have an angle in each equal, are as the products od the sides about the equal angles, we have ABE: CDE:: AE X BE: CE X DE. sin. A. sin. B sin. B ABE - ~.B. AE A AE-B.s sin. (A + B)' sin. E' sin. A sin. CDE BE = AB-. CE - DE. sin. E' sin. DOE Substituting these values in the preceding proportion, cancelling the common factors, observing that sin. (A + B) = sin. E, multiplying extremes and means, and //2. ODE. sin. DOES dividing, we get DE sin. E. sin. CDE! (515) The first formula is a consequence of the expression for the area of a triangle, given in the first paragraph of the Note to Art. (65). (51i) The reasons for the operations in this article (which are of very frequent occurrence), are self-evident. (51) The expression for DZ follows from Art. (65), Note. The proportion in the next paragraph exists because triangles having the same altitude are as their bases. (519) By construction, GPC = the required content. Now, GPC = GDC, since they have the same base and equal altitudes. We have now to prove that LMC = GDC. These two triangles have a common angle at C. Hence, they are to each other as the rectangles of the adjacent sides; i. e., GDC: LMC:: GO X CD:: LC X CM. Here CM is unknown, and must be eliminated. We obtain an expression for it by means of the similar' triangles LCM and LEP, which give LE: LC:: EP = CD: CM APP. B.] For Part Xl. 40o CD X LO Hence, CIM =.X Substituting this value of CM in the first proportion, LB and cancelling CD in the last two terms, we get LC2 GDC:LMC:: GC':; or GDO: LMC:: GC X LE: LC2. LE' LC = (LH + HC) = LH2 + 2 LH X HC + HC2. But, by construction, LHR = HK =H HE- EK2 = HE'- ECI 2 (HE- EC) (EII-EC) = I-I0 (HE-re). Also, GC = 2 HC; and LE =LH + HE. Substituting these values in the last proportion, it becomes GD: LM:: 2. HC (LH + HE): HG (HE —EC) + 2 LH X HC + HC2.:: 2 LH + 2 HE: HE- EC + 2 LH - H.:HE — EC + 2 LH + HE + EC.: 2 HE + 2 LH. The last two terms of this proportion are thus proved to be equal. Therefore, the first two terms are also equal; i. e., LMCG -GDC = the required content. Since HK = / (HE - EIK), it will have a negative as well as a positive value. It may therefore be set off in the contrary direction from L, i. e., to L'. The line drawn from L' through P, and meeting CB produced beyond B, will part off another triangle of the required content. (620) Suppose the line LM drawn. Then, by Art. (65), Note, the required content, c = ~ CL X CM. sin. LCM. This content will also equal the sum of the two triangles LGP and MGIP; i. e., c = — CL X p + CM M >X q. The first of 2 c these equations gives CM = sin. LM Substituting this in the second equation, we have cq X + CL. sin. LCM' Whence, - p. CL. sin. LCM +- cg = c. CL. sin. LCM. Transposing and dividing by the coefficient of CL2, we get 2 cQ 2 cq CL -—. GL —- - p p. sin. CLM' CTC2 /^ 2 e^ C p v p. sin. LOGM If the given point is outside of the lines CL and CM, conceive the desired line to be drawn from it, and another line to.join the given point to the corner of the field. Then, as above, get expressions for the two triangles thus formed, and put their sum equal to the expression for the triangle which comprehends them both, and thence deduce the desired distance, nearly as above. (522) The difference d, between the areas parted off by the guess line AB, and the required line CD, is equal to the difference between the triangles APC and BPD sin. A. sin. P By Art. (65), Note, the triangle APC =- APa'i. (A sin. (A ~+ P)' sin. B. sin, P Bimilarly the triangle BPD =- BP2 s sin. (B +- P)' sin. A. sin. P sin. B. sin. P. d=i' AP2 — ~ RBPa. sin. (A + P) sin. (B - P) 404 DEMNONSTRATIONS [APP. B By the expression for sin. (a + b) [Trigonometry, Art. (8)], we have d p.AP. sin. A. sin. P sin. B. sin. P sin. A. cos. P +- sin. P. cos. A sin. B. cos. P -+ sin. P. cos. B COS. a Dividing each fraction by its numerator, and remembering that - cot. a, w have d AP2 - BP' cot. P + cot. A cot. P + cot. B For convenience, let p = cot. P; a = cot. A; and b = cot. B. The above equatior will then read, multiplying both sides by 2, AP2 BPa 2 d= -- p+e p+b Clearing of fractions, we have 2 dp2 + 2 dap 2 dbp + 2 dab. =p. A2 + b. AP'-p. BP2- a. BP3. Transposing, dividing through by 2 d, and separating into factors, we get /+ (a AP2 - - BP2 \. APS- a. BP2,(X AP3-PBP2\, // b. AP2 - a.BP2 / AP2BP2 \ 2 2d 2d ~ 2d If A = 90~, cot. A = a - 0; and the expression reduces to the simpler form given in the article. (523) Conceive a perpendicular, BF, to be let fall from B to the required line DE. Let B represent the angle DBE, and 0 the unknown angle DBF. The angle BDF = 90~ -; and the angle BEF - 90~ - (B -) ) 90~ - B +. By Art. sin. BDE. sin. BED (65), Note, the area of the triangle DBE = DE2o i BDE- 3 sin. (BDE +- BED) DE in. (90~- ) sin. (90~ —B +- ) sin. B Hence DE2 2 X DBE X sin. B 2 X DBE X sin. B sin. (900 - 0).sin. ((90~ -B + p) cos. f. cos. (B- i)' Now in order that DE may be the least possible, the denominator of the last fraction must be the greatest possible. It may be transformed, by the formula, cos. a. cos. b = ~ cos. (a - b) +-. cos. (a - b) [Trigonometry, Art. (8) ], into j cos. B + 4. cos. (B - 2 i). Since B is constant, the value of this expression depends on its second term, and that will be the greatest possible when B - 2 =- 0, in which case ft = B. It hence appears that the required line DE is perpendicular to the line, BF, which bisects the given angle B. This gives the direction an which DE is to be run. Its starting point, D or E, is found thus. The area of the triangle DBE = ~ BD. BE. sin. B. Since the triangle is isosceles, this becomes V/[2 DBE DBE = BD2. sin. B; whence BD =/ ( Si B) DE is obtained from the expression for DE2, which becomes, making 3 = j B, D 2 DBE sin. B; whence,/(2. DBE. sin. B) cos, ^ B. cnos. s.B cs, j B KPP. B.] For Part XXI 4)5 (52j) Let a= value per acre of one portion of the land, and 5 that of the other portion. Let x = the width required, BC or AD. Then the value of xXBE xXAL BCFE =aX 10,and the value of ADFE = b X 1- Putting the sum of these equal to the value required to be parted off, we obtain value required X 10 a X BE + bX AE (652) All the constructions of this article depend on the equivalency of triangles which have equal bases, and lie between parallels. The length of AD is deiivad from the area of a triangle being equal to its base by half its altitude. ("27) Since similar triangles are to each other as the squares of their homologous sides, ABC: DBE:: AB': BD2; whence BD -=AB A AB /' AB AC I/T -7. The construction of Fig. 363 is founded on the proportion BF: BG::BG:: BA; when BD - BG _ V (BA X BF) BA m+ n. (52~) By hypothesis, AEF: EFBC:: m: n; whence AEF: ABC::: a n + A cc AC X DB cn and AEF=-ABC =. Also, AEF- =. AE X EF. i + — n 2 }? -t-?i DB X AE The similar triangles AEF and ABD give AD DB:: AE: EF =- ----- The DB x AE second expression for AEF then becomes AEF=- AE D Equating AD this with the other value of AEF, we have AO X DB m AE' X DB /m AC X DB ~AE2 XDB; whence AE = AC X AD X. 2 m -f nc 2.AD ( \m + n (530) In Fig. 366, the triangles ABD, DBC, having the same altitude, are to each other as their bases. In the next paragraph, we have ABD: DBC:: AD: DC::: n; whence AD: AC:: qnc: mn -n; and AC: DC: m-n: n; whence the expressions for AD and DC. In Fig. 367, the expression for AD is given by the proportion AD: AC::: m: — + n. Similarly for DE, and EC. (531) In Fig. 368, conceive the line EB to be drawn. The triangle AEB =- ABC, having the same altitude and half the base; and AFD- AEB, because of the equivalency of the triangles EFD and EFB, which, with AEF, make up AFD and AEB. The point F is fixed by the similar triangles ADB and AEF The expression for AF, in the last paragraph, is given by the proportion, ABC:ADF:: AB X AC: AD X AF; AB X AC ADF AB X AC n whence, AF ~ whence, APF AD ABC AD'm + n (532) The areas of triangles being equal to the product of their altitudes by half their bases, the c"^';ructim" in Fig. 369 and Fig. 370 follow therefrom. 106 DEMONSTRATIONS LAPP. D. (533) In Fig. 381, conceive the line BL to be drawn. The triangle ABL will be a third of ABC, having the same altitude and one-third the base; and AED is equivalent to ABL, because ELB -- ELD, and AEL is common to both. A similar proof applies to DOG. (S34) In Fig. 872, the four smaller triangles are mutually equivalent, because of their equal bases and altitudes, two pairs of them lying between parallels. (335) In Fig. 373, conceive AE to be drawn. The triangle AEC = -. ABC, having the same altitude and half the base; and EDFC = AEC, because of the common part FEC and the equivalency of FED and FEA. (53;) In Fig. 374, in addition to the lines used in the problem, draw BF and DG. The triangle BFC =- B ABC, having the same altitude and half the base. Also, the triangle DFG = DFB, because of the parallels DF and BG. Adding DFC to each of these triangles, we have DCG = BFC =-, ABC. We have then to prove LMC = DCG. This is done precisely as in the demonstration of Art. (519), page 402. (523) Let AE = x, ED = y, AH - x', HF - y', AK - a, KB b. The quadrilateral AFDE, equivalent to 1 ABC, but which we will represent, generally, by as2, is made up of the triangle AFH and the trapezoid FHED. AFH -. x'y'. FHED = (x - x') (y + y')... AFDE = m=. x'y' - ( - x') (y - y') - x (y- y') -- x'y The similar triangles, AHF and AKB, give bx' a: b:: x: y' =-. Substituting this value of y' in the expression for rnm, we have m2 =' A 6 _A whence (2 -xy) AK ( ABC AE X ED) ^ b" zbx-ay KB X AE-AK X ED~ The formula is general, whatever may be the ratio of the area mn to that of the triangle ABC. (53~) In Fig. 376, FD is a line of division, because BF = the triangle BDF divided by half its altitude, which gives its base. So for the other triangles. (539) In Fig. 377, DG is a second line of division, because, drawing BL, the triangle BLC = ~ ABC; and BDGC is equivalent to BLC, because of the common part BCLD, and the equivalency of the triangles DLG and DLB. To prove that DF is a third line of division, join MD and MA. Then BMA -= BGA. From BM]A take MFA and add its equivalent IFD, and we have MDFB == BGA - (ABDG - BDG) - ( ABGC BDG) -= ABC - BDG. To MDFB add MDB, and add its equivalent, i BDG, to the other side of the equa. tion, and we have MDFB + IDB =- ABC -- BDG - + BDG; or, BDF = ABC. (540) In Fig. 378, the triangle AFC = ~ ABC, having the same base and one" third the altitude. The triangles AFB and BFC are equivalent to each other, each being composed of two triangles of equal bases and altitudes; and each it therefore one-third of ABC. APP, B.] For Part Xl, 407 In Fig. 879, AFC: ABC:: AD: AB; since these two triangles have the common base AC, and their altitudes are in the above ratio. So too, BFC: ABC: BE: BA Hence, the remaining triangle AFB: ABC:: DE: AB. (541) By Art. (65), Note, ABC= ~ AC X CB X sin. ACB. But the angle AC B=ACD+DCB =- (180~-ADC)+- (180~-CDB) = S0~ — (ADC+CDB). Hence, ABC = I AC X CB X sin. - (ADC + CDB) = A AC X CB X sin. ~ ADB. Let r = DA = DB = DC. Since AB is the chord of ADB to the radius r, and therefore equal to twice the sine of half that angle, we have AB AB AB >X BC X CA Bin.. ADB =; whence, ABC Aa X O B X-; and r = )(BC 2;andrr 4. ABC Also, since the area of each of the three small triangles equals half the product of the two equal sides into the sine of the included angle at D, these triangles will be to each other as the sines of those angles. These angles are found thus: AB BC AC sin.; ADB 2; sin. ~ BDC =-; sin. ~ ADC = -. 2r 2' 2r (542) The formulas in this article are obtained by substituting, in those of Art. (523), for the triangle DBE, its equivalent qX AB X BC X sin. B. BD thus becomes = (- -+ ) sin/ ( XABXBC) V (,4,XABXBCX sin.B) sin. B ( e X C and DE -~- -t-xA BG) cos. ~ B cos. B -- X AB X BC). (543) The rule and example prove themselves. (544) In Fig. 383, conceive the sides AB and DC, produced, to meet in some point P. Then, by reason of the similar triangles, ADP: BCP:: AD2: BC2, whence, by " division," ADP - BCP = ABCD: BCP:: AD2 - BC2: BC2. In like manner, comparing EFP and BCP, we get EBCF: BCP: EF2-B1C2: BC2 Combining these two proportions, we have ABCD: EBCF:: AD2 - BC2: EF2 - BC2; or, m + rn: m: AD2~- BC2: EF2- BC2.'Whence, (vm + n) EF - BC2. B2 --- B mCn. AD2 m. m BC2; TEF /(n X AD2 +n X BC Also, from the similar triangles formed by drawing BL parallel to CD, we have AL: E:: BA: BE= BA AB(EF BC) AAL EAD — BG (545) Let BEFC= ABCD = a; let BC = 6; BH /h; and nm -- n AD —BC =c. Also let BG = x; anc EF =y. Draw BL parallel to CD. By sim. ilar triangles, AL: EK:: BA: BE:: BH: BG; or, AD-BC: EF-BC: BE:BG; n (p —,).,: y b:h: x; whence x= b) Als~~~~~~oa~2 the a. BG (E B) ( whence Aso, the area BEFC =ea.BG (EF -PBC)= x(y + b); whence y 408 DEM0ONTRATIONS. [APP. B Substituting this value of y in the expression for x, and reducing, we obtain 2 bi 2 ah 6b /b 2 al 6 b2h \ 2+ - X; whence we have x = — ~ - + 2 c c c \c C/ CX C The second proportion above gives y --; whence y =b - ~ X. Replacing the symbols by their lines, we get the formulas in the text. (546) ABEF =- ABCD. But ABRP = ABEF, because of the common part ABRF, and the triangles FRP and FRE, which make up the two figures, and which are equivalent because of the parallels Fit and PE. So for the other parts. (547) The truth of the foot-note is evident, since the first line bisects the trapezoid, and any other line drawn through its middle, and meeting the parallel sides, adds one triangle to each half, and takes away an equal triangle; and thus does not disturb the equivalency. (54~) In Fig. 285, since EF is parallel to AD, we have ADG: EGF:: Gi:2: GK2. EGF is made up of the triangle BOG-=a', and the quadrilateral BEFCO -. ABCD = (a- a'). Hence the above proportion becomes mpn n m + n a: a' + (a — a'): GH: GK2; or, m+ n (m+ n) a: ma+- a':: GII: GK2; whence G =- GH /( + na'-~ GK GE is given by the proportion GH: GK:: GA: GE = GA -.H In Fig. 886, the division into p parts is founded on the same principle. The triangle EFG = GBC + EFOB - a' + Q. Now AD;: EFG:: AG2: EG2; Qor, a' + Q: a' +:: AG(: EG; whence GaE -AG ( ). or, a'+: a'+: AG2:EG; whence G a GL is obtained by taking the triangle LMG = a' + ~; and so for the rest. (552) In Fig. 890, join FO and GO. Because of the parallels CA and BF, the triangle FCD will be equivalent to the quadrilateral ABCD, of which GCD will therefore be one half; and because of the parallels GE and CH, EHDO will be equivalent to GCD. (5N5) In Fig. 391, by drawing certain lines, the quadrilateral can be divided into three equivalent parts, each composed of an equivalent trapezoid and an equivalent triangle. These three equivalent parts can then be transformed, by means of the parallels, into the three equivalent quadrilaterals shown in the figure. The full development of the proof is left as an exercise for the student. In Fig. 8 9'2, craw GO. Then CBG -- ABOD. But CKQ = GGQ. Therefore CKQB - ~ ABCD. So for the other division line. (556) The division of the base of the equivalent triangle, divides the polygon similarly. The point Q results from the equivalency of the triangles ZBP and ZBQ PQ being parallel to BZ. APPENDIX C. INTRODUCTION TO LEVELLING. (1) The P'rineiples. LEVELLING 1S the art of finding how much one point is higher or lower than another; i. e., how much one of the points is above or below a level line or surface which passes through the other point. A level or horizontal line is one which is perpendicular to the direction of gravity, as indicated by a plumb-line or similar means. It is therefore parallel to the surface of standing water. A level or horizontal surface is defined in the same way. It will be determined by two level lines which intersect each other.,, Levelling may be named VERTICAL SURVEYING, or Up-and-dozwn Surveying; the subject of the preceding pages being Horizontal Surveying, or Right-and-left and Fore-and-aft Surveying. All the methods of Horizontal Surveying may be used in Vertical Surveying. The one which will be briefly sketched here corresponds precisely to the method of " Surveying by offsets," founded on the Second Method, Art. (6), "Rectangular Cc-ordinates," and fully explained in Arts. (114), &c. The operations of levelling by this method consist, firstly, in obtaining a level line or plane; and, secondly, in measuring how far below it or above it (usually the former) are the two points whose relative heights are required. (2) Tahe lrastrumenits. Alevel Fig. 415. line may be obtained by the following f — -- simple instrument, called a " Plumb-line level." Fasten together two pieces of wood at right angles to each other, so as to make a T, and draw a line on the upright one so as to be exactly perpendicular to the top edge of the other. Suspend a plumb-line as in the figure. Fix the T against a staff stuck in the ground, by a screw through the middle of the crosspiece. Turn the T till the plumb-line exactly covers the line which was drawn. Then will the upper edge of the cross-piece be a level line, and the eye can sight across it, and note how far above or below any other point this level line, prolonged, would strike. It will be easier to look across sights fixed on each end of the cross-piece, making them of horsehair stretched across a piece of wire, bent into three sides of a square, and stuck into each end of the cross-piece; taking care that the hairs are at exactly equal heights above the upper edge of the cross-piece. * Certain small corrections, to be hereafter explained, will be ignored for the present, and we will consider ~Ivpl lines as straight lines, and level surfaces as planes. 410 LEVELLINGS LAPP. A modification of this is to fasten a common Fi. 416. carpenter's square in a slit in the top of a staff, —.. by means of a screw, and then tie a plumb-line at the angle so that it may hang beside one arm. When it has been brought to do so, by turning the square, then the other arm will be level. Another simple instrument depends upon the urinciple that "water always finds its level," corresponding to the second part of our definition of a level line. If a tube be bent up at each end, and nearly filled with water, the surface of the water in one end will always be at the same height as that in the other, however the position of the tube may vary. On this truth depends the " Water-level." It may be easily constructed with a tube of tin, lead, copper, &c., by bending up, at right angles, an inch or two of each end, and supporting the tube, if too Fig. 41 flexible, on a wooden bar. la these 4 — --- ends cement (with putty, twine dipped in white-lead, &c.), thin phials, with their bottoms broken off, so as to leave a free communication between them. Fill the tube and the phials, nearly to their top, with colored water. Blue vitriol, or cochineal, may be used for coloring it. Cork their mouths, and fit the instrument, by a steady but flexible joint, to a tripod. Figures of joints are given on page 134, and of tripods on page 133. To use it, set it in the desired spot, place the tube by eye nearly level, remove the corks, and the surfaces of the water in the two phials will come to the same level. Stand about a yard behind the nearest phial, and let one eye, the other being closed, glance along the right-hand side of one phial and the left-hand side of the other. Raise or lower the head till the two surfaces seem to coincide, and this line of sight, prolonged, will give the level line desired. Sights of equal height, floating on the water, and rising above the tops of the phials, would give a better-defined line. The "Spirit-level" consists essentially Fig. 418. of a curved glass tube nearly filled with _ __ alcohol, but with a bubble of air left within, which always seeks the highest spot in the tube, and will therefore by I its movements indicate any change in the position of the tube. Whenever the bubble, by raising or lowering one end, has been brought to stand between two marks on the tube, or, in case of expansion or contraction, to extend an equal distance on either side of them, the bottom of the block (if the tube be in one), or sights at each end of the tube, previously propelly adjusted, will be on the same level line. It may be placed on a board fixed to the top of a staff or tripod. When, instead of the sights, a telescope is made parallel to the level, and vari. ous contrivances to increase its delicacy and accuracy are added, the instrument becomes the Engineer's spirit-level APr. c.] The Practice. 411 (3) The Practice. By whichever of these various means a level line has been obtained, the subsequent operations in making use of it are identical Since the " water-level" is easily made and tolerably accurate, we will suppose it to be employed. Let A and B, Fig. 419, represent the two points, the Fig. 419, difference of the heights of which is [j required. Set the instrument on any spot from which both the points can be seen, and at such a height i that the level line will pass above the highest one. At A let an assist- 2 - -- - - ant hold a rod graduated into feet, tenths, &c. Turn the instrument to- wards the staff, sight along the level line, and note what division on the staff it strikes. Then send the staff to B, direct the instrument to it, and note the height observed at that poant. If the level line, prolonged by the eye, passes 2 feet above A and 6 feet above B. the difference of their heights is 4 feet. The absolute height of the level line itself is a matter of indifference. The rod may carry a target or plate of iron, claspod to it so as to slide up and down, and be fixed, at will. This target may be variously painted, most simply with its upper half red and its lower half white. The horizontal line dividing the colors is the line sighted to, the target being moved up or down till the line of sight strikes it. A hole in the middle of the target shows what division on the rod coincides with the horizontal line, when it has been brought to the right height. If the height of another point, C, Fig. 420, not visible from the first station, be required, set the instrument so as to see B and C, and proceed exactly as with A Fig. 420.'~ ti:.....[ — C and B. If C be 1 foot below B, as in the figure, it will be 5 feet below A. T it were found to be q feet above B, it would be 3 feet above A. The comparative height of a series of any number of points, can thus be found in reference to any one of them. The beginner in the practice of levelling may advantageously make in his notebook a sketch of the heights noted, and of the distances, putting down each as it is observed, and imitating, as nearly as his accuracy of eye will permit, their pro 412 LEVELLING. [APP. C. portional dimensions.* But when the observations are numerous, they should be kept in a tabular form, such as that which is given below. The names of the points, or "Stations," whose heights are demanded, are placed in the first column; and their heights, as finally ascertained, in reference to the first point, in the last column. The heights above the starting point are marked +, and those below it are marked -. The back-sight to any station is placed on the line below the point to which it refers. When a back-sight exceeds a fore-sight, their difference is placed in the column of "Rise;" when it is less, their difference is a "Fall." The following table represents the same observations as the last figure, and their careful comparison will explain any obscurities in either. Stations. Distances. Back-sights. Fore-sights. Rise. Fall. Total Heights. A 0.00 B 100 2.00 6.00 -- 4.00 - 4.00 C 60 3.00 4.00 -1.00 - 5.00 D 40 2.00 1.00 + 1. -4.00 E 70 6.00 1.00 + 5.00 + 1.00 F 50 2.00 6.00 - 4.00 - 3.00 15.00 18.00 - 3.00 The above table shows that B is 4 feet below A; that C is 5 feet below A; that E is 1 foot above A; and so on. To test the calculations, add up the back-sights and fore-sights. The difference of the sums should equal the last " total height." Another form of the levelling field-book is presented below. It refers to the same stations and levels, noted in the previous form, and shown in Fig. 420. Stations. Distances. Back-sights. Hit. Inst. above Datum. Fore-sights. Total Heights. A 0.00 B 100 2.00 + 2.00 6.00 - 4.00 C 60 3.00 - 1.00 4.00 - 5.00 D 40 2.00 - 3.00 1.00 - 4.00 E'70 6.00 + 2.00 1.00 + 1.00 F 50 2.00 + 3.00 1 6.00 - 3.00 15.00 18.00 - 3.00 In the above form it will be seen that a new column is introduced, containing the Height of the Instrument (i. e., of its line of sight), not above the ground where it stands, but above the Datum, or starting-point, of the levels. The former columns of "Rise" and "Fall" are omitted. The above notes are taken thus: The height of the starting-point or " Datum," at A, is 0.00. The instrument being set up and levelled, the rod is held at A. The back-sight upon it is 2.00; therefore the height of the instrument is also 2.00. The rod is next held at B. The fore-sight to it is 6.00. That point is therefore 6.00 below the instrument, or 2.00 - 6.00 = -4.00 below the datum. The instrument is now moved, and again set up, and the back-sight to B, being 3.00, the Ht. Inst. is -4.00 + 3.00 - 1.00 * In the figure, the limits of the page have made it necessary to contract the horizontal distanoes to one-tenth of their proner proportional size. APP. o.] The Practice. 413 and so on: the Ht. Inst. being always obtained by adding the back-sight to the height of the peg on which the rod is held, and the height of the next peg being obtained by subtracting the fore-sight to the rod held on that peg, from the Ht. Inst, The level lines given by these instruments are all lines of apparent level, and not of true level, which should curve with the surface of the earth. These level lines strike too high; but the difference is very small in sights of ordinary length, being only one-eighth of an inch for a sight of one-eighth of a mile, and diminishing as the square of the distance; and it may be conpletely compensated by setting the instrument midway between the points whose difference of level is desired; a precaution which should always be taken, when possible. It may be required to show on paper the ups and downs of the line which has been levelled; and to represent, to any desired scale, the heights and distances of the various points of a line, its ascents and descents, as seen in a side-view. This is called a "Profile." It is made thus. Any point on the paper being assumed for the first station, a horizontal line is drawn through it; the distance to the next station is measured along it, to the required scale; at the termination of this distance a vertical line is drawn; and the given height of the second station above or below the first is set off on this vertical line. The point thus fixed determines the second station, and a line joining it to the first station represents the slope of the ground between the two. The process is repeated for the next station, &c. But the rises and falls of a line are always very small in proportion to the dis tances passed over; even mountains being merely as the roughnesses of the rind of an orange. If the distances and the heights were represented on a profile to the same scale, the latter would be hardly visible. To make them more apparent it is usual to "exaggerate the vertical scale" ten-fold, or more; i. e., to make the representation of a foot of height ten times as great as that of a foot of length, as in Fig. 420, in which one inch represents one hundred feet for the distances, and ten feet for the heights. The preceding Introduction to Levelling has been made as brief as possible; but by any of the simple instruments described in it, and either of its tabular forms, any person can determine with sufficient precision whether a distant spring is higher or lower than his house, and how much; as well as how deep it would be necessary to cut into any intervening hill to bring the water. He may in like manner ascertain whether a swamp can be drained into a neighboring brook; and can cut the necessary ditches at any given slope of so many inches to the rod, &C., having thus found a level line; or he can obtain any other desired information which depends on the relative heights of two points. To explain the peculiarities of the more elaborate levelling instruments, the precautions necessary in their use, the prevention and correction of errors, the overcoming of difficulties, and the various complicated details of their applications, would require a great number of pages. This will therefore be reserved for an. other volume, as announced in the Preface. 414 APPENDIX D. MAGNETIC VARIATIONS IN THE UNITED STATES. [From a Report by C. A. SCHOTT, Assistant U. S Coast Survey]. See Silliman's Journal, May, 1860, p. 335; and U. S. Coast Survey Report for 1859, App. 24, p. 296. TW and E. indicate West and East Declinations. They are given below in Degrees and tenths. A: p o o. - 5 1' ~ I ~ ~ S: ~. W. W. W W. W. W. W. W... W.. 1680 8.8 4.8 W. 1690 8.7 4.8 W. 1700 9.7 8.5 8.8 1710 9.0 10.4 8.0. 8.4 1720 8.3 9.5 7.6 7.9 1730 7.8 8.9 7.0 7.1 1740 7.4 8.3 6.4 6.3 1750 7.2 7.7 5.8 5.3 1760 8.1 7.0 6.9 6.1 5.2 4.4 1770 8.1 6.8 6.3 5.5 4.7 3.5 1.2 W. 1780 8.3 6.8 6.1 5.2 5.0 4.4 2.8 0.7 W. 1790 8.5 6.8 7.8 6.3 6.3 5.0 4.8 4.2 3.0 2.2 0.2W. 1800 8.9 7.0 7.5 6.2 6.4 5.0 4.6 4.2 3.0 2.0 0.4 0.2E. 1810 9.4 7.3 7.3 6.3 6.5 5.2 4.7 5.4 4.3 3.1 1.9 0.5 0.4 E. 1820 10.0 7.8 7.6 6.7 6.8 5.6 5.0 5.8 4.7 3.4 2.2 0.8 0.4 E. 1830 10.6 8.4 8.3 7.3 7.5 6.1 5.4 6.3 5.2 3.8 2.7 1.1 0.2E. 1840 11.2 9.1 9.1 8.1 8.4 6.7 6.0 7.0 5.7, 4.4 3.4 1.5 0.1 W. 1850 11.8 9.9 9.7 8.9 9 1 7.4 6.7 7.7 6.4 5.2 4.3 2.0 0.6W. 1860 12.3 10.6 10.3 9.9 9.7 8.1 7.5 8.3 7.0 6.0 5.2 2.6 1.2W. 1870 12.7 11.1 11.2 10.9 10.2 8.9 8.3 9.0 7.6 6.8 6.1 3.1 1.9 W. Minimum 1765 1782 11813 1800 1779 1794 1801 1787 1795 1799 1805 1798 1815 t n ^~~~~~.- n 8 n'' -. ~. -., -- - 1- 1 E1 E. E. E. E. E. E. E. E. 1770 3.7 1780 4.0 1790 4.1 11.1 11.4 13.6 15.1 18.9 1800 4.1 4.1 7.1 11.4 12.3 14.1 15.4 19.1 1810 4.0 4.2 7.2 11.7 13.0 14.5 15.7 19.3 1820 3.6 4.2 7.3 12.0 13.6 14.8 16.0 19.5 1830 3.2 4.1 7.2 12.2 14.2 15.1 16.'3 19.7 1840 2.8 4.0 7.1 12.3 14.6 15.4 16.6 19.8 1850 2.2 3.7 7.0 12.5 15.0 15.6 16.9 20.0 1860 1.7 3.5 6.8 12.6 15.3 15.8 17.2 20.2 1870 1.2 3.2 6.6 12.6 15.4 15.9 17.2 20.4 Maximum. 1794 1817 1820 __ _ __I Not yet atta in ed. ANALYTICAL TABLE OF CONTENTS. PART I. GENERAL PRINCIPLES AND FENDAIENTAL METHODS. CHAPTER I. Definitions and lethods. alTIOLE PAGE ARTICLE PA G (1) Surveying defined........... 9 Division of the subject. (2) When a point is determined.. 9 (12) By the methods employed.. 14 (3) Determining lines and surfaces 10 (13) By the instruments........ 14 To determine points. (14) By the objects............. 14 (5) First Method............... 10 (15) By the extent............. 15 (6) Second do................ 11 (16) Arrangement of this book... 15 (7) Third do................ 11 (17) The three operations common (~) Fourth do................ 12 to all surveying........ 16 (10) Fifth do............. 13 CHAPTER II. Making the leasurements. Measuring straight lines. (25) Chaining on slopes......... 21 (19) Actual and Visual lines.... 16 (28) Tape..................... 23 (20) Gunter's Chain............ 16 (29) Rope, &c.................. 24 (21) Pills..................... 19 (30) Rods.................... 24 (22) Staves................... 19 (32) Measuring-wheel.......... 24 (23) How to chain............. 19 (33) Measuring Angles........ 25 (24) Tallies................... 21 (34) Noting the MIeasurements... 25 CHAPTER III. Drawing the Map. (35) A Map defined............ 25 (45) Scales for farm surveys..... 29 (36) Platting.................. 25 (46) Scales for state surveys..... 31 (37) Straight lines............. 26 (47) Scales for railroad surveys.. 32 (38) Arcs..................... 26 (49) How to make scales........ 33 (39) Parallels.................. 26 (50) The Vernier scale.......... 35 (40) Perpendiculars............ 27 (51) A reduced scale............ 36 (41) Angles................... 28 (52) Sectoral scales............. 36 (42) Drawing to scale........... 28 (53) Drawing scale on map...... 37 (44) Scales..................... 29 (54) Scale omitted............. 37 CHAPTER IV. Calculating the Content. (55) Content defined........... 38 (6S) Quadrilaterals............ 44 (56) Horizontal measurement.... 38 (69) Curved boundaries....... 45 (57) Unit of content............ 40 (70) Second Method, Geometrically 45 (5~) Reductions................ 40 (71) Division into triangles..... 45 (59) Table of Decimals of an acre. 41 (72) Graphical multiplication.. 47 (60) Chain correction........... 41 (73) Division into trapezoids.... 48 (61) Boundary lines............ 42 (74) Do. into squares...... 48 (75) Do. into parallelograms 49 METHODS OF CALCULATION. (76) Addition of widths....... 50 (63) First Mlethod, Arithmetically. 43 (77) Third Method, Instrumentally 50 (64) Rectangles.............. 43 (7) Reduction to one triangle.. 50 (65) Triangles................ 43 (~4) Special instruments....... 54 (66) Parallelograms.......... 44 (S7) Fourth Method, Trigonometri. (67) Trapezoids............... 44 cally.................... 56 416 CONTENTS. PART II. CHAIN SURVEYING. CHAPTER 1. Surveying by Diagonals. ARTICLB PAGE ARTICLE PAGO (90) A three-sided field......... 58 Keeping thefield-notes...... 62 (91) A four-sided field.......... 59 (94) By sketch.............. 62 (92) A many-sided field......... 60 (95) In columns............. 62 (93) How to divide a field...... 61 (90-97) Field-books........... 64 CHAPTER II. Surveying by Tie-lines. (98) Surveying by tie-lines.... 66 (101) Inaccessible areas..... 67 (100) Chain angles........... 67 1 (102) Without platting...... 67 CHAPTER III, Surveying by Perpendiculars. To set out Perpendiculars. Offsets. (104) By Surveyor's Cross..... 69 (114) Taking offsets...7...... 75 (107) By Optical Square...... 70 (117) Double offsets.......... 76 (10~) By the Chain........... 72 (11~) Field work............. 77 Diagonals and Perpendiculars. (119) Platting............... 79 (110) A three-sided field...... 72 (120) Calculating content..... 80 (111) A four-sided field....... 73 (121) When equidistant..... 80 (11 ) A many-sided field...... q4 (122) Erroneous rules....... 81 (113) By one diagonal........ o5 (123) Reducing to one triangle 81 (124) Equalizing............ 81 CHAPTER IV. Surveying by the methods combined. (125) Combination of the three (132) Exceptional cases.......... 92 preceding methods...... 82 (134) Inaccessible areas......... 93 (127) Field-books.............. 83 (136) Roads.................. 95 (130) Calculations.............. 88 (137) Towns................. 95 (131) The six-line system....... 90 CHAPTER V. Obstacles to Measurement in Chain Surveying. (13S) The obstacles to Alinement and MIeasurement.................... 96 (139) LAND GEOMETRY.............................................. 96 Problems on Perpendiculars. (140) PROBLEM 1. To erect a perpendicular at any point of a line......... 97 (143) 2. " " when the point is at or near the end of the line............ 98 (148) 3. " " when the line is inaccessible... 99 (150) 4. To let fall a perpendicular from a given point to a given line 99 (153) 5. " " when the point is nearly opposite to the end of the line... 100 (156) 6. "' when the point is inaccessible.. 101 (158) 7. " " when the line is inaccessible... 101 Problems on Parallels. (160) PROBLEM 1. To run a line from a given point parallel to a given line. 102 (165) 2. Do. when the line is inaccessible............ 103 CONTENTS. 417 OBSTACLES TO ALINEMENT. ARTICLE PAOH A. To prolong a line........................................ 105 (169) By ranging with rods..... 105 (174 B) By transversals.......... 107 (17 1) By perpendiculars.... 106 (175) By harmonic conjugates... 108 (172) By equilateral triangles.. 106 (176) By the complete quadri(173) By symmetrical triangles.. 107 lateral................108 B. To interpolate points in a line........................... 109 (177) Signals................. 109 (181) With a single person..... 111 (178) Ranging................ 109 (182) On water............... 111 (179) Across a valley.......... 110 (183) Through a wood......... 112 (180) Over a hill............. 110 (184) To an invisible intersection. 112 OBSTACLES TO MEASUREMENT. A. When both ends of the line are accessible..................... 113 (186) By perpendiculars........ 113 (I19) By transversals.......... 114 (I87) By equilateral triangles... 113 (190) In a town.......... 114 (lS~) By symmetrical triangles.. 114 B. When one end of the line is inaccessible...................... 115 (191) By perpendiculars........ 115 (198) By transversals.......... 117 (194) By parallels............. 116 (199) By harmonic division..... 117 (195) By a parallelogram...... 116 (2OO) To an inaccessible line.... 118 (196) By symmetrical triangles.. 116 (201) To an inacc. intersection. 118 C. TW7hen both ends of the line are inaccessible.................... 119 (202) By similar triangles....... 119 (204) By a parallelogram....... 119 (203) By parallels............. 119 (205) By symmetrical triangles.. 120 INACCESSIBLE AREAS...................... 121 (207) Triangles............... P1211 (208) Quadrilaterals........... 121 PART III. COIPASS SURVEYING. CHAPTER I. Angular Surveying in general. (210) Principle................ 122 (217) The Compass............ 124 (211) Definitions.............. 122 (219) Methods of Angular Sur(2 13) Goniometer............. 123 veying................ 125 (214) How to use it........... 123 (220) Subdivisions of Polar Sur-'215) Improvements........... 124 veying................ 125 CHAPTER I,. The Compass. (221) The Needle.............. 127 (22~) Tangent Scale.......... 132 (222) The Sights.............. 128 (229) The Vernier............ 132 (223) The Telescope........... 128 ( O30) Tripods.................. 133 (224) The divided Circle........ 128 (231) Jacob's Staff............. 134 (225) The Points.............. 129 (232) The Prismatic Compass... 135 (226) Eccentricity............. 180 (234) The defects of the Compass. 137 (227) Levels.................. 132 27 i 8 CONTENTS. CHAPTER III. The Field-work. A.inTColdS PAGE ARTICLE PAt1 (235) Taking Bearings......... 138 (242) Angles of deflection.. 144 (236) Why E. and W. are re- (243) Angles between courses. 145 versed............. 139 (244) To change Bearings.... 146 (237) Reading with Vernier.. 140 (245) Line Surveying.......... 1i7 (23 ) Practical Hints....... 140 (246) Checks by intersecting bearings........... 148 Mark stations. Set beside (247) Keeping the Field-notes 149 fence. Level crossways. Do (2l) New York Canal Maps 149 not level by needle. Keep 14 same end ahead. Read from (252) Farm Surveying......... 150 same end. Caution in read- (254) Field-notes........... 151 ing. Cheek vibrations. Tap (256) Tests of accuracy...... 153 compass. Keep iron away. Electricity. To carry corn- (25) Method of Radiation... 154 pass. Extra pin and needle. (259) Method of Intersection. 154 (260) Running out old lines.. 154 (239) To magnetize a Needle. 142 (261) Town Surveying......... 155 (240) Back-sights.......... 143 (262) Obstacles in Compass Sur(241) Local Attraction...... 143 veying............... 156 CHAPTER IV. Platting the Survey. 263) Platting in general........ 157 (273) Drawing-board protractor. 166 (264) With a protractor........ 157 (27 ) With a scale of chords.... 166 (25) Platting bearings......... 158 ( 775) With a table of chords.... 167 (26S) To make the plat close.... 161 (2.76) With a table of natural sines 168 (269) Field platting........... 162 (277) By Latitudes and Depart(272) With a paper protractor.. 164 ures................... 168 ICHAPTER V. Latitudes and Departures. (27~) Definitions.............. 169 Applications. (279) Calculation of Latitudes (262) Testing survey.......... 175 and Departures......... 170 (283) Supplying omissions..... 176 (2~0) Formulas............... 171 (284) Balancing............... 177 (2~1) Traverse Table.......... 171 (26~5) Platting................ 178 CHAPTER VI. Calculating the Content. (2~6) Methods................ 180 (292) General rule........... 184 (2S7) Definitions.............. 180 (293) To find east or west station 184 (2i~) Longitudes.............. 181 (294) Example 1............. 184 (2~9) Areas.................. 182 (296) Examples 2 to 13........ 186 (290) A three-sided field...... 182 (297) Mascheroni's Theorem.... 188 (291) A four-sidedfield........ 183 CHAPTER VII. The Variation of the Mlagnetic Needle. (296) Definitions.............. 189 (306) Table of Azimuths....... 196 (299) Direction of the needle.... 189 (307) Setting out the meridian.. 197 To determine the true meridian. To determzine the variation. (300) By equal shadows of the (30S) By the bearing of the star. 198 sun................. 190 (309) Other methods........... 199 (301) By the North Star when in (310) Magnetic variation in the the meridian.......... 191 United States........ 199 (302) Times of crossing the me- Line of no variation..... 199 ridian.............. 193 Lines of equal variation.. 200 (303) By the North Star when at Magnetic Pole.......... 200 its extreme elongation.. 194 (3]1 ) To correct magn. bearings. 200 (304) Table of times......... 195 (312) To survey a line with true (305) Observations........... 196 bearings............. 202 CONTENTS. 419 CHAPTER VIII. Changes in the Variation..tTICLE PGE ARTICLE PAGf (314) Irregular changes........ 203 (3l~) By interpolation...... 205 (315) The Diurnal change...... 203 (319) By old lines.......... 206 (316) The Annual Change...... 204 (320) Effects of this change... 207 (317) The Secular change....... 204 (321) To run out old lines..... 208 Tables for United States. 205 (322) Example............. 208 To determine the secular (323) Remedy for the evils of change.............. 205 the secular change.... 210 PART IV. TRANSIT AND THIEODOLITE SURVEYING. (BY THE 3d METHOD.) CHAPTER I. The Instruments. (324) General description of the (333) Supports................ 221 Transit and Theodolite.. 211 (334) The Indexes. Eccentricity. 221 The Transit.......... 212 (335) The graduated circle...... 223 The Theodolite...... 213 (336) Movements. Clamp and (325) Distinction between them. 214 Tangent screw......... 223 (326) Sources of their accuracy. 214 (337) Levels............... 224 (327) Explanation of the figures. 215 (338) Parallel plates........... 225 (328) Sectional view.......... 216 (339) Watch Telescope......... 226 (329) Telescopes.............. 217 (340) The Compass............ 226 (330) Cross hairs.............. 218 (341) Theodolites............. 226 (331) Instrumental parallax..... 220 (342) Goniasmometre.......... 227 (332) Eye-glass and object-glass.. 221 CHAPTER II. Verniers. (343) Definition............... 228 (351) Circle divided to 20'.... 235 (344) Illustration.............. 228 (352) Circle divided to 15'....... 236 (345) General rules............ 229 (353) Circle divided to 10'..... 237 (346) Retrograde Verniers...... 230 (354) Reading backwards....... 237 (347) Illustration.............. 231 (355) Arc of excess............ 238 (348) Mountain Barometer...... 231 (356) Double Verniers......... 238 (349) Circle divided into degrees. 232 (357) Compass Verniers........ 239 (350) Circle divided to 30'....... 233 CHAPTER II1. Adjustments. (353) Their object and necessity. 240 Rectification........ 243 (359) The three requirements in (362) In the Theodolite...... 245 the Transit........... 240 (363) Third Adjustment. To cause (360) First Adjustment. To cause the line of collimation to the circle to be horizontal revolve in a vertical plane 246 in every position........ 241 Verification (plumb-line; Verification.......... 241 star; steeple and stake) 246 Rectification......... 241 Rectification......... 246 (361) Second Adjustment. To (364) Centring eye-piece........ 247 cause the line of collima- (365) Centring object-glass...... 247 tion to revolve in a plane 242 Adjusting line of colliVerification.......... 242 mation............. 248 420 CONTENTS. CHAPTER IV. The Field-work. ARTICLE PAGE ARTICLE PAGI (366) To measure a horizontal (372) Line-surveying........... 254 angle................. 250 (373) Traversing, or surveying (367) Reduction of high and by the back angle... 254 low objects......... 251 (374) Use of the Compass.... 255 (368) Notation of angles..... 252 (375) Measuring distances with (369) Probable error.......252 a telescope and rod.. 256 (370) To repeat an angle.... 252 (376) Ranging out lines........ 257 (371) Angles of deflection.... 253 (377) Farm-surveying.......... 258 (37S) Platting................. 259 PART V. TRIANGULAR SURVEYING. (BY THE 4th METHOD.) (379) Principle................ 260 (385) Observations of the angles.. 2 7 (380) Outline of operations...... 260 (386) Reduction to the centre.. 268 (381) Measuring a base......... 261 (387) Correction of the angles... 270 Materials.............. 261 (388) Calculation and platting... 270 Supports............... 262 (389) Base of Verification....... 271 Alinement............. 262 (390) Interior filling up........ 271 Levelling.............. 262 (391) Radiating Triangulation... 272 Contacts............... 262 (392) Farm Triangulation....... 272 (382) Corrections of Base....... 263 (393) Inaccessible Areas....... 278 (383) Choice of stations......... 263 (394) Inversion of the Fourth U. S. Coast Survey Ex- method................ 273 ample............... 265 (395) Defects of the Method of In(384) Signals................. 266 tersections............. 274 PART VI. TRILINEAR SURVEYING. (BY THE 5th METHOD.) (396) The Problem of the three (398) Instrumental Solution. 277 points................ 275 (399) Analytical ". 277 (397) Geometrical Solution..... 275 (400) Maritime Surveying...... 278 PART VII. OBSTACLES IN ANGULAR SURVEYING. CHAPTER I. Perpendiculars and Parallels. (402) To erect a perpendicular to a line at a given point................ 279 (403) To erect a perpendicular to an inaccessible line, at a given point of it 280 (404) To let fall a perpendicular to a line, from a given point........... 280 (405) To let fall a perpendicular to a line, from an inaccessible point.. 280 (406) To let fall a perpendicular to an inaccessible line from a given point.. 281'407) To trace a line through a given point parallel to a given line........ 281 (408) To trace a line through a given point parallel to an inaccessible line.. 281 CONTENTS. 421 CHAPTER II. Obstacles to Alinement. AIOCLE PAGa A. To prolong a line......................... 282 (409) General method......... 282 (413) When the line to be pro(410) By perpendiculars....... 282 longed is inaccessible... 283 (411) By an equilateral triangle. 282 (414) To prolong a line with only (412) By triangulation........ 283 an angular instrument... 283 B. To interpolate points in a line.............. 284 (415) General method.......... 284 (418) By Latitudes and Depart(416) By a random line........ 284 ures, with transit......285 (417) By Latitudes and Depart- (419) By similar triangles...... 286 ures, with compass... 285 (420) By triangulation......... 286 CHAPTER III. Obstacles to IMeasurement. A. When both ends of the line are accessible........... 287 (421) Previous means.......... 2871 (423) A broken base........... 28 (422) By triangulation......... 287 (424) By angles to known points. 288 B. When one end of the line is inaccessible............ 288 (425) By perpendiculars....... 288 (429) To find the distance from a (426) By equal angles.......... 288 given point to an inacces(427) By triangulation........ 289 sible line............. 289 (428) When one point cannot be seen from the other.... 289 C. When both ends of the line are inaccessible.......... 290 (430) General method.......... 290 (433) When no point can be found 431) To measure an inaccessible from which both ends can distance, when a point in be seen............... 292 its line can be obtained.. 291 (434) To interpolate a base... 292 (432) When only one point can be (435) From angles to two points.. 293 found from which both (436) From angles to three points 293 ends of the line can be (437) From angles to four points. 294 seen................. 291 (43~) Problem of the eight points 296 CHAPTER IV. To Supply Omissions. (439) General statement................................ 297 (440) CASE 1. When the length and bearing of any one side are wanting.... 298 C.ASE 2. When the length of one side and the bearing of another are wanting............................................ 298 (441) When the deficient sides adjoin each other............... 298 (442) When the deficient sides are separated from each other..... 299 (443) Otherwise: by changing the meridian.................. 299 CASE 3. When the lengths of two sides are wanting................. 300 (444) When the deficient sides adjoin each other............... 300 (445) When the deficient sides are separated from each other.... 301 (446) Otherwise: by changing the meridian................. 301 CASE 4. When the bearings of two sides are wanting................. 302,447) When the deficient sides adjoin each other........... 302 (448) When the deficient sides are separated from each other..... 302 422 CONTENTS. PART VIII. PLANE TABLE SURVEYING. ARTIOLS PAG. (449) General description....... 303 (455) Method of Resection...... 308 (450) The Table............... 303 (456) To Orient the Table...... 308 (451) The Alidade............. 304 (457) To find one's place on the (452) Method of Radiation...... 305 ground............... 309 (453) Method of Progression.... 306 (45S) Inaccessible distances... 310 (454) Method of Intersection... 307 PART IX. SURVEYING WITHOUT INSTRUMENTS. (459) General principles........ 311 (463) Distances by sound....... 313 (460) Distances by pacing...... 311 (464) Angles................. 314 (461) Distances by visual angles. 312 (465) Methods of operation... 4 314 (462) Distances by visibility.... 313 PART X. MAPPING. CHAPTER I. Copying Plats. (466) Necessity............... 316 (474) Reducing by squares..... 319 (467) Stretching the paper...... 316 (475) " by proportional (468) Copying by tracing....... 317 scales..... 320 (469) " on tracing-paper.. 317 (476) " by a-pantagraph 321 (470) " by transfer-paper 317 (477) " by a camera luci(471) " by punctures.... 318 da......... 321 (472) " by intersections. 318 (478) Enlarging plats......... 321 (473) " by squares...... 319 CHAPTER II. Conventional Signs. (479) Object.................. 322 (483) Signs for water.......... 325 (40s) The relief of ground..... 322 (484) Colored topography..... 325 (481) Signs for natural surface... 324 (485) Signs for detached objects. 327 (4S2) Signs for vegetation...... 324 CHAPTER III. Finishing the Map. (486) Orientation............ 328 (489) Joining paper............ 329 (47~) Lettering.............. 328 (490) Mounting maps...,, 329 (488) Borders................. 328 CONTENTS. 423 PART XI. LAYING OUT, PARTING OFF, AND DIVIDING UP LAND. CHAPTER I. Laying out Land. JRTIOLE PAGE ARTICLE PAGE (491) Its object.............. 330 (496) To lay out circles........ 332 (492) To lay out squares...... 330 (497) Town lots.............. 333 (493) To lay out rectangles..... 331 (49S) Land sold for taxes...... 333 (494) To lay out triangles...... 332 (499) New countries........... 334 CHAPTER II. Parting off Land. (500) Its object................................................... 334 A. By a line parallel to a side. (501) To part off a rectangle......................................... 335 (502) " " a parallelogram..................................... 335 (503) " " a trapezoid........................................ 335 B. By a line perpendicular to a side. (505) To part off a triangle.......................................... 336 (507) " " a quadrilateral................................ 337 (50S) " " any figure........................................ 337 C. By a line running in any given direction. (509) To part off a triangle.......................................... 337 (511) " " a quadrilateral...................................... 338 (51.3) " " any figure.......................................... 339 D. By a line starting from a given point in a side. (514) To part off a triangle....................................... 339 (516) ". a quadrilateral................................... 340 (517) " " any figure.................................. 340 E. By a lise passing through a given point within the field. (519) To part off a triangle.......................................... 342 (520) " " a quadrilateral..................................... 343 (522) " " any figure................................ 344 F. By the shortest possible line. (523) To part off a triangle..........................................45 (524) G. Land of variable value......................... 345 (525) H. Straightening crooked fences.................... 346 CHAPTER IlI. Dividing up Land. (526) Arrangement................................................ 347 Division of Triangles. (527) By lines parallel to a side.............................. 8347 (528) By lines perpendicular to a side............................... 348 (529) By lines running in any given direction..............3...... 348 (530) By lines starting from an angle................................. 349 (531) By lines starting from a point in a side................. 349 (535) By lines passing through a point within the triangle............. 851 (540) Do. the point being to be found........................... 353 (541) Do. the point to be equidistant from the angles............... 353 (5412) By the shortest possible line................................... 354 Division of Rectangles. (543) By lines parallel to a side..................................... 354 424 CONTENTS. AI"TICLB PAGQ Division of Trapezoids. (544) By lines parallel to the bases.................................. 35 (546) By lines starting from points in a side........................... 55 (547) Other cases................................................ 356 Division of Quadrilaterals. (54 ) By lines parallel to a side...................................... 356 (549) By lines perpendicular to a side................................. 58 (550) By lines running in any given direction.......................... 58 (551) By lines starting from an angle............................... 358 (552) By lines starting from points in a side....................... 358 (554) By lines passing through a point within the figure............. 359 Division of Polygons. (555) By lines running in any given direction........................... 360 (556) By lines starting from an angle.................................. 360 (557) By lines starting from a point on a side.......................... 361 (55S) By lines passing through a point within the figure................. 361 (559) Other Problems.............................................. 361 PART XII. UNITED STATES' PUBLIC LANDS. (560) General system..........,363 Meandering............ 371 (561) Difficulty............... 364 (565) Marking lines.......... 372 (562) Running township lines.... 366 (566) Marking corners.......... 372 (563) Running section lines... 368 (567) Field-books............. 376 (564) Exceptional methods...... 370 Township lines......... 377 Water fronts........... 370 Section lines........... 378 Geodetic method....... 871 Meandering............ 378 APPENDIX. APPENDIX A. Synopsis of Plane Trigonometry. (1) Definition................. 379 (7) Their mutual relations...... 883 (2) Angles and Arcs............ 379 (~) Two arcs................ 383 (3) Trigonometrical lines........ 380 (9) Double and half arcs....... 384 (4) The lines as ratios........ 381 (10) The Tables.............. 384 5) Their variations in length... 381 (11) Right-angled triangles...... 385 (6) Their changes of sign........ 82 (12) Oblique-angled triangles.... 385 APPENDIX B. Demonstrations of Problems, &c.'Theory of Transversals........... 387 Proofs of Problems in Part V.... 397 Harmonic division.............. 890 " " in Part VI..... 398 The Complete Quadrilateral...... 91 " " in Part VII... 399 Proofs of Problems in Part II., " " in Part XI..... 401 Chapter V................... 393 APPENDIX C. Introduction to Levelling. (1) The Principles............. 409 (3) The Practice.............. 411 (2) The Instruments........ 409 [ADVERTISEMENT.] W. & L. E. GTURLEY, TROY, New York.:PlIC3 LISSrT, 1867. In common with all other manufacturers, we have been compelled by the great advance in the cost of labor, the war tax, and the materials used, to increase our old established prices for Instruments, &c. We believe, however, that in most cases they are still far below those of other makers of established reputation. Compasses. Plain, with Jacob Staff mountings, 4 inch needle........................$30 00 " " " 5 "........................ 38 00 " *" " 6 "........................ 42 00 Vernier, " " 6 "........................ 53 00 Railroad, " ".......................83 00 Extras. Compass tripod, with cherry legs.....................................$ 8 00'" " levelling screws and clamp and tangent movements... 18 00 I" " " without " "... 16 00 " mountings without legs.............................. 7 00 Compound tangent ball............................................. 7 00 Brass cover for compass glass..................................... 5 Outkeeper, for keeping tally....................................... 1 75 Transits. Vernier, plain telescope,* 6 inch needle, with compass tripod............$ 90 00 Surveyors' " 4 " " adjusting "............. 160 00 " " 5 " " " "............. 165 00 c" " Ii " " " "............. 165 00 Engineers' " 4 " " " "............. 180 00 5 " " " ".............185 00 " " 5 " with watch telescope............. 225 00 " " 5 " with theodolite axis............. 225 00 " " 5 " with two telescopes............. 285 00 Extras to Transits. / Vertical circle, 31 inch diameter, vernier reading to five minutes...........$ 9 00 " 4~ " " " single "....... 15 00 Clamp and tangent movement to axis of telescope...................... 8 00 Level on telescope, with ground bubble and scale..................... 15 00 Rack and pinion movement to eye-glass............................... 5 00 Sights on telescope, with folding joints................................ 8 00 Sights on standards at right angles to telescope........................ 8 00 Solar Compasses. Solar compass, with adjusting sockets and tripod......................$215 00 Solar telescope compass, with adjusting socket and tripod............... 240 00 Micrometer telescope, 16 to 20 inches long, with rack movement to object glass, and with movable clips to attach the sights to No. 1............. 28 00 * A plain telescope is one without any of the attachments or extras, as we term them, such as the clamp and tangent, vertical circle, and level. 426 ADVEERTISEMENT.-W. & L. E. GUNLEY, TROY, N. Y. Levelling'g lodis. Yankee or Boston........................................... $18 00 New York, with improved mountings................................... 18 00 ]Levelling Iastrumienls. Sixteen inch telescope, with adjusting tripod...........................$135 00 Eighteen " "....................... 135 00 Twenty-two " ".......................... 135 00 Twentv'B..135 00 Twenty-two " ".135 00 Builders' level, with adjusting tripod................................. 50 00 " " leveling screws, and clamp and tangent movements....... 60 00 Chains. 100 feet, with oval rings, No. 5 refined iron wire........................$12 00 " " " 6 " "......................... 00 50 feet, " " "......................... 6 50 50 feet, " " 6 " "...................... 5 00 66 feet, " " 8 "......................... 4 75 33feet, " " 8 "......................... 2 5 66 feet, " " 10 " "......................... 400 83feet, " " 10 " "......................... 2 50 100 feet, " " 8 best steel wire........................ 12 00 100 feet, " " 10 "......................... 10 50 0Ofeet, " " 8 " "................. 6 50 50 feet, " " 10 " "........................ 5 75 66feet, " " 8 " ".........................1050 66feet, " " 10 "........................ 8 00 33feet, " " 8 " "......................... 5 5 33 feet, " " 10 " "......................... 4 50 100 feet, brazed links and rings, No. 12 best steel wire................... 15 00 50 feet, " " 12 "................... 8 00 66 feet, " " 12 "................. 14 00 33 feet, " " 12 "................... 7 00 Mtarking Pins. Set of 11 pins, iron wire, No. 4....................................... $1 50 " "' steel wire, No. 6'........................ 2 00 i" " brass wire, No. 4....................................... 3 00 Imported iSeasuring Tapes. Gold price. Chesterman's steel, 33 feet.........................................$10 50 " 50 feet.................................. 14 25 " 66 feet.......................................... 18 00 " 100 feet.......................................... 25 00 "metallic 33 feet.......................................... 3 00 " 50 feet........................................... 3 75 " " 66 feet........................................... 4 25 " " 70 feet........................................... 4 50 " " 80 feet........................................... 5 25 " " 100 feet......................................... 6 25 Pocket Compasses. With folding sights, 2~ inch needle, very serviceable for tracing lines once surveyed....................................................$ 9 00 With folding sights, 2J inch needle, with Jacb Staff mounting...........11 50 " 3 "............ 13 50 " 3 ~ " without " 11 00 Without sights, 1 to 2 inch needle........................from 25 cents to 5 00 Miners' compass, or Dipping needle, for tracing iron ore, a new and beautiful article, glass on both sides...................................... 10 00 INFORMATION TO PURCHASERS. MANUAL. —TO those who may wish to purchase any of the Instruments mentioneJ in the previous pages of this Advertisement, we will send our Manual-a Book ol 125 pages, containing a full description of the same, with the adjustments, &c., free of charge, (postage included,) on application to us at Troy, N. Y. INSTRUMENTS WANTED.-In regard to the best kind of Instruments for particular purposes, we would here say, that where only common surveying, or the bearing of lines in the surveys for County Maps is required, a Plain Compass is all that is necessary. In cases where the variation of the needle is to be allowed, as in retracing the lines of an old survey, &c., the Vernier Compass, or the Vernier Transit, is required, Where, in addition to the variation of the needle, horizontal angles are to be taken, and in cases of local attraction, the Rail Road Compass is preferable; and for a mixed practice of Surveying and Engineering, we consider the Surveyor's Transit superior to any instrument made by us or any other manufacturers. In the surveys of U. S. public lands, the county and township lines are required to be run by such instruments as the Solar Compass. Where Engineering is the exclusive design, the Engineers' Transit and the Leveling Instruments are of course indispensable. WARRANTY.-All our instruments are examined and tested by us in person, and are sent to the purchaser adjusted and ready for immediate use. They are warranted correct in all their parts-we agreeing in the event of any defect appearing after reasonable use, to repair, or replace with a new and perfect instrument, promptly and at our own cost, express charges included, or we will refund the money, and the express charges paid by the purchaser, Instances may sometimes occur, in a business as large and widely extended as ours, where, owing to careless transportation, or to defects escaping the closest scrutiny of the maker, instruments may reach our customers in bad condition. We consider the retention of such instruments in all cases an injury very much greater to us than to the purchaser himself. TRIAL OF INSTRUMENTS.-It may often happen that this statement of the prices and quality of our instruments, may come into the hands of those who are entirely unacquainted with us. or with the quality of our work, and who therefore feel unwilling to make a final purchase of an article, of the excellence of which they are not'perfectly assured. To such we make the following proposition: We will send the instrument to th,express station nearest the person giving the order, and direct the Express Agent on delivery of the same, to collect our bill, together with charges of transportation. and hold the money on deposit until the purchaser shall have had-say two week.s actual trial of its quality. If not found as represented he may return the Instrument before the expiration of that time, and receive the money paid, in full, including express charges, and lirect the Instrument to be returned to us, 428 INFORMATION TO PURCHASERS. Low PRICES OF OUR INSTRUMENTS.-It is often urged by other makers, and per sons prejudiced in their favor, that it is impossible to make first rate instruments, at the prices charged by Lis, and which are so very far below those of other skillful manufacturers. We have only to reply, in addition to what we have stated in our Warranty, that a visit to our works, and a comparison of our facilities, with those of our competitors, would dispel all questions as to our ability to surpass them, not only in the cheapness, but also in the superior quality of our work. PACKING, &c.-Each instrument is packed in a well finished mahogany case, furnished with lock and key and brass hooks, the larger ones having besides these, a leather strap for convenience in carrying. Each case is provided with screw drivers, adjusting pin, and wrench for centre-pin, and, if accompanied by a tripod, with a brass plumb-bob; with all instruments for taking angles, without the needle, a reading microscope is also furnished. Unless the purchaser is already supplied, each instrument is accompanied by our "Manual," giving full instructions for such adjustments and repairs as are possible to one not provided with the ordinary facilities of an instrument maker. When sent to the purchaser, the mahogany cases are carefully enclosed in outside packing boxes, of pine, made a little larger on all sides to allow the introduction ol elastic material, and so effectually are our instruments protected by these precautions that of several thousand sent out by us during the last thirteen years, in all seasons, by every mode of transportation, and to all parts of the Union and the Canadas, not more than three or four have sustained serious injury. MEANS OF TRANSPORTATION.-Instruments can be sent by Express to almost every town in the United States and Canadas, regular agents being located at all the more important points, by whom they are forwarded to smaller places by stage. The charges of transportation from Troy to the purchaser are in all cases to be borne by him, we guarranteeing the safe arrival of our instruments to the extent of Express transportation, and holding the Express Companies responsible to us for all losses or damages on the way. TERMS OF PAYMENT are uniformly cash, and we have but one price. Our prices for instruments are nearly one-third less than those of other makers of established reputation. They are as low as we think instruments of equal quality can be made, and will not be varied from the list given on the previous pages. Remittances may be made by a draft, payable to our order at Troy, Albany, New York, Boston or Philadelphia, which can be procured from Banks or Bankers in almost all of the larger villages. These may be sent by mail with the order for the instrument, and if lost or stolen on the route, can be replaced by a duplicate draft, obtained as before, and without additional cost. Or the customer may pay the bill on receipt of the instrument to the express agent taking care to send funds bankable in New York or Boston. The cost of returning bills collected by express of amounts under $15,00 will be charged to the customer. W. & L. E. GURLEY, Mathematical Instrument Makers FULTON-ST., OPPOSITE NORTH END OF UNION R. R. DEPOT, TROY, N. Y. TRAVERSE TABLES: OR, LATITUDES AND DEPARTURES OF COURSES, CALCULATED TO THREE DECIMAL PLACES: FOR EACH QUARTER DEGREE OF BEARINKG LATITUDES AND DEPARTURES. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. j 40C 1.000 0000 2.000 0*000 3-00ooo o.ooo 4ooo o00ooo 5.000ooo 90 o0 I.ooo 00 004 2.000 0o009 3.ooo o.oi3 40ooo 0.017 5.000ooo 89 o0 Iooo 0000oo 2.000 0.017 3.000 0.026 4.ooo 0.035 5.ooo 893 o0 I.ooo.oI 3 2.000 0.026 3.000 0.039 4.000 0.052 5.ooo 893 1 1i.000 0.017 2.000 0.035 3.000ooo 0.052 3.999 0.070 4-999 S90 4.000 0.022 2.000 0.044 2. 999 o o65 3.999 0-087 4-999 884 1 i1. oo 0.026 1.999 0.052 2.999 0.079 3.999 o.io5 4.998 884 i i.*ooo 0.o31 1.999 o.o6i 2.999 0.092 3.998 o.122 4-.998 884 2 o0999 o. 35 1 999 0.070 2.998 o.io5 3.998 o0-40 4-997 88 2 0.99 o39.03 998 0079 2.998 0.ii8 3.997 0157 4.996 874 24 0.999 0-044 1.998 0.087 2-997 -.J3i 3.996 0.I74 4-995 877 24 0.999 0oo48 1.998 0.096 2.997 o.if4 3.995 0.192 4-994 874 3o 0.999 0-052 1.997 0.I05 2.996 0.157 3.995 0.209 4.993 S70 34 0.998 0.057 1.997 o0.i3 2.995 0.I70 3.994 0.227 4.992 864 34 0.998 o.o6I I 996 o.122 2.994 o i83 3.993 0.244 4-99I 864 34 0.998 0.065 1.996 o.i3i 2.994 o.196 3.991 0o262 4.989 864 o0.998 0.070 1.995 o.i40 2.993 0.209 3.990 o0279 4.988 860 44 0-997 0.074 -1995 o.I48 2.992 0.222 3.989 o0296 4.986 854 44 0.997 0.078 I-994 o0I57 2-99I o0235 3.988 o-3I4 4.985 854 44 0-997 o.o83 1.993 o-.66 2.990 0.248 3.986 o-33I 4-983 854 50 o 996 0o087 1-992 0.174 2.989 0-261 3.985 0.349 4-981 85 5 0.996 0.092 1.992 0o.83 2.987 0.275 3.983 0.366 4-979 84i4 5 0.995 o.096 1.991 0.I92 2.986 0.288 3.982 0.383 4-977 844 54 o0.995 0.10 o I.990 0.200 2.985 o.30o 3.980 o.40 o 4-975 841 60 0.995 o.iob.989 o.o9 2.984 o.314 3-978 o.4I8 4-973 8~4 6 0-994 0.109 I.988 0.218 2.982 0.327 3-976 0.435 4-970 834 6. 0.994 o.II3 1.987 0.226 2.98I 0o340 3-974 o.453 4.968 834 64 0o993 o.x18 1.986 0.235 2.979 o0353 3-972 0.470 4.965 834 70 0o993 01I22 1.985 0.244 2-978 0.366 3-970 0.487 4.963 830 74 0.992 0.126' I984.0.252 2~976 0.379 3-968 o0505 4-960 824 74 0.99I o0.13 1.983 0.261 2.974 o0392 3-966 0.522 4.957 682 74 0.99I o.035 I982 0.270 2.973 0.405 3.963 0.539 4.954 824 8 o-990 0.139 1I98I 0.278 2-97I o-4I8 3-961 0o557 4-951 82~ 84 0.990 o.-43 1.979 0.287 2.969 o.430 3-959 0.574 4'948 814 8o 0.989 0.-48 1.978 0.296 2.967 0.443 3-956 0.591 4-945 8I1 84 0.988 o0.52 I.977 o.304 2.965 0.456 3-953 0.608 4-942 8I1 90 o0.988 o0.56 1.975 o.3I3 2.963 0.469 3-95I 0-626 4-938 810 94 0o987 o-.i6 I 974 o0321 2-96 o0.482 3-948 o0643 4.935 80o 9$ 0o986 o.r65 1.973 o0330 2.959 0.495 3-945 0.660 4-93I 80o 94 0.o986 0.169'971 0o339 2.957' 0.508 3-942 0.677 4-928 80; 1O: 0.985 0.174 I-970 o0347 2.954 0.52i 3.939 0-695 4-924 0~ io~ 0.984 0.178 1.968 0.356 2.952 o.534 3.936 0.712 4.920.794 io0 0.983 0.182 1.967 o.364 2.950 0.547 3.933 0.729 4-91I 794 ic0 o0982 0.187 1.965 0.373 2.947 0.560 3.930 0.746 4-912 794 110 o.982 o0.91 1.963 0.382 2.945 0.572 3.927 0-763 4-908 790 ii o.98I o.195 1.962 o0390 2.942 o0585 3.923 0.780 4.904 784 ii o-980 0o.99 1.960 0.399 2.940 o0598 3.920 0.797 4-900 784 iIx 0.979 0.204 1.958 0.407 2.937 o.6ii 3-916 o.815 4.895 784 L2~ 0o978 0.208 1.956 o.4i6 2.934 0.624 3.913 0.832 4.891 7S~ 121 0-977 o.212 1.954 0.424 2.932 0.637 3.g909 0849 4.886 -771 124 0.976 0.216 1.953 0.433 2.929 o.649 3.905 o-866 4-88I 77^ 12 0.975 o.22I 1.95I o.44I 2.926 0.662 3.,9oI 0o883 4.877 774 S~I| 0.974 0.225 1.949 0.450 2.923 0.675 3.897 0.900 4-872 70 i31 0.973 0.229 1.947 0.458 2.920 o0688 3.894 0.917 4.867 764 13 0-972 6.233 i.945 0.467 2-.917 0.700 3.889 0.934 4.862 764 1i34 0-971 0.238 1.943 0 o.475 2.914 0.713 3.885 0.95I 4.857 76. 140 0.970 0.242 1.941 0.484 2.911 0.726 3.88I 0.968 4.85I 760 i41 0.969 o.246 I.938 6o492 2.908 0.738 3-877 0.985 4.846 754 144 o-968 0.250 1.936 0o.50 2.904 0.75I 3.873 1.002 4-841 754 144 o967 0.255 1.934 o0509 2-90o 0.764 3.868 I.or8 4.835 754 15" o.966 0.259 1.932 o.518 2.898 0.776 3.864 1.o35 4.830 5~o't LP tep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. - 2 LATITUDES AND DEPARTURES. 6 7 8 = eE=Dep. Lat. Dep. Lat. Dep. Lat. Dep. p 0 o0oo 6-ooo 0ooo 7 ooo oooo. ooo 8. 0 oooo 9.ooo oooo 0 0 o 0-022 6ooo 0.026 7000 oo3 8ooo o. o35 98000 035 03 89 oA o.o44 6.ooo 0.052 7.0oo o.o6I 800 0.o 070 9.000 0.079 89A oj o.o65 5-999 0.079 6.999 0.092 7-999 o.io5 8.999 o.ii8 8910 o0o87 5-999 o-io5 6.999 0o.22 7-999 o-i4o 8.999 o0.57I S9~ 14 o.io9 5.999 o.-i3 6.998 o.i53 7.998 0. 175 8.998 ci96 88 I o.0i3i 5.998 o-157 6.998 0.83 7-997 0.209 8-997 0.236 88 i o0.153 5-997 o.i83 6.997 0.214 7.996 0.244 8.996 0*275 884 02o.174 5-996 0.209 6.996 0.244 7.995 0.279 8.995 o-.34 ~S~ 24 0.I96 5.995 0.236 6.995 0.275 7.994 o - 3 4 8.993 0.353 87i 24 0o248 5-994 0-262 6.993 o.305 7-992 o.349 8.99I 0.393 87 24 o024o 5.993 0-288 6.992 o.336 7-992 0.384 8.990 0.432 874 30 0o262 5-992 o.34 6.990 o0.366 7-989 0-419 8.988 0.471 S| ~ 34 0.283 5.990 o.0-34o 6.989 0.397 7.987 o.454 8.986 o0.5o 864 3 o0.305 5-989 o.366 6.987 0.427 7.985 o.488 8.983 0.549 86 o3 0.327 5.987 0.392 6.985 o.458 7.983 0.523 8.981 0.589 86 40 o.349 5-985 0o.49 6.983 o0.488 798i o.558 8.978 o.628 6~ 44 0.371 5-984 0o445 6-98I 0-.59 7-978 0.593 8-975 o.667 85 41 o-392 5.982 0-471 6-978 0-549 7-975 o0628 8.972 o 706 854 44 o0-414 5-979 0-497 6-976 o-58o0 7973 o0662 8.969 0-745 85. 50 o-436 5977 0o-523 6.-973 o.6o 7-970 0.697 8966 0-784 ~5'1 54 o0458 5.975 0.549 6-971 0.641 7.966 0.732 8.962 0.824 844 54 0-479 5.972 0-575 6.968 0.67I 7.963 0-767 8.959 o0863 84 5- o.5o[ 5-970 -.6o01 6.965 0.701 7-96o 0-802 8-955 0-902 844 6~ o0523 5-967 0o.627 6-962- 0.732 7-956 o-836 8.951 0.941 S4~0 64 o.544 5.964 0.653 6.958 0.762 7.952 0.87I 8-947 0o980 834 6 o.566 5-96I 0.679 6.955 0.792 7-949 o0906 8-942 1.019 834 64 o.588 5.958 0.705 6.951 o0.823 7.945 o0940 8.938 I.o58 834 7 0.609.5-955 0.731 6.948 o-853 7-940 0.975 8.933 1-097 S3 7 o-63i 5.952 0-757 6.944 o-883 7-936 0-oio 8-928 1-i36 8247 o.653 5.949 0-783 6.94/o 0.9I4 7-932 I-o44 8-923 1-175 824 7 0-674 5'945 0-809 6.936 0.944 7-927 1-079 8-918 1.214. 824I c| o-696 5-942 o-835 6.932 0.-974 7-922 I-II3 8-912 I -253 S20 8 0-717 5.938 0-86i 6.928 -00oo4 7-917 i48 8-907 I-29I 8| I 8 0-739 5.934 0.887 6.923 -.o35 7-92 *1-82 8-gI901 330 81 84 0o76I 5-930 o0913 6.919 -.o65' "7 907 1-2I7 8-895 I.369 8i4 90 0-782 5.926 0.939 6.914 I.095 7-902 1-25I 8-889 i-4o8 S1~ 94 0.804 5-922 0-964 6.-909 I.25 7896 1-286 8883 -447 809 94 0-825.5-918 0-990 6.904 i.155 7-890 I-32o 8-877 1-485 80o 94 0o847 5-9I3 1-016 6.899.i. 85 7-884 I.355 8-870 I.524 80o 100 o-868 5-909 3-042 6-894 I-2I6 7-878 I.389 8.863 I.563 S8W i |o o. 0890 5-904 i.o68 6.888 I.246.7-872 I -424 8.856 I-601 79 oI.0.g911 5-900.o093 6.883 1-276 7-866 1.458 8-849 i.640 794 i o 0-933 5.895 I-119 6.877 i.306 7-860 1.492 8.842 I.679 79 110. o0.954 5.890 I.-45 6.871 i.336 7.853 r.526 8.835.717 7 9~ iij 0-975 5.885 1.171 6.866 i.366 7-846 i.56i 8-827 1756 784 Il4 0-997 5.880 I.-96 6.859 I.396 7-839 I-595 8-819 1.794 784 i i.oi8 5.874 1.222 6.853 1.425 7.832 1-629 -8818i i.833 784 10o.0o4o 5.869 1-247 6.847 I.455 7.825 i.663 8-803- i.87I 7~~ 12-4 i.o61 5-863 1-273 6-84i i.485 7-818 1.697 8795 1-910 774 124 2.082 5.858 1-299 6.834 I-515 7.810 1.732 8-787 1-948 77.4 124 i.io3 5.852 i.324 6.827 i.545 7-83 1.766 8-778 1.986 774 o30 1.1i25 5.846 i.35o 6.82I 1.575 7795 1.8oo 8-769 2-025 770o 1i34 I146 5-840 I375 6.814.-604 7787 i.834 8-760 2.063 76 13~ 1-167 5.834 i14o0 6.807 i-634 7779 i-868 8.75 2-.101 764 i34 1i.88 5.828 1-426 6.-799 i664/ 7-771 1 902 8-742 2.139 764 140 1-2IO 5.822 -I.452 6-792 I.693 7.-762 I.935 8-733 2.-77'o76 14-4. 231 5-8i5 I477 6-785 I.723 7-754 I.969 8.723 2.215 754 I44 1.252 5-809 I.5o2 6-777 1.753 7745 2-003 871i3 2-253 754 144 1.273 5.802 I.528 6.769 1-782 7-736 2-o37 8-703 2291 754 10 1-294 5-796 I-553 6-761 1-812 7-727 2.071 8-693 2-329 75~ I Lat. Dep. Lat. Dep.' Lat. Dep. Lat. Dep. Lat. C 001 e 7 0 ___ 3s LATITUDES AND DEPARTURES. - CD E.. _ _ ^ Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. p 15" 0.966 0- 259 I.932 o5i8 2.898 o0 776 3.864 i.o35 4 -830 y7o 54 0.965 o0.263 1.930 o.526 2.894 0.789 3.859 1-052 4.824 744 I5A 0.964 0.267 I'927 o.i34 2.89I 0.802 3.855 1.0o69 4.8T8 744 I154 0.962 0.271 I1.925 o.543 2.887 o.8i4I 3.850 i.o86 4.812 744 160 0.96i 0.276 1.923 o.55i 2.884 o0.827 3.845 i o3.4.8o6 740 i6i 0.960 0.280 1.920 o56o 2.880 D0.839 3-84o 1.119 4.800 734 16 0.o959 0o284 1-9I8 o.568 2.876 o0.852 3.835 I.i36 4 794 734 i64 0.958 0.283 i.915 o0.576 2.873 o.865 3.830 1.153 4.788 73I J1.~ o.956 0.292 1.913 o.585 2.869 0.877. 3.825 1.169 4.782 730 17- 0.955 0.297 1.910 0.593 2.865 o0.890 3.820 i.86 4.775 724 I7 0o.954 0o3oi 1.97 0o60I 2.86I 0.902 3.8I5 1.203 4-769 721 17i 0.952 o.3o5 I-*905 o.6io 2.857' 0.915 3.8io 1.220 4.762 724 1~o 0.951 0.309 1-.92 o.6i8 2.853 0.927 3.804 1-236 4.755 72~ 84 0.950o o.3I3 1.899 0.626 2.4g/9 0.939 3-799 i-253 4-748 714 i84 0.948 0o.37 1-897 o.635 2.845 0.952 3.793 1.269 4-742 71I I8-/ 0.947 0o321 1.894 o643 2.841 0.964 3.788 1.286 4-735 71Ii 19~ 0.946 o0.326. 1.89 o.65i 2.837 0.977 3-782 1.302 4.728 71~!19 0o.944 o.33o0 1888 0.659 2.832 0.989 3.776 1.319 4.720 701 19/ 0o943 o'334 I 885 o.668 2.828 /i.ooi 3.77I i.335 4-713 704 I19 0.941 o.338 1.882 0.676 2.824 Io014 3.765 1.352 4.706 704 20~ 0.94o 0.342 I 879 o.684 2.81.9 1-026.3. 759 i 368 4.698 7oo 20i o.:38 o.346 I 1.876 o.692 2.815 io38 3.753 I.384 4 69, 694 204 0-937 o.350 1.873 0.700 2.810 i o5i 3.747 1401 4.683 694 204 0 935 0.354 1. 870 0. o709 2-805 io63 3.741.417 4-676 69 210 0-934 o.358 1.867 0. o717 2.801 1.075 3.734 I 433 4.668 go 21 0.O932 0.362 i.864 0. o725 2.796 1.0o87 3.728 i.45o 4,660 681 21 0.930 o0.367 i.86I 0.733 2.79.I i.ioo 3.722 i.466 4.652 684 21 o0.929 0.371 i.858 0.741 2.786 1i.Ii 3.7i5 1 482 4.644 68 22~ 0.927 o0375 I 854 0.749 2-782 1.124 3.709 I.498 4.636 o6S 22 0o-926 0.379 i. 85i 0.757 2-777 i.36 3.702.5I5 4.628 674 224 0.924 o.383 i.848 0.765 2.772 i.i48 3.696 i.53i 4.619 674' 2241 0.922 0.387 I.844 0-773 2.767 i.6o 3.689 1 - 547 4.6 16 674 230 0-92I o.39i i.84i 0.781 2.762 1.172 3.682 i.563 4.603 (o;~ 23 0.919 o 395 i.838 0.789 2.756 1.184 3.675 1-579 4-594 664 234 0-917 0o399 i 834 0 o.797 2-751 1 -96 3.668 1-595 4.585 664 234 0-.915 o.4o3 i.83i o.8o5 2.746 I.208 3.66i 1-.61 4.577 664 2~ 0-9 o 4 0-407 1827 0.83 2.741 1-220 3.654 1-627 4.568 66~ 244 0-912 0-o.4 1 824 0.82I 2.735 I.232 3.647 i-643 4.559 654 244 0-9I0 0o.45 1.820 0.829 2.730 1.244 l 3.64o0 I659 4-55o 654 244 0.908 0.419 i-8i6 o0.837 2-724 I-256 3.633 I-675 4-54i 654 25 0og906 0.423 i.8i3 o.845 2.719 1-268 | 3.625 1.690 4.532 6-, 0 254 0.904 0.427 I.809 o.853 2.713 1.280o 3.6i8 1.706 4.528 644 25 0o.903 0o.43i.8o5 o.861 2.708 1.292 3.6io 1.722 4-513 644 254 0.901 0o.434 i.8oi 0.869 2-702 i.3o3 3.6o3 1.738 4-503 644 26~ o0.899 o.0-438 I.798 o0.877 2.696 i.3i5 3.595 1.753 4-494 640 264 0.897 0-442 1.794 0o.885 2.691 1.327 3.587 1.769 4.484 634 26 o0.895 o.446 1.790 o0.892 2.685 1.339 358o0 1.785 4-475 63 264 o0893 o 45o.7j86 0.900oo 2.679 i-35o 3.572.800oo 4.465 634 27 0o89i o.454 1.782 o.9o8 2.673 1.362 3.564 /.8I6 4-455 6 3o 274 0.889 0-o.458 1.778 0.916 2.667 1.374 3.556 i.83i 4-445 624 274 0.887 0o462 1.774 0 o-.923 2.661 i.385 3.548 1.847 4-435 624 274 o.885 0 466 1.770 0.931 2.655 1.397 3.540 1.862 4-425 624 28~ o0883 0.469 1.766 0.o939 2.649 I.4o8 3.532 1.878 4.4i5 620 284 o0881 0.473 1.762 0.947 2.643 1.420o 3.524.893 4 4o4 61j 28 o0.879.477 i 758 o 954 2.636 1.431 3.55I 1.909 4394 61o 28 0. o877 o.48 I.753 0.962 2.630 I.443 3.507 1-924 4.384 6iI 290~ 875 o.485 1.749 0-970 2.624 1.454 3.498 1.939 4.373 6 1~ 291 0.872 o0.489 -745 0.977 2.617 I-466 3.490 i.954 4-362 604 294 0.870 0.492 I.741 0.985 2.611 1.477 3.48I 1.970 4.352 6oj 291 o.868. o0.496 I.73b 0.992 2.605 1.489 3.473 i.985 4-34I 604 30 o0-866 o500oo 1.732 i.ooo 2.598 1.500 3 464 2000oo 4-33o 60o | Dep. Lat. Dep. Lat. Dep. La t Pep. Lat' Dep. LATITUDES AND DEPARTURES. _ 0 0_ 7 0 0_ Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 15" I-294 5-796 i.553 6.76I 1.812 7.727 2.07i 8.693 2.329 750 75 i.31i5 5.789 1-578 6.754 1i841 7-718 2.104 8.683 2.367 744 i54 i.336 5.782 i.603 6.745 1.871 7-709 2.-38 8.673 2.405 74[ i5- 1.35-7 5775 1.629 6.737 1.900 7.700 2.172 8.662 2.443 744i 160 1-378 5.768 i.654 6.729 I-929 7.690 2.205 8.65i 2,.481 740 i64 I.399 5/760 /-679, 6.720 1.959 7.680 2.239 8 640 2.518 73/61 1.420 5.753 1.704 6.712 I.988 7.671 2.272 8.629 2.556 731 6.64 i-44i 5.745 1-729 6.703 2.017 7-667 2.306 8.618 2.594 731 1/ 1.462 5.738 1.754 6.694 2.047 7.-650 2.339 8.607 2.63I ~30 17 1i.4831 5.730 1.779 6.685 2.o076/ 7.640 2.372 8.595 2.669 7241 I7 i.50o4 5.722 i.804 6.676 2.o05 7.630 2.406 8.583 2.706 724 174 7.524 5.714 1-829 6.667 2 7.34 7.6r9 2.439 8.572 2.744 72' /lO 1.545 5.706 1.854 6.657 2.-63 7.608 2.472 8.560o 2.78I 72 184 i.566 5.698 1.879 6.648'2.192 7.598 2.505 8.547 2 818 711 i8/ 1.587 5-690 o.904 6.638 2.221 7.587 2.538 8.535 2.856 711 i81.-607 5.682 1.929 6.629 2.250 7.575 2.572 8.522 2.893' 7 1 I90 7.628 5.673 1.953 6.619 2.279 7.564 2.605 8.5io 2.93o0 / 1 9.4 i-648 5.665 1.978 6.609 2.308 7.553 2.638 8.497 2.967 704'91.-669 5.656 2-003 6.598 2.337 7-541 2-670 8.484 3-004 701 793 7.690 5.647 2.028 6.588 2.365 7.529 2.703 8.477 3.o41 702O. 1.7Io 5.638 2.052 6.578 2.394 7.518. 2.736 8.457 3.078 00 204 I.731 5.629 2.077 6.567 2.423 7.506 2.769 8.444 3.ii5 693 201 1.751 5.620 2/101 6.557 2.457 7.493 2.802 8.43o0 3.52 691 20J 1.77I 5-.6i 2.126 6.546 2.480 7-48s 2.834 8.4i6 3.189 694 1~0 I-792 5.6o0 2.150 6.535 2.509 7-469 2.867 8.402 3.225 690 214 7.812 5.592 2.I75 6.524 2.537 7-456 2.900 8.388 3.262 684 21 1i.833 5.582 2.199 6.5J3 2.566 7-443 2.932 8.374 3.299 681 274 i.853 5.573 2.223 6.502 2.594j 7.430 2.964 8.359 3.335 684 %f~O 1.873 5.563 2.248 6.490 2.622 7-417 2.-997 8.345 3.371 6/ 224 1.893 5.553 2.272 6.479 2.651 7-404 3.029 833o 3.40o8 671 221 7.913 5.543 2.296 6.467 2.679 7-391 3-o61 8-3i5 3.444 67-i 229 1.934 5.533 2.320 6.455 2.707 7.378 3.094 8.3oo 3.480 67 230 1-954 5.523 [ 2.344 6.444 2.735 7.364 3.726 8.285 3.5I7 6'~ 234 1'974 5.513 2-368 6.432 2.763 7-350 3.158 8-269 3.553 664 23 I-'994 5.502 2.392 6.419 2-79I 7-336 3.g90 8.254 3.589 661 234 2.074 5.492 2.416 6.407 2~.89 7-322 3.222 8.238 3.625 664 240 22.034 5.48i 2 440 6.395 2.847 7-308 3.254 8.-222 3.66i 6 60 244 2.054 5.471 2-464 6.382 2.875 7-294 3.-286 8.2o606 3.696 654 241 2.073 5.460 2-488 6.370, 2.903 7-280 3.3i8 8.o90 3-732 651 241 2.093 5.449 2-5I2 6.357 2-937 7-265 3-349 8.173 3.768 654 20Q 2.113 5.438 2.536 6.344 2.958 7-250 3.38r 8.157 3:804 65~ 254 2.133 5.427 2.559 6.331 2.986 7-236 3.41i3 8.I4o. 3.839 64 251 2.153 5.4i6 2.583 6-3i8 3.oi4 7-221 3.444 8-123 3.875 644 251 21.72 5.404 2.607 6.3o5 3o041 7-206 3.476 8.io6 3.910 64i f2;0 2.192 5.393 2.630 6.292 3.069 7.-90 3.507 8.089 3.945 640 26 2-.2II 5.38r 2.654 6.278 3.096 7-175 3.538 8.072 3.981 634 261 2.237 5.370 2.677 6-265 3.123 7.160 3.570 8.o54 4.0o6 631 261 2.250 5.358 2.70I 6.251 3.15I 7-.44 3.6o01 8.037 4-o5 63j 2'0 2.270 5.346 2.724 6.237 3.-178 7.-28 3.632 8o.01 4-086 6~0 27i 2.289 5.334 2.747 6.223 3.205 7.7-.2 3.663 8.ooi 4-.2r 621 271 2.309 5.322 2.770 6.209 3.232 7.096 3.694 7.983 4-.56 621 27 2.328 5.3o10 2.794 6.195 3.-259 7.080 3.725 7.965 4-.90 621 ~So 2.347 5.298 2.817 6.i8i 3.286 7.064 3.756 7.-947 4.225 620 28- 2.367 5.285 2.840 6.I66 3.313 7.047 3-787 7-928 4.26c' 6i1 281 2.386 5.273 2.863 6.752 3.340 7-o3I 3.817 7.909 4-29.4 6j1 284 2.405 5.260 2.886 6.137 3.367 7-014 3.848 7.891 4.329 6i 290 2.424 5.248 2.909 6-I22 3.394 6.997 3.878 7.872 4-363 61~ 294 2.443 5.235 2.932 6.107 3-420 6.980 3.909 7.852 4.398 6o4 291 2.462 5.222 2.955 6.093 3.447 6.963 3.939 7.833 4-432 60o 291 2-48I 5.209 2-977 6.077 3.474 6'946 3.970 7.814 4-466 6o0 300 2.500 51796 3.oo000 6.062 3.500oo 6.928 4-ooo 7`794 4-5oo 60~ ^ Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat..7 0_ _'0 ____I, 5 LATITUDES AND DEPARTURES.. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep., Lat. pi 30~ o.866 o05oo 1.732 io000 2.598 500oo 3 464 2000oo 4.33 600 30o o.864 o.504 1.728 1.oo8 2.592 I.5II 3.455 2.015 43I19 594 30 o0.862 o.508 1.723 I.oI5 2.585 I.523 3.447 2-o30 4-308 591 30o 0.859 o*0-5 1 719 1o023 2.578 I.534 3.438 2.045 4-297 591 3.0t 0.857 o-.55 I.714 i o3o 2.572 I.545 3.429 2.060 4.286' 0 31I o.855 o0519 I70 i-o38 2.565 I.556 3.420 2.075 4-275 584 31 0o.853 o0.522 1.705 i.o45 2.558 I.567 3-4II 2o090 4.263 58J 3i1 o.85o 0.526 1.701.052 2.551 I.579 3.40o 2o105 4-252 58' 32~ o0.848 0o530 i.696.o60 2.544 I 590 3.392 2.120 4.240 e ~8 32 0o.846 0o534 1.69I I 067 2.537 ir6oi 3.383 2.1-34 4.229 574 324 o.843 0-537 i.687 I 075 2.530 1.612 3.374 2.149 4.217 57I 324 o.84i 0.54i I-682I.082 2.523.1623 3.364 2.164 4.205 57-t 33 0.839 0.545 I 677 I -o89 2.516.I634 3.355 2-I79 4-.93 57 331 o.836 o.548 I 673 I 097 2.509 645 3.345 2'193 41I8i 561 33 o.834 0o552 I.668 iio04 2.502 I.656 3.336 2.208 4-I69 56] 33, o.83I o0556 i.663 i.ii 2.494 I-667 3.326 2.222 4.157 564 34~ o.829 o 0559 i.658 I 1II8 487 I.678 3.3I6 2.237 4-I45 5 60 34- 0 o827 o563 I.653 I1.26 2.480 I.688 3.306 2.251 4-I33 55 341 0.824 0o566 i.648.33 2.472 I.699 3-297 2.266 4-I21 551 344 0.822 0.570 i.643 J-14o 2.465 1.710 3.287 2.280 4-o108 554 85~ o0.8I9 o0-574I -638 I.147 2-457 I172I 3.277 2.294 4.096 550 354 0o-87 0577 I.633 I i.I54 2450 1.731 3.267 2.309 4-.83 5435| 0o.84 o.58I I. 628 I.i6i 2 442 1.742 3'257 2-323 4-071 544 354 0-812 o584.623 i.68 2.435 1753 3.246 2.337 4.o58 544 3g0 o080 o-588 i.6i8 1.176 2.427 1763 3.236 2.351 4-045 541 36 o080o6 0.591 i.6i3.i83 2.419 I 774 3226 2.365 4-.32 534 364 o0804 0.595.6o8.190o 2.412 784 3.215 2.379 4-oI9 53 364 o080o 0.598 I.603' I97 2-404 1-795 320o5 2.393 4oo6 534 370 0 799.o602 1.597 1 204 2-396 I-805 31I95 2.407 3.993 53 374 0 796 0o605 1.592 1.2i 2.388 1-816 3 i84 2.42I 3.980o 524 1 374 0-793 o0.609 1587 I218 2.380 I826l 3I73 2.435 3.967 52 374 0-79 0o.612 i.58j I.224 2.372 1-837 3.I63 2.449 3.953 521 30 o0788 0.616I. 76 1.576 I.23 2.364 I847 3-.52 2.463 3940 5 2~ 381 0o785 0.619 1I 57 I -238 2.356 1-857 3-I41 2.476 3.927 514 384 0o783 0o623 I.565 1.245 2.348 I-868 3-I30 2-490 3.913 51i 381 0-780 o.626 1.56o 1.252 2.340 r878 3.120 2.504 3899 5 39~ 0.777 o.629 i-554 1-259 2.33r J888 3.0o9 2.517 3.886 51~ 39i 0.774 o633 1.549 1 265 2.323 I-898 3.098 2.53I 3-872 5oj 394 0-772 0o636.543 1.272 2.3I5 I-908 3.o86 2.544 3-858 504 394 0-769 0o639 I538 1.279 2.307 1-918 3.075 2.558 3.844 5o04 40~ 0.766 0o643 1.532 1.286 2.298 1.928 3.064 2-57I 3-83o 5i 0o 40o 0.763 o.646.526 1.292 2.290 1.938 3.o53 2.584 3.816 494 40 0.760 o.649 I 521 1299 2.28I 1.948 3.042: 2.598 3.802 49 40o 0.758 0o653 I.5i5 i.306 2.273 1.958 3.o30 2.611 3.788 49i.41' 0.755 o.656 r.5o9 1-312 2.264 1.968 3.019 2.624 3774 490~ - 4I |0.752 o0659 -504 1 319 2.256 1.978 300oo7 2.637 3-759 48- 41 0-749 / o-663 4498 1-325 2.247 1.988 2.996 2.650 3.745 484 411. 0.746 o.666 r.492.332 2.238.998 2.984 2664 3.730 48 41~ 0.743 0.669 1.486 I338 2-229 2.007 2973 2.677 3716 4 S~ 421 0.740 0o672 I 480 - 345 2.221 2.017 2.961 2.689 3.701 471 42~ 0/.-737 o676 I.475 -I35I 2-2I2 2.027 2-949 2-702 3.686 47^ 42J.0. 734 o.679 I 469 I.358 2.203 2.036 2-937 2.7I5 3.672 474 43~,o.-73 o.682 I.463 I-364 2.194 2.046 2.925 2.728 3.657 4~ 0 434 o0718 o.685 1457 i 370 2i.85 2.056 2.9I3 2.741 3.642 461 43 o.,725 o.688 I.451 1377 2.176 2.065 2.901 2.753 3.627 464 43 o0.722 0.692 1.445 1 383 2I.67 2.075 2.889 2-766 3.6I2 464 440 0,719 o695 I'439 1.389 2. i58 2.o84 2.877 2-779 3597 46~ 444 0o.76 o.698 I.433 1.396 2.I49 2.093 2.865 2.79I 3-582 454 L44 0.713 0.701 1.427 1-402 2.140 2.I03 2.853 2.804 3.566 45"i 444 0.710 0.704 1-.420 1-408 2.131 2.112 2.841 2.816 3~55i 4545 1 0.707 0.707 1414I| 1414 2.12I 2.121 2.828 2.828 3.536 L 45 0' Dep. at. Dep. Lat. Dep. L Dep. Lat Dep. 1 Q e 211.11'g- 11 - LATITUDES AND DEPARTURES. I' Dep. Lat. Dep. Lat. Dp Lat. Dep. Lat. Dep. 300 2-500 5. i196 3ooo 6062 3.5oo 6.928 4-000 7-794 4o500 600 3o0 2.519 5'i83 3-023 6-047 3.526 6.911 4-o3o 7-775 4.534 594!304 2.538 5-170 3.o45 6-o3I 3.553 6.893 4-o6o 7-755 4.568 594 3o0 2.556 5.i56 3-068 6.oi6 3-579 6.875 4.090 7-735 4.602 59~. 310 2-575 5.143 3-09o 6.000ooo 3605 6.857 4-120 7.715 4.635 9~0 31i 2.594 5.-129 3-113 5.984 3-63I 6.839 4-.50 7.694 4.669 581 31& 2.612 5.ii6 3-135 5.968 3.657 6-821 4-.80 7.674 4-702 584 03i4 2.631 5.102 3.157 5.952 3.683 6.8o3 4.210 7.653 4.736 584 32 2.650 5o.088 3-.8o 5.936 3-709 6.784 4-239 7.632 4-769 S0 324 2.668 5.074 3.202 5-920 3-735 6-766 4-269 7-612 4.802 571 324 2.686 5.060 3.224 5.904 3.76i 6-747 4-298 7-59I 4-836 574 324 2.705 5'046 3.246 5,887 3.787 6-728 4.328 7.569 4'869 574 833 2.723 5.032 3.268 5.871 3.8i2 6-709 4-357 7.548 4.902 /70 334 2.741 5.oi8 3-290 5.854 3.838 6-690 4-386 7.527 4-935 564 334 2.760 5-oo3 3.312 5.837 3.864 6.671 4.4i6 7-505 4-96.7 564 334 2.778 4-989 3-333 5-820 3.889 6.652 4-445- 7-483 5000ooo 56i 340 2-796 4-974 3.355 5-8o3 3-914 6.632 4.474 7-461 5-o33 5/ 0 341 2.8,4 4-960 3.377 5.786 3.940 6.613 4.502 7.439 5.065 551 341 2.832 4-945 3.398 5.769 3.965 6.593 4-531 7-417 5-098 554 344 2.850 4.930 3.420 5-752 3.990 6.573 4-560o 7.395 5-.i30 554 350 2.868 4.915 3.44i 5-734 4.oi5 6.553 4.589 7372" 5-162 5-~ 351 2.886 4.900oo 3.463 5.716 4.o4o 6.533 4.617 7-350 5.194 544 354 2o904 4.885 3.484 5.699 4.065 6-513 4.646 7.327 5.226 544 354 2-921 4-869 3.505 5.68i 4.090 6.493 4.674 7.304[ 5.258 544 30~ 2.939 i 4.854 3.527 5-663 4.ii5 6.472 4-702 7-281 5-290 5~40 361 2.957 4.839 3.548 5.645 4-I39 6.452 4-730 7-258 5.322 531 364 2-974 4.823 3.569 5.627' 4-I64 6.431 4-759 7-235 5-.353 534 364 2.992 4-808 3.590 5.609 4.i88 6.4io 4-787 7-21' 5.385 53i 3I 0 3oo009 4-792 3.6ii 5-590 4.213 6.389 4.8i5 7-188 5.416 ~ 30 374 3-026 4-776 3.632 5.572 4.237 6.368 4.842 7-I64 5.448 524 374 3.o44 476o0 3.653 5.554 4-261 6.347 4-870 7.140 5-479.524 374 3-o6i 4-744 3.673 5-535 4.-286 6.326 4-898 7.116 5.5Io 524 3~ 3-078 4.728 3.694 5.516 4-310 6.304 4-925 7-092 5.54i 52~ 381 3-095 4-712 3.715 5.497 4.334 6-283 4.953 7.068 5.572 514 384 3r113 4.696 3-735 5.478 4.358 6.261 4.980 7-043 5-6o3 51[ 384 3-.3o 4.679 3-756 5.459 4-381 6-239 5o~07 7-019 5.633 5I~ 39~0 3.147 4-663 3.776 5.440 4.4o5 6.217 5.035 6.994 5.664 51~ 391 3.i64 4.646 3-796 5-.421 4-429 6.195 5-062 6-970 5694 501 394 3-.80 4-63o 3-8i6 5.40o 4-453 6-173 5-089 6.945 5-725 5o0 39Z 3-197 4-6i3 3.837 5.382 4476 6-i5i 5.ii6 6.920 5.755 50o 490 3.214 4.596 3.857 5.362 4.500oo 6.128 5.142 6.894 5-785 500 4o0 3.23,I 4-579 3.877 5.343 4.523 6-io6 5-169 6.869 5.815 494 40o 3.247 4.562 3.897 5.323 4.546 6-083 5-196 6.844 5.845 494 40o 3.264 4-545 3.917 5-303 4.569 6.o6i 5.222 6.8i8 5.875 494 ~410 3.280 4.528 3.936 5.283 4.592 60o38 5.248 6.792 5-905 490 414 3-297 4.51I 3.956 5.263 4-6i5 6.oi5 5.275 6.767 5.934 481 414 3-313 4.494 3.976 5.243 4.638 5-992 5.30o 6.741 5.964 484 411 3.32.9 4-476 3.995 5-222 4-661 5.968 5.327 6.715 5.993 484 42~ 3.346 4.459 4-.o5 5-202 4.684 5.945 5-353 6.688 6.022 4I~ 424'3.362 4-441 4-o34 5-.82 4.707 5.922 5.379 6.662 6.o5i 474 424 3.378 4-424 40o54 5-161 4.729 5.898 5-405 6.635 6.o080 47 42 3.394 4.406 4-073 5.140 4-752 5.875 5.43o 6.609 6.109 474 430 3.4io 4.388 4-092 5.119 4-774 5-85i 5-456 6.582 6.i38 4,Q1 4314 3.426 4.370 4i-111 5-099 4-796 5.827 5-48i 6.555 6.167 46 43 j 3.442 4-352 4-i3o 5-078 4.818 5.8o3 5.507 6.528 6.195 41, 434 3.458 4.334 4-I49 5-057 4-84i 5-779 5.532 6.50o 6.224 46'f 440 3.473 4.316 41i68 5-o35 4.863 5-755 5.557 6-474 6-25 460 444 3-489 4.-298 4-187 5-014 4.885 5-73o 5.582 6.447 6.280, 451 t44 3.505 4-280 4-206 4.993 4-go6 5-706 5.607 6.-419 6...3 45S 441 3-520 4.261 4.224. 497I 4-928 5.68.5-632 6.392 6.336 44 140 3.536 4-243 4-243 4-950 4-950 5.657 5.657 6.364 6-.364 450 t2 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep.. Lat tk - ~~~~~~. = El ~~~~~~~~~~~~~~~' TABLE OF CHORDS: [RADius1.-O0000]. M. 0 0~ 2 2 30 4~ 5 60o 70 So 90 1o0~ 0o.0ooo *0175.o349 0524.0698.0872.1047.I221 139. 5.1569 -I743 o' I.ooo3.OI77 0o352.o526.0701.0875.IO50.1224 -1398.1572 I746 I 2.0ooo6.o80o 0355.0529.074.0878.053.1227 -i401.1575' 749 2 3.ooog.oi83.o358 o0532.0707.088I.Io55.1230 1I404.1578.1752 3 4 OI2!.0I86.o36r.o535.07O1.o884 -ro58.1233.1407 i'58i 4I755 4 5.ooI5.oi189.o364 *o538.07I3.0887.-061.1235.1410.I584 -1758 5 6 OOI7 -1OIg2.0366.o54.O0715 o.08g.' 064.1238.1413.1587 *I76I 6 7 *0020 0oI95 o36g 0544.07-18.o8.93 ~.I67'1241 -14I5 *I589.I763 7 8 o0023 OI018.0372.0547 ~072I o896.1070 1.244 1418 I1592 I1766 8 9 0oo26 -020o.0375 o0550.o724 *0899.IC73.I247 -1421.1595'1769 9 10.0029'0204.o378.0553.0727 *0901 10G76.1250.1424.-I598 *I772 10 ii.0oo32 0o207'o38I.o556.0730.0^04 -1079 1253.1427 *I6oi. I775 II 12 o.0035 0209.o384.o558.0733.0907.1082.1256.430.I604.1778 12 I3.oo38.02I2.0387 Oc56i.0736.010.I.io84 1I259.433.1607 I178I 13 14.oo004io0215.039o 0564.0739.0913.1087 -I262.1436.16IO -1784 14 5 oo0044.02I80.o393.0567.0742.0916 I.19o.1265.1439 11613 *1787 I5 I6 o00477 0221.0396.0570.0745 1.0919.093.1267.1442.16I6.1789 I6 17.0049.o224 o0398.0573.0747.0922 i1096.1270.1444 1I618'1792 17 I8.00521.0227 "040I.0576.075.0o925.1099.1273. 1447 1I621.1795 i8 19.oo55.0230 o0404.0579.0753.0928.1102 1276 1I450 1624.1798 19 20 00oo58.0233.0407 o0582.0756.093 I.105 I1279 -I453.1627.I80o 20 21 oo 6.0236o 04Io.o585.0759l 0933 11oo8.1282.456 1 630.1804 21 22.0064/ 0239 o04I3.o588.0762.0936.II11.1285.1459.I633 1I807 22 23.0oo67 024I.0416.0590.0765 o0939.II14'1288 1462.I636.18Io 23 24.0070.0244.0419.0593.0768.o942. I1I6.1291.i465.639.1813 24 25 0oo73 0247.0422 o0596 1077I.o945.111. 29 4.I468.1642.18i6 25 26.0076 10250.0425.0599g 0774.o948.1122.91296 1471 I.645 1i8i8 26 27'0079 0253.0428.0602.0776 1095I.I125'.299 1473'1647 1I82I 27 28 00810 o0256.0430 oo605.0779 -o954'!II228 1302 1476 I 65o 1I824 28 29 o0084.02591 o433.o608.0782.0957.113I 130o5 1479.i653.1827 29 30.0087 -02621.0436.061I.0785 0960.1134 1I308 I1482.1656'I830 30 31.oo9o 0265.0439.614.o788.0o62.1I37.i3ii I.1485 1I659 1.833 31 32.oo93.0268 10442.06I7'0791 -I965'I401.13I4.1488.1662.1836 32 33.0096.0271.o445.0619.o794.o968. I143.1317 1491.665 I1839 33 34.0099 o 0273 o448.o622.0797 10971.I45.I320.1494 1.668 1.842 34 35.010.0276.o451.o625 18ool.0974.1148.1323.1497 I1671.I845 35 36 ooIo5.0279.o454 o0628.o803.0977 11.51.1325 I 50o.1674 I.847 36 37.oro8.0282.0457.o63i. o806.og80.1154.1328.1502.1676.1850 37 38.OII1.0285.046o.o634.o808.0983.1157 1.331i i5505 o 679. I853 38 39.OII3'0288.0462'0637 -08II.0986.1160o.334.15o8.1682.i856 39 40.0oI6.0o291.o465.o640.o8i4 o0989.1I63.1337 I5.II.1685.1859 4o 4I.0II91.0294.0468 o0643.0817.0992.II66.i3401.I54.i688.1862 41 42.0122.0297.0471.o646.o820 0994.1169 I343.1517.I69g1 I865 42 43.o25 o.0300.o474.0649 o.823 0997 1172. 346.1520.1694 868 43 44.0128.0303.0477.o65I.o826.1Ioo.1175.13491.1523 1.697.1871 44 45.oi3I o0305.0480.0654.0829 oo003.1177 I352.1526.1700.1873 45 46.oi34 o0308.0483.0657.0832.0oo6 1180o.. 355. 529.170o3 1876 46 47 1.oJ37 -o03I.o486.o660o 5.08 83..35 109 5.137 1531.17o0.879 47 48.oI40o.o314.0489.o663.o838.1012.II86.136o.i534.1708 1I882 48 49 0.o43.0317.0492.o666..84o.1io5.1189.i363.1537.171 1'1885 49 50 S o045 o0320 o 0494 o0669'o843'1oi8.II92. 366.I540'17I4 1I888 50 51.oi48 o0323.0497 o0672.0846.1021I.1195.13691 I543.I717 1891 5I 52 -.o5I.0326.0500.0675.0849 1I023.1198.1372.r546.1720.1894 152 53.o054.0329 o0503.0678.0852.10261.1201.1375.1549 1723. 1897 53. 54.0157 0332 o0506.o68i.o 855.10291.1204.1378.S552.1726.19oo 54 55.oi6o o0335 o0509.o683.o858.1032.1206.138i.I555.1729.1902 55 56.oi63.0337 -o512.o686.o861.Io35.I209.384 [I558 1732.1905 56 57'o166.o340 o05I5.06890.o864 1.o38.12I2.I386 [.56o I1734.1908 57 58 -.o69 10343.0o58.0692.0867 Io4I. 1215.1389.-563.1737 1I9I1 58 59.0172 0346 o0521.o695.0869.Io44.1218 1I392 i1566'1740.1914 59 60 0I75 o0349 o524.0698'0872.1047.1221.1395.1569.1743.1917 6o 8 TABLE OF CHORDS: [RADIUS-1.000 ]. 1M. 1 12~ 13~ 140 15 160 170 1S~ 19~ S0 2~ 10.:' 1917.2091.2264 -2437.2611.2783.2956.3129.33oi.3473.3645 o i.1920.2093.2267.2440.263I.2786.2959.3I32.304.3476.3648 I 2.1923.2096.2270.2443.2616.2789.2962.3134.3307.3479.3650 2 3.I926.2099.2273.2446.2.69 2792.2965.3I37.33io.3482.3653 3 4.1928.2102.2276.2449.2622.2795.2968.340o.33I2.3484.3656 4 5.193I.2105.2279'2452.2625.2798.297I.3143.3315.3487.3659 5 6.1934.20o8.2281.2455.2628.2801.2973.3I46.33i8.3490.3662 6 7'1937.2111 -2284.2458.2631.2804.2976.3149.332I.3493.3665 7 8.1940.2II4.2287.2460.2634.2807.2979.3152.3324.3496.3668 8 9 1I943.2I17.2290.2463.2636.2809.2982.3i55.3327.3499.3670 9 I0 I1946'.2119 *2293.2466.2639.2812.2985.3I57.333o.3502.3673 10 II.1949.2I22 -2296. 2469.2642.28Ir.2988.3160.3333.35o4.3676 II 12.I952.2I25.2299.2472.2645.2818 2991.3163.3335.3507.3679 12 i3.1955.2I28.2302.2475.2648.2821.2994.3166.3338.35io.3682 13 I4 I.957.2I3I.2305.2478.2651.2824.2996.3169.3341.3513.3685 14 I5.1960.2I34.2307 -2481.2654.2827 29999 3172.3344.35I6.3688 I5 16.1963..2137.2310 *2484.2657.2830.3002.3175.3347.3519.3690 16 17.I966.2I40.23I3.2486.2660.2832 3o005.3178.335o.3522.3693 17 I8.1969 2f43.2316 -2489.2662.2835.3008.380o.3353.3525.3696 18 19 I972.2146.2319 -2492 26.6 2838.30II 383.3355.3527.3699 19 20.I975'2148.2322.2495.2668.2841.30o4.3i86.3358.3530.3702 20 21 1978.2151.2325.2498.2671 2844.30I7.3189.336.3533.3705 21 22 I.98I.2154.92328.2501.2674.2847.3019.3I92.3364.3536.3708 22 23.1983.-257.233I.2504.2677.2850.3022.3195.3367.3539.37o1 23 24.I986.2I60.2333.2507 *2680.2853.3025.3198.3370.3542.3713 24 25 xI989 -2I63 *2336.2510.2683.2855.3028.3200.3373.3545.3716 25 26'1992'2I66.2339 -2512 -2685.2858.3031.3203.3376.3547'3719 26 27 -1995.2169.2342.25I5.2688.2861.3o34.3,206.3378.3550.3722 27 28 1I998.2172.2345.2518'2691.2864 3037.3209.338i.3553.3725 28 29 -2001'2I74.2348.2521 *2694.2867.3o4o.3212.3384.3556.3728 29 30.2oo4.2177.2351.2524 -2697 -2870.3042.32I5.3387.3559.3730 30 3i.2007.2180.2354.2527.2700.2873.3045.3218.3390.3562.3733 31 32.200 o.2183.2357.2530.2703.2876.3o48.322I1 3393.3565.3736 32 33!2012.2I86.2359 -2533.2706.2878.305I.3223.3396.3567.3739 33.34.20I5.2189.2362.2536.2709.2881.3054.3226.3398.3570.3742 34 35.2018 21I92.2365.2538.2711.2884.3057.3229.34o0.3573.3745 35 36.2021.2I95.2368.2541.2714.2887.3o6o.3232.3404.3576.3748 36 37.2024.2198.2371 -2544 -2717.2890.3063.3235.3407.3579.3750 37 38.20271 2200 -2374 -2547.2720.2893.3065.3238.34io.3582.3753 38 39.2030o 2203.2377 *2550.2723.2896.3068.3241.34I3.3585.3756 39 40.2033.2206.2380.2553.2726.2899.307I.3244.3416.3587.3759 40 41'2036'2209.2383.2556.2729.2902.3074.3246.34193 3590.3762 41 42..2038.2212.2385.2559,.2732..2904.3077.3249.3421.3593.3765 42 43.12041.22I5.2388.2561.2734..2907.3080.3252.3424 -3596.3768 43 44.2044.2218.2391.2564.2-37.2910.3083.3255.3427.3599.3770 44 45 -2047 1222I.2394.2567- 2740.2913.3086.3258.3430.3602.3773 45 46.2050.2224.2397 -2570.2743.2916.3088.326I.3433.3605.3776 46 47.2053.2226.2400 2573.2746.2919.3091.3264.3436.3608.3779 47 48.2056.2229.2403.2576'2749.2922.3094.3267'34391 36Io.3782 48 49.2059.2232.2406.2579.2752.2925.3097.32691 344I.36I3.3785 49 50.2062.2235.2409.2582.2755.2927 3100o.3272.3444.3616.3788 5 o, 51.2065.2238.2411.2585.2758.2930.3io3.3275.3447.36i9.3790 5I 52.2067.224I1.2414 -2587' 2760.29.33.3o06.3278.3450.3622.3793, 512 53.o2070.22441 24I7.-2590.2763.2936.3o09.328I.3453.3625.3796 53. 54.2073.2247 1 2420.2593.2766.2939.3iii.3284.3456.13628.3799 54 55 -.2c76.2250.2423.2596.2769.2942.3II4.3287.3459.3630,3802 55 56'2079.2253.2426.25991 2772.2945.3117.3289.3462.3633.38o5 56 57.12082.2255.2429. 2602.2775.29948.320.3292. 3464.3636;.3808 57 58 20o85.2258.2432.2605.2778.2950.3123.3295.3467 3.639.3810 58 59 2088.2261.2434.2608.278I.2953.3I26.3298.3470.3642.38I3 59 60.2091.22641.2437 ~261I.2783.2956.3I29.330I.3471 3645.38I6 6 9 11:2953~~~~~~~~~~ TABLE OF CHORDS: [RADIUS =1.0000]. M. 22 230 240 2~0 26~ 2~ 2S~ 290 300 310 32~ M. o'.38i6.3987 -4I581 4329..4499.4669.4838.5o08.5176 *5345'55I3 0 I.38I9.3990.4I6I.4332.4502.4672.4841.50Io.5179.5348.55I6 I 2.3822.3993.4164 *4334.4505.4675.4844.50I3.5182.535o.55i8 2 3.3825.3996.4I67 *4337.4508.4677.4847.50o6.5i85.5353.5521 3 4.3828.3999 -4170.4340.45io.468o.485o.50o 9.5i88.5356.5524 4 5 *.3830.4002.4172.A343.45I3.4683.4853.5022.5190.5359.5527 5 6.3833.4004 14175.4346.45I6.4686.4855.5024.5193.5362.5530 6 7.3836.4007.4178.4349 -45I9.4689.4858.5027.5I96.5364.5532 7 8.3839.4o0I.4181.4352. 4522.4692.486i.503o.5199'5367.5535 8 9.3842.40o3.4I84.4354.4525.4694.4864.5033.5202.5370.5538 9 IO.3845.40o6.4187.4357.4527.4697.4867.5036.5204.5373 [ 554I 10 II.3848.4019 *4190.4360.4530.4700.4869.5039 5207.5376.5543 ii I2.3850.4022.4192.4363.4533.4703.4872.504I.52IO.5378.5546 12 I3.3853.4024.4I95.4366.4536.4706.4875.5044.5213.538I.5549 I3 I4.3856.4027 -.498.4369.4539.4708.4878.5047.52I6.5384.5552 14 I5.3859.4030o 420I.4371.4542.471 488I.5050o 5219.5387.5555 15 I6.3862.4033.4204.4374.4544.47I4.4884.5o53.522I.5390.5557 z6 I7.'3865.4036.4207 *4377 1.4547.4717.4886.5o55.5224.5392.5560 1I7 18.3868.4039 ~4209 4380 11455o.4720.4889.5058 15227.5395.5563 8 I9.3870.4042.4212.4383.4553.4723.4892.5061.5230.5398.5566 19 20.3873 -4~441 4215.4386.4556.4725.4895.5o64 5233.5401.5569 20 21.3876.4047 -4218.4388.4559 14728 -4898.5067.5235.54o4.5571 2I 22.3879.4050.4221.4391.456i.4731.490o.5070.5238.54061.5574 22 23 1.3882.4053.4224.4394 -4564.4734.4903.5072.5241.5409.5577 23 24.3885.4056.4226.4397.4567 *4737 -49o06 5075.5244 ~5412.5580 24 25.3888.40591 4229'4400.4570'4740.4909.5078.5247 15415.5583 25 26.3890.4061.4232.44o3.4573.4742.49I2.5081.5249 1.548.5585 26 27'.3893.4064.4235.44o5.4576.4745.4915.5084.5252.5420.5588 27 28 1.3896.4067.4238.44o8.4578.4748.4917 15086.5255.5423.5591 28 29.3899.4070.4241.44II.458i.4751.4920.5089.5258.5426.5594 29 30.3902.4073.4244'4414 14584.4754 *4923.5092.526I.5429.5597 30 31.3905.4076.4246.4417.4587.4757.4926.5095.5263.5432.5599 3I 32 1.3908.4079.4249.4420 /4590.4759.4929.5098.5266!.5434.5602 132 33 |.3910.4o08.4252.4422..4593.4762.4932.50oo.5269.5437.56o5 33 34.39i3.4084.4255.4425.4595.4765.4934.5103.5272 1.5440.5608 34 35.33916.4087.4258.4428.4598.4768.4937.50o6.5275 1-5443.56II 35 36 |.3919.4090.426l.443I.460I.4771.4940.5o09 -5277.5446.56I3 36 37.3922.4093.4263 4434 -4604.4773.4943 -5112.5280.5448.56i6 37 38.3925.4096.4266.4437.'4607.4776'4946.5r51.5283.545i.56I9 38 39.3927 -4098.4269.44339.4609 *4779.4948.5I7.5286 1.5454.5622 39 40 | 3930.410I.4272 4442.4612.4782.495I 5.I20.5289 15457.5625 40 4I.3933.4I04.4275.4445.46i5.4785.4954.5123.5291I 5460.5627 41 42.3936.4I07.4278.44448.46I8.4788.4957 -5126.5294.5462.5630 42 43 -.3939.4II1.4280.445.462 I. 4790.4960. 5129.5297.5465.5633 43 44 -.394 411I3. 4283.4454.4624'4793.4963.5131.5300.5468.5636 44 45.3945.41i6 4286.4456.4626.4796.4965.5I34.5303 J.5471.5638 45 46.3947 -'4I8'4289.4459.4629 *4799 -4968.5137'5306.15474 564 46 47 -.3950 -412I.4292.4462.4632.4802.497.5I40o.53o8.5476.5644 47 48 1.3953.4124.4295.4465 1.4635.4805.4974 -5i43.53I 1 5479.5647 48 49.3956 -4127.4298.4468.4638.4807 *4977.5145.53I4'5482.5650 49 50.3959.4i30.43oo.4471.464I *48io.4979 -5i48.5317.5485 5652 5.3962.4133.43o3.4474.14643.4813.4982.515I.530o.5488.5655 1 5 52'.3965.4I35.4306.4476.44646.48i6.4985.5154.5322.5490.5658 52 53.3967.4I38.4309.4479.4649.48I9.4988'5i57'5325 15493.566I 1153 54.3970 41I41.4312.4482.4652.4822.499'51 60.5328.5496.5664' 54 55.3973.4144'43I5 14485 J.4655.4824.4994.'562.5331.5499.5666 55 56.3976.4I47.4317.4488 1.4658.4827.4996.5i65.5334.5502.5669 56 57.3979 14i5o.4320.449I.4660.483o0 4999.5168.5336.5504.5672 57 58.3982.4153.4323 14493.4663.4833.5002.5171.5339 5507.56751 58 59.3985.4I55.4326.4496.4666.4836.5005.5174.5342.5510.5678 59 6o.3987 -4I58.'4329 *4499.4669'4838'5oo8'5176'5345'5513.568o 16 10 TABLE OF CHORDS [RADIUS 1.0000].. 330 34~0 350 360~ 370~ 38~ 39~ 400 41~ 420 430~ o'.5680.5847.60o4 680o.6346.65 II.6676.6840.7004 7167.7330 o I.5683.5850.6017.6i83.6349.6514.6679.6843.7007 -7170.7333 I 2.5686.5853.6020.6I86.6352.6517.6682.6846.7010.7173 7335 2 3.5689.5856.6022.6189.6354.6520.6684.6849.7012.7176 7338 3 4.5691.5859.6025.6I91.6357.6522.6687.685i.7015 7178.7341 4 5.5694.586I.6028.6I94.636o0 6525.6690.6854.70I8.7181.7344 5 6.5697.5864.603 r.6I97.6363.6528..6693.6857 /7020.7184.7346 6 7 5700 5867.6034 6200.6365.653i.6695.6860 7023.786.7349 8.5703.5870 60o36.6202.6368.6533.6698.6862.7026.7189.7352 - 9. 5705 5872.6039.6205.6371.6536.6701.6865.7029 7192 7354 9 io.5708.5875.6042.6208.6374.6539.6704 6868 *7031.7195 7357j 10 II.57II.5878.6045.6211.6376.6542.6706.6870.7034.7197.7360! iI I2.5714.588i.6047.6214.6379.6544.6709.6873.7037.7200 7362 12 I3.5717.5884.6050o 6216.6382.6547.6712.6876 *7040 ~7203.7365 13 i4.5719.5886 60o53.6219.6385.6550.6715.6879.7042 17205.7368 14 [5.5722.5889.6056.6222.6387.6553.6717.6881I.7045.7208 7371 I5 16.5725.5892.6o58.6225.6390.6555.6720.6884.7048 17211.7373 16 17.5728.5895.6o6i.6227.6393.6558.6723.6887.7050 *7214 7376 17 I8.5730.5897.6064.6230.6396.656.6725.6890.7053.72I6.7379 8 19.5733.5900 -6067 1.6233.6398.6564.6728.6892 7056 72I9 17381 19 20.5736.5903.6070.6236.6401.6566.6731 6895.7059.7222.7384 20 21.5739.5906.6072.6238 *6404.6569.6734.6898 -7061.7224.7387 /21 22.5742.5909.6075.6241.6407.6572.6736.69o0I 7064.7227.7390 2 23.5744 15911.6078.6244.64o0.65751 6739.6903.7067.7230.792 23 24.5747.59I4.608I.6247.6412.6577.6742.6906.7069.7232.7395 24 25.5750 o 5917.6o83.6249.64i5.650o1 6745.6909 |.7072.7235 7398 1 25 26.5753.5920.6086.6252.64I8.6583.6747.69 11 7075.7238.7400 26 27.5756.5922.6089.6255.6421.6586.6750.6914 I7078 17241.7403 i 27 28.5758.5925.6092.6258.6423.6588.6753.69I7 I7080 7243.7406 28 29.576I.5928.6095.6260.6426.6591.6756.6920 1.7083.7246.7408 29 30.5764.5931.6097.6263.6429.65941 6758 6922.7086.7249.7411 3o 31.5767.5934.610oo 6266.6432.6597.6761.6925 |.7089.7251.74[14 31 32.5769.5936.6o03.6269.6434.6599.6764.6928 1[7091 -7254. 74I7 1 32 33.5772.5939.6o06.6272.6437.6602.6767.6931 [7094.7257. 74'9 33 34 i.5775.5942.6Io8.6274.6440.6605.6769.6933.7097.7260.7422 34 35 1 5778.5945.6111.6277.6443.6608.6772.6936 [.7099.7262.7425 I35 36.578I.5947.61r4.6280.6445.66ol.6775.6939.7102.7265.7427 36 37.5783.5950 -6117.6283.6448.66I3.6777.6941.7105.7268.7430 37 38.5786.5953.6119.6285.645I.66i6.6780.6944 7I108.7270.7433 38 39.5789.5956.6122.6288.6454.6619.6783.6947 -7IO.7273.7435 39 40.5792.5959 6125 |.629I.6456.662I.6786.6950 71I3 17276.7438 40 41.5795.5961.6I28.6294.6459.6624.6788.6952.7116.7279.744I1 141 42 5797.5964.6130.6296.6462.6627.6791.6955 1-718 7281.7443 142 43.5800o 5967.6I33.6299.6465.6630.6794.6958 1.7121.7284.7446 i 43 44.5803.5970 6I36.163021 6467.6632.6797 6961 -71724'7287 17449 44 45.5806 5972.6I39.6305.6470.6635.6799.6963 117I27.7289.7452 45 46.58081 5975.6142.6307.6473.6638.6802.6966 -17129.7292. 7454 46 47.58I1 5978.6i44.63Io.6476.6640o.68o5.6969 -17132.7295 7457 47 48.5814.5981.6I47.63i3.6478.6643.6808.6971 -7I351 7298 17460 48 49.58I7.5984.650 o.63i6.6481.6646.68io.6974 -1737.7300o 7462 49 50.582> 5986.6i53.63i8.6484.6649.68i3.6977.7140 l7303.7465 50 51 5822 5989.6155.6321.6487.665i.68i6.6980.7143.7306 7468 5 52.5825.5992.6I58.6324.6489.6654.68I9.6982 I 7146 73081 7471 [ 52 53.5828.5995.6i6i.6327.6492.6657.6821.6985 I7148.73II1 7473 153 54.5831.5997.6i64.633o.6495.666o.6824.6988.7151i.7314.7476 54 55.5834.6000o.666.6332.6498.6662.6827 1699I 754 17316 17479 i 55 56 5836.6003.6669.6335.6500.6665.6829.6993 7156 73 9 748 56 57.5839.6006.6172.6338.6503.6668.6832.6996 -7159.7322.7484 57 58.5842.6009 o.675.634I.6506.6671.6835.6999.7162 -7325.7487 58 59.5845.60.6I78,.63431.6509.6673.6838.700I1 765.7327.7489 59 6o~ 5847.6o14.68o0.6346.65i.6676.6840 7004 o 7167 7330 7492 60 II TABLE OF CHORDS: [RADIUS=1.0000]. M. 440 450 464~ 470 48~ 49~ 50O 510~ 520 530 540~ 0.o7492.7654. 7815.7975.8135.8294.8452.86io.8767.8924.9080 0' I.7495.7656.7817.7978.8137.8297.8455 8613.8770 18927.9082 1 2 7498.7659.7820.7980.8I40.8299.8458.86I5.8773.9929.9085 2 3.7500.7662.7823.7983.8I43.8302.846o0 86I8.8775.8932.9088 3 4 1.7503 7664.7825.7986.8I45.8304.8463.8621.8778.8934.9090 4 5.7506.7667.7828.7988.8i48.8307.8466.8623.8780.8937.9093 5 6. 7508 7670.7831.7991 8i1i 5 83o0.8468.8626.8783.8940.9095 6 7.75T11 7672.7833 *7994.8I53.8312.847I' 8629.8786.8942;9098 7 8 -7514.7675 -7836.7996.8156 -83I5.8473.863I- 8788.8945 o9101 8 9 1175I6.7678.7839 7999'859.83i8.8476.8634 18791.8947 19o3 9 9 o 1.7519.768I1 784I.8002.8i6i.8320.8479.8636.8794.8950.91061 I II *7522 p7683.7844 800o4.8i64.8323 8481.8639 8796.8953.908 iI I2 7524 7686 7847.8007 8167.8326.8484.8642.8799.8955.9111 12 13 *7527 7689.7849.8oio0 8169.8328.8487.8644 1880o.8958.9113 I3 14.7530.769I.7852.8012.8172 1833i.8489.8647.8804.8960.9116 14 I5.7533.7694 7855 18o05.8175.8334.8492.865o0 8807.8963. 9119 I5. I6.7535.7697.7857.80o8.8177.8336.8495.8652.8809.8966.9121 16 17.7538.7699.7860 8020 8i8o10 8339.8497.8655.88I2.8968.9124 17 I8.7541.7702 78631 8023.8i83.834I.85oo.8657.88I4 18971.9f26 I18 9 7543 7705 7865.8026.8i85.8344.8502.8660.8817 18973 929 I 9 20.7546.7707 *7868.8028.-8188. 8347.8505.8663.8820.8976* 9132 20 21I 7549.7710'787I -803I.'890.8349.8508.8665.8822 8979 9I 34 21 22.7551.7713.7873.8o34 -8193.8352. 85o0.8668.8825 18981' 9137 22 23.7554 177t5 7876.8036.8196.8355.85i3 867I.8828.8984 19139 23 24.7557 -7718.7879.8039.8198.8357.85i6.8673.8830.8986 9I142 24 25.7560 ]7721.7882.8042.8201.836o0 85i8.8676.8833.89889 9145 25 26 17562 17723.7884.8044.8204.8363 18521.8678.8835.8992.9147 26 27 17565.7726.7887 -8047 8206.8365.8523.8681.8838.8994.9150 27 28.7568 17729.7890.8050o 8209.8368.8526 8684.884I.8997 91I52 28 29.7570.7731 7892.8052 8212.8371.8529 8686.8843.8999.9155 29 30o.7573 17734 7895.8055 8214.8373.8531I 8689.8846 9002 go 9157 30 31.7576.7737.7898.8058..8217.8376.8534.8692.8848 *9005 9I601 31 32.7578 7740 17900 o8060.8220.8 878 3 7.8537 8694.885I 9007.9163 32 33 *.7581 7742.7903.8063.8222.8381.8539I 8697.8854. 90o0 i 965 33 134.7584.7745 7906.8066.8225.8384.8542.8699.8856 9012.9I68 34 35 7586.7748 17908.8068 8228.8386.8545.8702.8859 o9015,9170 35 36 l7589 g77-50 7911 807I1 82301 8389.8547. 8705 886i.90I 8.9I73 36 37.7592.7753 17914 *8074 8233.8392.8550. 8707.8864 19020.9176 37 38.7595.7756 7916.8076 18236.8394.8552.8710.8867.9023.9178 38 39.75971 7758 7919.80799 9 8238'8397.8555.8712.8869.9025,.9I8I 39 4o 7600o 7761 7922 8082.82418.84o00 8558.8715.8872 *9028.9183 40 41.7603.7764.7924.8084.8244.8402.8560 I87I8.8874 g9031.9186 41 42.7605.7766 79.27.8087.8246.84o5.8563.8720.8877 *9033.9188 42 43.76081-7769.7930 8090o.8249.84o8.8566.8723.888o0 9036.9I91| 43 44.7611 7772 7932.8092 8251.84io.8568 ~8726.8882.9038 *9194 44 45.7613 17774.7935 8o095.8254.84i3.8571 8728 88885.9041.9I96 l45 46.76I6.7777.7938'8098'8257.84i5.8573.8731.8887.9044 19I991 46 47 l7619'7780 -7940 810oo.8259.8418.8576.8734.8890.9046 19201 47 48.7621 7782.7943 18o3.8262.8421 8579.8736.8893'9049 9204 48 49 7624 7785.7946.8o15.8265.8423.858I.8739.8895. 905I 9207 49 50.7627.7788 17948 8io8 82671 8426 8584.8741.8898.9054 -9209. 50 51.7629.7791 -795-.81 II.8270.8429.8587.8744.8960 o9056.9212 I5I 52.7632.77931 7954 -81i3.8273.843i.8589.8747 *89o03.959.92I4 52 53 17635.7796,7956.81 6.8275.8434.8592.8749 18906.9062.9217 53 5411 7638'7799'79591 81I91 8278.8437.8594 8752 18908.9064.9219 54 55 11764o0 78o01 7962.8121.8281'8439.8597.8754.8911.9067.9222 55 561 7643.7804.7964.8124.8283.8442.8600oo 8757.8914 1.9069.9225 56 57.7646 7807 7967.8127.8286.8444.8602.8760.89I6.9072.9227 57 58.7648.78091 7970.8129.8289~ 8447 18605 ~8762.89I9. 9075.9230 58 59 765i.7812.7972.8I32' 8291.8450.8608.8765.8921.9077.9232 59 6o0 7654.7815 17975.8i35'82941 8452.86Io.'8767.8924 9080 * 0235 60'..._ - TABLE OF CHORDS: [RADIUS 1.0000]. M. 55~ 560 570 5 1590 60~ 610 62 63~ 641 M. 0'.9235.9389.9543.9696.9848.0000oooo I.o5 I.o3oi i.o450.0598 o' I.'9238.9392.9546.9699.985.00ooo3 I.oi53 i.0303 I.o452.ic6oi I 2.9240.9395.9548.97o0.9854.ooo5 I. oi 56 i.o306 1.o4555.o603 2 3 i.9243.9397.955i.9704.9856I.ooo8 I.oI58.o3o08 1.o457 i.o6o6 3 4.9245.9400.9553.9706.9859 i.0oo0o Ioi6i i.o3ii i.o460.o608 4 5. 9248.9402o.9556 -9709.9861.o00o3 I. 63.o3i3 I 0462. 6 I 5 6.9250.9405.9559.97I. 9864.ooi5.oi66.0o3i6. 0465 i.o6i3 6 7 j9253.9407.956r1 9714.9866 I.ooI8 I.o68 I.o318 I.o467 I0.616 7 8.9256.9410.9564 *9717.9869 1.0020 I*0171.0321 I.0470 1.0618 8 9.9258.9413.9566 *9719.9871.0023 3.73 I0323.I 0472 I.0621 9 Io.9261.9415.9569.9722.9874 1.0025 II0176 o.0326 1o0475.o623 io II.9263.9418 19571'9724 -9876 1.0028 1.0178 1.0328 1.0477 1.0626 ii 12.9266.9420.9574.9727.9879 1.o003Q 1.181.o33i1.o48o0.0628 I2 13.9268.942.3.9576.9729.9881 I.0033 i.o83 i.o333 I.0482 1.o630 13 I4 -9271 -9425.9579 -9732.9884 i.oo35 i.o86 I.o336 I.o485 1.o633 14.15.9274 -9428.958 1.9734.98861.oo38 I.oI88I.o338 I.o 487 1.o635 I5 16.9276.943o.9584 -9737.9889 1x.oo40 11o91 1.034, I-o49o i.o638 16 17.9279.9433.9587 -9739.9891 i.oo43 1.0193 I.0343 -.0492 i o640 17 i8.9281.9436.9589 -9742.9894 1oo0045 01 96 I.o346 I.0495 I-o643 18 19.9284.94381.9592.9744.9897.-oo48 i.oi98 io.348 1.o497 I.o645 19 20.9287 -9441.9594 -9747.9899.o0050 1.020I Io35 i io5o00 1o648 20 21j.9289-.9443 19597 -9750.9902 1.0053 1.0203.o353 I.oo502 1.0650 21 22.9292.9446.9599 -9752.9904 I.oo55.oo206 1.0356 I.o504.1o653 22 23 192941 9448.9602.9755.9907 ioo58.0208 i.o358 1o.0507 i.o655 23 24 1'9297 9451.9604.9757 -99o9 I-oo60 I o0211 I.o36I.o509 I o658 24 25.9299.9454.9607 -9760 -19912 aIoo6 3 1 o023 1.363 I.o512 I.o660 25 26.9302.9456.96Io1.9762 -9914 I.oo65 I z0216 i.o366 i.o5i4 6 662 26 27'9305.9459.9612 -9765 99717 00 i 68 1.0281 io368 i.o57 i.o665 7 27 28.9307.946r1 9615.9767'9919 1.0070 0221 I.o370 I 051o g I 0667 28 291 9310.9464.9617'9770'9922 1-0073.0o223.0o373 1.0522 1.0670 29 30o 93I2.9466 -9620'9772'9924 1-0075 1-0226 I'0375 i.524 I'.672 30 31.9315.9469.9622.9775 1997 00oo 78 I 0228 1.0378 I.0527 i.0675 31 32.9317.9472.9625 19778.9929 r. oo8o I.0231 1.0380o.0529 I.o677 32 33.9320 *9474.9627'9780.9932 I.oo83 1.0233 I.o383 1I.532 1.o 680 33 34.9323.9477.9630 -9783.9934 i.oo86 I-0236 1.0385 i.o534 10o682 34 35.9325 *94791 9633 -97851 9937 I.oo88 1'0238 i.o388 1.0537 i.0685 35 36.9328.9482.9635.9788.9939 1.0091.0o2411o3go 0539.o539 687 36 37.9330.9484.9638.9790 -9942.oo0093 3 I o243 jo 393.0542 1069g0 37 38j 9333.94871 964o0 9793'9945'oo096 I10246 1i03951'i544 1.0692 38 39'9335 ~9489.9643.97951 9947 1o0098 0 o248 o398 I o547 1. 0694 39 40.9338.9492.9645 -9798.9950 I.oioi I-0251 10400 o0549 1 o0697 40 4i'9341'9495.9648.9800.9952 i-1oo3 1-0253 -0o4o3 1-.55i 1.0699 41 42 -9343 *9497.9650.9803.9955 1oo.06 1.0256.1.0405 i.o554 I-0702 42 43.9346.9500.9653.9805.9957.oo08 1- 0258 I.0408 1.o556 1-0704 43 44.9348.9502.9655.9808.9960 1.o II 1.0261 1I.~41l 1.0559 1o0707 44 45.9351.95051 9658.o8o1.9962 i. oi3.02o63 1.o4I31.o56I I.0709 45 46'9353.9507.966I.98I3.9965 i'oii6 i o266 i.o4i5 1.0564 1.07I2 46 47.9356.95o1.9663.9816 -9967 I I8 i 8 I 0268 I0o4i8 I-o566 1-0714 47 48.9359.9512.9666.9818.9970 0I I21 1 0271 1.0420 1.0569 1 0717 48 49.936I.95151.9668.9821'9972 1O123 1.0273.0o423 I.0571 I.07I9 49 50 j9364.95I8.9671.9823 -9975 1o0126 1I0276 -.o425 1o574 1-072I 50 5i.9366.952o0 9673.9826 -9977 1.0128 1-0278 1.0428 1-0576 1x0724 51 52.9369.9523.9676.9828.9980 1.o3I I.0281 Io430o I0579 10726 52 53.9371,9525.9678.9831.99821.oi33.o0283 i.o433 i.58 1.o0729 53 54?9374.9528.968i.9833.9985 i..o36.o0286 i.o435 i.o584 1-0731 54 55.9377.953.9683.9836.9987I.o38.o0288.o438.o586.o734 55 56.9379.9533.9686.9838 -9990 I.o14 I -0291 I.o440 i.o5891g.0736 1 56 57.9382'9536.9689'984 9992 1.o43 1.o293 -.o443 I.-o59g 1o739 57 58.9384.9538.969I.9843.99951.o46 1 0296 t.o445.0o593 1.074I 58 59.9387.954I.9694.9846.9998J.oI48.o0298.o447 i.o596 1o0744 59 60.9389 195431 9696.984810ooooJ.oi5i.o3o0I.o45o0.o0598 r.0746 60 13 6 TABLE OF CHORDS: [RADIUS=1.0000].. 650 66~ 60 6S~ 690 700 71~ 720 730. o' 1.0746 1.0893 I.I0391 i-i84.1328 I.1472 I.I6I4 I.1756I I.896 O' I 1.0748 I.0895 I. o4I I. I 86 I. 33 I.1474 I I616 I.758 I.899 2 1.0751 I.0898 I.I044 1.1189 i.333 I.1476.1 619 1.1760.1901 2 3 1o0753 I.0ogoo I.o46 1.119I I.I335I.1479.1621 I 1763 I 1903 3 4 1.0756 I.0903 I 048. I 194 I.338 I. 1481. I624 1 765 1 I906 4 5 1.0758.o09o 5 I.051 II.196 I.340 I.483 I.626 I.I767 1I908 5 6 1.0761.0907 I I53.11I98 I.1 34211 I486 I.628 1I 770 11910 6 7 1.0763.o 910o I.Io56 I.120oI i. 345 I 488 I.163 I I 1772 11913 7 8 I 0766 1 0912g I. Io58 I.1203 I.1347I I49I I I633 1.1775.195 8 9 1-0768.I 915 I o061 I.20 6 i.350 I 1493 I.635 I.1777 - I917 9 10 1.0771 I 09I7 I.10o63 I.'1208 1.1352 1.1495 I1.638 -1 779 -I 1920 I0 II 1.0773.0920o I.Io65 I.1210 i.i354 1-1498 I I.64o 1 I782 11 922 a I 12 1.0775 1o 0922.I0o68 I.12I3 I.1357 5o500 1.1642 1.1784 1. 1924 i I3 1.0778 I.o0924 I.1070 I.1215 I.1359 1I502 I. 645 I 1786.1 927 13 I4.0780 I.0927 Io1073 1.1218 I.1362 I I5o5 1.647 1. 789 I 1929 i4 I5 1.0783 I.929 -10o75 I.I220 i. 364 I.1507 I.1650 1.1791 I.93I 15 i6.o0785 I.932 I 1078 I.1222 I.i366 i.15io I.1652 11793 I 1934 16 17 1 0788 I 0934 1..080 I. 225.1 369 I 152 1.1654 1 1796 -1936 17 18 1.0790 I 0937 1082I.12 27 1. 371 i.i5I4 1. I657 1.798 I 1938 i8 19 1 0793 I 0939 I.Io85 1.230 I 1374 1 517.1659 I 1800.1 94I 19 20 1.0795.0942.z1087 1.232 I 1376 I 1519 I 1661 I I803 I *1943 20 21 I.0797 10944.090o I.1234 11378 I.522 I.1664 I.8o5 1.1946 21 22 I.08oo.o 946 1092 1.1237 11381 I.524 I.666 I.807 I.I948 22 23.Io802 I o949I -1094 1.1239 i -383 1.526. I668 I1810 1.I950 23 24 I.805 I.g951 I 097 I I242 I. I386 I.1529 I.671 I.1812 1.1952 24 25 Io0807 I.o954 I.1099 I 1244 I.388 I.1531.I 1673 i.i8i4.1955 25 26 I.o81o I 0 956 I.102. I I246 I 1390 I.1533 I.676 I.817 1.1957 26 27 I 0812 I.0959 -III04 1 1249 1I393 i.536 I.678 I.I89 1.I959 27 28 I.o815 o096I II1107 1-1251 1.1395 I.538.I 680 1 1821 1 I 962 28 29 I08I7 1.09~63 I09II 2541138 I.2541 I.I683 1.824 1-1964 29 30.0o820 I.0966I 1.111 II256 I.4oo I. 543. 685 1.826 1.966 30 31 1.0822 1.0968 I.III4 11258.1402 1.1545 1. 687 I1.829 II1969 31 32. 0824.0971 i.ii6 1.1a261 I. 4o5 I.548 1.690 I.I83i II.97I 32 33 I.0827 Io0973 IIg1191I2.63 I.1407 i.i550. I1692 i.i833 I.1973 33 34.o0829 1-0976 1.1121 1.1266 I.1409 1.1552 1.1694 I.I836 I.1976 34 35.0832 1 0978 1 1123 1 1268.14I2 i.i555 1.1697.1i838 I.1978 35 36 i.o834 1o0980o 1.26 1.1271 1.1i44 11i557 1.1699 I.I84o 1.1980 36 37.0837 10983 1.1128 1.1273 1.1417 II1560 1.1702 i.i843 I.1983 37 38.0839 1o0985 i-i3 1.12775 I.1419 1.I562 1.704 1ii845 i.1985 38 39 I.o84I 1.0988 i.ii33 11I278 1.1421 i.i564 I.i706 1 847 1.1987 39 40 i.o844 1.0990o I 136 1 1280 1.1424 1.1567 1.17091 i..-85o 011990 40 4I 1.o846 1.0993 i.1I381..1283 I.1426 I1.569.I7I 1I-.852 1.1992 i 4 42 1.0849 I.o995 I.II4o0 11285 I.I429 1-1571 1.17I3 1.1I854 1-I994 42 43 i.o085i 1 0997.II43 1.1287 i-I431 r1.574 I.1716 1.1857 11997 43 44 i.o854 i.iooo.II45 1.1290 I.1433 1.1576 1I1718I1.18591-1999 44 45 i.o856.1002 1 II48 1.1292.1436 1.1579.17201.86i 1.2001 45 46.0859 Ixioo1r5.II5o 1.1295 i.438 1.1581 I.1723.L I 864 1.2oo4 46 47 I.o86I I.1o07 i.-152 1.1297 1-1441 i.i583. I725 i.i866 1.2006 47 48 I.o863 1.oIIo.IzI55 1.1299 -II443 i.i586 1.17271 I868 1.2008 48 49 i.o866 II.012o l.157 1.I302.-1445 i.i588 1.17301o.871.I201I 49 50 I.o868 I. io4 I I.60 I.i13o4 1.1448 1.1590 1.1732 1.1873 1.2013 50 51. 0871 I.10I7 I.1162 1.1307 i.i450.i1593 I.I735 i.i875 1.2015 51 52.0873 i1IOi9 1.ii65 1.1309 1.452 I.1595 1.I737 -1I878 1.2018 52 53.Io876 1.1022 zI.167 i.i3ii I.I455.1598 1.1739 i.88o.2020 53 54 t.0878 1.1024.1II69 i.i3i4 1.I457 i.i6oo I.I742 1.1882 1.2022 54 55. Io881 I 1027 1-1172 1.i36 i.i460 I.1602 1.I744 i.i885 I.2025 55 56 i.o883 I.1029 I.1174 1319.1462. 65.462 i6 I.887 I-2027 56 57.o0885 i.io3i II.I77 I1.321 I.i464 1.1607 1*749 1.1889 I-2029 57 58 i.o888 i.io34 I 1179 I. 3231 1 74i I-I1609 1-I75Ii.1892 1.2032 58 59 1.089o0 Iio36 i.ii8i 1.1326 II.469 I.612 a I 1753 I.1894 1.2034 59 60 10o893 I.1o 39 i.ii84 1.i328z 14721.I6I4 17561i. 1896 I'2036 60 14 TABLE OF CHORDS: [RADIUS= I.00001. 7I 4 75 01 *so 790,SO sio.- 20 6 I.2036 I.2189 I.2323 1.2464 1.2600 1.2735 1.2869 1 0890 I.32134 I.2053 12I 1I 1.23129 1246653 162589 2737 1.28758 I.304 13137 7 8 1.204551.2948.2332 1I2468551.2641 I27426 12874 I2973 1.33926 2 3 I.20573.219682 233420 24571 I26259 7 2728.2876 I32996 i.3142 9 4 I.2060 I-21984 1233622 2473 1-26095 I27344 128678 I2998I.3430 4 5I I2062 I.2207 23251.24625 I2598 1.2733 I.2867 i.30o0I I.332 5 6: I.20564 I.283 I.2327 I'2464 1.26o4 1.27485 12868 I.30025 i31347 73 I.205663.225 I.234329 124861 2662 1.275371 2885 I.30I48 31537 7 8 I.205569 1.228.23532.2482 1.2618 r.2753 1.2887 I13007 1-31352 9 1-2057 21-2196 I.23348 24784 I2620 1274255 288976 1.302092 -3i4 9 IO6.2073 6.2298 I.23536 124873 126230 1275744 28978.3024i.3463 i6 I7 1.2762 1.2204 I.2338 1.24789 12625 1.2760 I.288094 30327.3145 Ii 12i8 120648 1223 I'.2341 I 24791 -26427 1274862 I2882 I30295 i.361 12 13.20660 1.2295 I.234357.24893 1.262 1.275164 1.2885 I308.3150 13 24 I 20983 12228 I.23459.24829.2632 1.2766 1.2987.30233.3I652 4 15 I.20785.2224 I.2368 I.24984 12634. I2769551.293 I-30322.31674 5 6221 12087 1.2226 I.23645 I2487 1.2636 1I2757 12895 I.302384 3569 6 17 I1.2090 7.2228 I.2366 52 53 28912638 1.2773 1.2894 I.3027 I-3158 172 18..I20789 12217 1235684 1255 2.1.64271 277521.2896.30429 13674 84'1 I.:20809 12293 1.23570 1.2495 126439 127786 1.28981 303441 I.376 2 206 I.2083 12232 I.237359 2496 I1264532 2780 1.2904 I.303346 -378 20 271 12085 I.2237 I.23675 1259812 2648 I.2782 912906 I.30935 -3167 27 28; 212087 2264 I.23477 12504 I.26536 I27784 129085 I3038 i.3i83 22 29 I.2104 I.2228 I.23866 12516 i.2658 1-27871 I2907 I.3053 i31785 29 24.I20926.22344.2368 1.2558 I.2654 I278975 290922 304255.3187 32 25.20948 I.2247 1.238470 25207 I26563 I279781 2925 I30574 318976 25 26 I2097 1.223549 I237386 252309 126459 I2793 i291427 I3046 I3178 26 233 I20993.22537 I2375 I.2512 I.266481 27952 I2962 13062 13I930 332 3428 2101 I.2254 I-239771 252814 26635 127984.293I I30564 i18396 28 329 I2174 I.2242 1.23938 I2563 1.2665 1.28787 I29234 I30536 I.3198 5 360 I.2I06 I.22584 I23968.2532 I.266854 I27890 29362.305685.387 36 371 I.2I182.22647 I23984 1252I4 I2657 1.279804 2985 I.30571 -3202 37 382 I2I24 I.226349 I23864 2537 I126759 12793 I-2927 I.3073 1.3920 32 339 I.2127 12265.2389 4 2 25539 126674 I2795 I2929 I 30675 I 3293 3 34 I-2I15 9 I225467 I2390 -252 I.26631.779281I 2945 I 30764 I-329 43 35I 21 3I-22756.240793 2543 1I26796 I2813 1I2947 I-3066 1.3198 35 36, I-220 34I2258.2396 49125462 I268 -1.28602.29349 I30682.320 36 43.2'l 22 I.22740 I2398 4.2548 1.26837012848 I.2953 I.30784.3250 37 4381 I2I8.22677 I.2440 I.2537 1.268672 2807 I-294 I-307863.3204 38 39' I-212741227965 24602 I255392 26887412822.-2942 I.307885 3207 43 40 I.212439 12267 1-2485 I.2545 I.26977 12812 1 2958 I1309770 3202 46 471 1.2I13 I12283 1.242071 1-257 I2679 I-2823 1-2947 1307931-32241 47 42 1.2I348 1 227286 2423 I12546 I.26958.281629 1.962 30.9521I3263 48 43 9112I3610 2274 I.24125 125648 1269783 2838 I.2965 I.308974 3285 43 44 1.21382.22077 I248 1. 25564 I.26998 I2833 1.2954 67 I3086 3 3231 44 45 I-214 11.2279 1-24306 255266.267088 28226.2956.30I88.3233 45 52 1.21437 I.2289i.2438 1.2555 68I 270 I.283825.2958I i 300 I.322 46 53 1i21459 1228397 24341 I2557[ 1-2769 128270 I29673 1.39o63 3234 47 48 1 2i648.I22996 I.2437 1.25573 -2695 I28 29 76 I.3 o8 I395.3226 48 495 1I21564 123028 I.243925 1I2567 1.-27 128345.296578.3I097 3242 49 561 12I52166 12304 I24284.25774.2699 I-284733 29867 1.3092 -3244 56 5 12I5684 I.2936 I.24431 1.256 I.2701 I.8284936.2968 I-3II51 32336 5 5821 1.2157I -2309 1.243462 25682 12747.I2838 1-2978 131041 1324835 52 53' 12159 1-2297 I.2434 1-2571 1.2706 i..2840 1.2973 1.3106.32371 53 54 1-2161 1.2299 I-2437 1-2573 1.2708.-2842 1.2976 1.3108.3i339:54' 559 I 27364 I 2302 I. 243981 I 2575 1. 2710 28451.2978 I 3 I9.3242 55 5611 I.27566 I2304 I-24415L-25786 I2713 i284567 2989'1.3I2.i3244 56 5711 I-268.-2306 i.2443.-25801.12715 1.2849 I-2982 1.3115.-3246 57 581! 1-2171 I-2309 1-2446. ~25821.27171 I-2851.2985 1-3117.3248 i58 5 1.21731 I.2311 1.2448 125841.2719 1-2854 I.2987 1-3119 13250 59 601 I.2175 I.2313 1.2450 I.2586 I.2722 1.2856 I.2989 1-3121 1.32521 60 15 TABLE OF CHOR DS: [RADIUS=1.0000]. m. 3o. 84 ~~ 50 0 S7O sso 89o M. o' 1.3252 I.3383 I.3512 1.3640 1.3767 1.3893.40oi8 o' I.3255 I.3385.35I4 i.3642 1.3769 1.3895 1.402. I 2.3257.3387.35I6 i.3644 1.3771 1 3897 1.4022 2 3 I.3259.3389.35I8 I.3646 1.3773.3899 14024 3 4 6.326i 1.339I 1.3520 1.3648 1I3776 I.3902 1.4026 4 5 i 1.3263 1.3393.3523.3651 1.3778 I.3904 1.4029 5 6 1.3265.3396 1.3525 I.3653 1. 3780 z.39o6 1.4o31 6 7 1. 3268.3398 1.3527 1..655 1.3782 I.3908 I.4o33 7 8 1.3270 1.3400 1.3529 1.3657 1.3784 1.3910 1.4o35 8 9 1.3272 1.3402.353 I.3659 1.3786 1.3912 I.4037 9 I1. I 3274.34o4 1.3533 1 3661 1.3788 1.3914 I 4039 10 ii 1.3276 I 3406 1.3535 I 3663 I.3790 1.3916 i.404I 1 I 2 1.3279 13409.3538.3665t 1.3792 1.3918 i.4o43 12 13 1.3281 I.34i1 1.3540 I.3668 I.3794 1.3920 I.4045 13 I4 1.3283 1.34i3 1.3542 1.3670 I-3797 1.3922 1.4047 I4 i5 1.3285 I-34I5 I.3544 I1.3672 I.3799 1-3925.I4049 15 r6 1.3287 I 3417 1-3546 1.3674 I 38oi 1.3927.405I I6 17 1.3289 I 134I9 1.3548 1.3676 1.3803 1.3929 I.4o53 17 I8 1-3292 I-342 1.I3550 1.3678 I.38o5 1.3931 I-4055 18 I9 1.3294 13424 1.3552 x.3680 1.3807 1.3933 i.4o58 19 20 1-3296 -13426 1I3555. 3682 1.3809 I13935 I-406o 20 21 1 I-3298 1-3428 I-3557' I.3685 I.381 I3937 I1-4062 21 22 1.33o00.3430.3559 I.3687 I.3813.3939 iz4o64 22 23 1.3302 1.3432 I.3561 I.3689 I.38I6 I*3941 i-4066 23 24 I3305 I.3434 I 3563 1.3691 1.3818 1.3943 I.4068 24 25 1 I3307 13437 I.3565 1.3693 1.3820 1.3945 1.4070 25 26.3309 13439, 1.3567 1.3695 1.3822 1.3947 1.4072 26 27 1.33 [ 1.344I I.3570 1.3697 1.3824 1.3950 1.4074 27 28 1.3313 rI3443 13572 I.3699 -13826 1-3952 1.4076 28 29 I.3315 1.3445 I 3574 1.3702 1.3828 1-3954 1.4078 29 30 I-33t8 1.3447 1.3576 1.3704 i.383o 1-3956 i.4080 30 31 1.3320 I-3449.-3578 1-3706 1.3832 I.3958 1.4082 31 32 1.3322 1.3452 1.3580 1.37o8 1.3834 1.3960 I.4o84 32 33 I.3324 i -3454 I -3582 I.3710 1.3837 I.3962 I.4086 33 34 1.3326 I 3456 I.3585 1.3712.3839 I.3964 1.4089 34 35 1.3328 i 3458 I.3587 1.3714 1-3841 I.3966 I 4091 35 36 I.3331 I 3460 1.3589 I.37i6 I.3843 1.3968 1.4093 36 3- I.3333 1.3462 I.3591 1.3718 1.3845 1.3970 I.4095 37 38 I-3335 1.3465 1.3593 1.3721 13847 1-3972 1.4097 3 39 I.3337 13467 13595 1.3723 1-3849 1-3975 1.4099 39 40 I.3339 I.3469 I*3597 1I3725 1-3851 1I3977 I-410I 40 41 1-334i 1.3471 1-3599 1.3727 I'.3853 1.3979 I-4103 41 42 1.3344 1 3473 13602 1.3729 I.3855 1.398I I.4I05 42 43 1.3346 1.3475 I.3604 1-3731 i.3858 1.3983 I-4107 43 44 1 3348 1.3477.3606 1.3733 I-3860 1.3985.41o09 44 45 I-3350 I-3480 1.3608 1.3735 1.3862 1.3987 I-41I 45 46 1.3352 1.3482 1.36Io 1.3738. 3864 i.3989 1.41i3 46 47 1-3354 1.3484 1.3612 1.3740.386 1.3991 I-4115 47 48 1 3357 1.3486 1 36i4 1.3742 I-3868 1.3993 I 4117 48 49 1.3359.3488 13617 1.3744 13870 I-3995 I'4II9 49 50 i 336I I 3490 1.3619 I-3746 1.3872 I 3997 1.4I22 50 51 i.3363 -3492 1.3621 1.3748 1.3874 1.3999 1.4124 51 52 I3365 1.3495 1.3623 1.3750 1.3876 I.4002 i.4126 52 53 I.3367 I-3497 1-3625 I3752 i 3879 i 4oo4 1.4128 53 54. 3370.-3499 1-3627 1.3754 1.3881 i.4006.4i30o 54 55 1.3372 1 350I I3629 1.3757 I.3883 I.4008 I.4I32 55 56 1 I3374 -35o3 1.3631 I.3759 I.3885 I.4010o.4I34 56 57 1 I-3376 I3505 I.3634 1.3761 1.3887 1.4012 I.436 57 58! I3378 i 3508 I.3636 1.3763 1.3889 1.4014 1.4138 58 59 I.3380 i-35Io 1.3638 I.3765 1.3891 I.4oi6 I-.440 59 60 I 3383.35I2 I.3640 1.3767 13893 1.40o8 I.4142 6 16 TABLE I., OF LOGARITHMS OF NUMBERS FROM 1 TO 10000 N.' Log. N. Log. N. Log. N. Log. 1 o-oooooo 26 I -44973 51 I 707570 76 I 880814 2 o-30Io30 27 I 1-43 I364 52 1 -716003 77 1886491 3 0-477121 28 1 447158 53 1.724276 78 I 892095 4 o0602060 29 I 462398 54 1.732394 79 1897627 5 0o698970 30 1I477121 55 1 740363 80 1.903090 6 0-778I5I 31 I149I362 56 1-748I88 81 I 908485 7 0.845098 32 I 505150 57 1-755875 82 1.913814 8 0-903090 33 I 51i85I4 58 1763428 83 1919078 9 0.954243 34 1-53I479 59 1-770852 84 1-924279 io Ioooooo000000 35 544068 60 1-77815I 85 1-929419 11 1.041393 36 1-556303 61 1 785330 86 1-934498 12 I 079181 37 1568202 62 1-792392 87 1-9395I9 I3 1-113943 38 1-579784 63 1 j9934I 88 1-944483 I4 1I.46128 39 1-591065 64 i-806180 89 1I949390 I5 1.176091 40 1 602060 65 1-812913 90 1-954243 i6 I1204I20 41 I 612784 66 1-8I9544 91 1.959041 17 1-230449 42 1-623249 67 1-826075 92 1963788 i8 1 255273 43 1633468 68 832509 93 1 968483 19 1-278754 44 I -63453 69 1-838849 94 1-973I28 20 -30oo30 o 45 1-653213 70 1-845098 95 I'977724 21 1-322219 46 1 662758 71 1I 851258 96 1982271 22.342423 47 1.672098 72 1857333 97 1.986772 23 1-36I728 48 1-68124I 73 1-863323 98 1-991226 24 13 3802II 49 I 690196 74 1-869232 99 1-995635 25 1 397940 50 1 I 698970 75 I 87506I I00 2-000000 of the O's, to catch the eye, and to indicate that from thence the annexed first two figures of the Logarithm in the second column stand in the next lower line, directly under the asterisk. I 2 LOGARITHMS OF NUMBERS. TABLE I. N. 1 1 2 3 4 5 6 8 9 D. 1oo00 0 ooooo 434 o858 3o 1734 2166 2598 3029 3461 3891 432 10I 4321 4751 5181 5609 6038 6466 6894 7321 7748 8174 428 102 *86oo 9026 9451 9876 *3o 0 724 1147 1570 1993 2415 424 io3 OI 2837 3259 J680 4100 4521 4940 5360 5779 6197 66i6 419 104 7033 7451 7868 8284 8700 g916 9532 9947 1361 0775 416 105 021189 6o3 2016 2428 2841 3252 3664 4075 4486 4896 412 o16 5306 5715 6125 6533 6942 7350 7757 8i64 8571 8978 408 107 *9384 9789.195 0600 1004 1408 1812 2216 2619 3021 404. 108 03 3 44 8264227 4628 5029 5430 5830 6230 6629 7028 400 Io09 7426 7825 8223 8620 9017 9414 9811 *207 o602 0998 396 iio 04 393 I787 2182 2576 2969 3362 3755 4148 4540 4932 393 III 5323 5714 61o5 6495 6885 7275 7664 8053 8442 8830 389 12 92I18 960o6 a9993 38o 766 53 I 538 I924 2309 2694 386 II3 05 3078 3463 3846 4230 46I3 4996 5378 5760 6142 6524 382 I4 *6905 7286 7666 8046 8426 88o5 9185 9563 9942 +32o 379 115 o6 698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 ii6 4458 4832 5206 5580 5953 6326 6699 707I 7443 7815 372 117 *8i86 8557 8928 9298 9668 o38 0407 0776 1145 I5I4 369 118 071882 2250 2617 2985 3352 3718 4085 4451 4816 5182 366 119 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 120 *g181 9543 9904.266 0626 0987 I347 1707 2067 2426 360 121 082785 3144 3503 386I 4219 4576 4934 529I 5647 6004 357 122 6360 6716 7071 7426 7781 8136 8490 8845 9g98 9552 355 123 *9905 +o58 061.0963 I3I5 I667 2018 2370 2721 307I 351 124 093422 3'172 4122 447I 4820 5169 55i8 5866 6215 6562 349 125 6910 7257 7604 7951 8298 8644 8990 9335 9681 o026 346 126 100371 0715 1059 403.1747 2091 2434 2777 3119 3462 343 127 3804 4146 4487 4828 5169 5510 5851 6191 653I 6871 340 128 -*72I0 7549 7888 8227 8565 8903 9241 9579 9916 +253 338 129 11059g 0926 1263 1 599 1934 2270 2605 2940 3275 3609 335 130 3943 4277 4611 4944 5278 5611 5943 6276 6608 6940 333 131 *7271 7603 7934 8265 8595 8926 9256 9586 9915.245 330 132 12 574 0903 1231 I560 I888 2216 2544 2871 3198 3525 328 133 3852 4178 4504 4830 5i56 5481 5806 6t13 6436 6781 325 134 * 7io5 7429 7753 8076 8399 8722 9045 9368 9690 +oi2 323 I35 i3 o334 o655 0977 1298 1619 1939 2260 2580 2900 3219 321 i36 3539 3858 4177 4496 4814 5133 545I 5769 6086 64o3 318 137 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 315 i38 *9879 +I94 o058 0822 1136 I450 1763 2076 2389 2702 314 139 14 301 3327 3639 3951 4263 4574 4885 5196 5507 5818 311 140 6128 6438 6748 7058 7367 7676 7985 8294 8603 89g11 309 141 *9219 9527 9835 4142 0449 0756 Io63 1370 1676 1982 307 142 15 2288 2594 2900 3205 35o 3815 4120 4424 4728 5032 305 I43 5336 5640 5043 6246 6549 6852 7154 7457 7759 8061 303 144 * 8362 8664 8965 9266 9567 9868 i68 0469 0769 I1o68 3o0 145 i6 i368 1667 1967 2266 2564 2863 3161 3460 3758 4055 299 I46 4353 4650 4947 5244 554I 5838 6I34 6430 6726 7022 297 147 73I7 76I3 7908 8203 8497 8792 9086 9380 9674 9968 295 148 170262 0555 o848 1141 1434 1726 2019 2311 2603 2895 293 149 3I86 3478 3769 4060 435I 4641 4932 5222 55I2 5802 291 150 6091 6381 6670 6959 7248 7536 7825 8II3 8401 8689 289 151 8977 9264 9552 9839 +126 o043 0699 0985 1272 1558 287 152 18 1844 2129 2415 2700 2985 3270 3555 3839 4123 4407 285 153 4691 4975 5259 5542 5825 6Io8 6391 6674 6956 7239 283 154 * 7521 7803 8084 8366 8647 8928 9209 9490 9771 +o5I 281 155 190 332 0612 0892 1171 1451 1730 2010 2289 2567 2846 279 I56 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 278 157 5900 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 i58 * 8657 8932 9206 9481 9755 4029 0303. o577 o850 1124 274 i59 20 1397 1670 1943 2216 2488 2 761 3033 33o5 3577 3848 272 N. 0 1 2 3 4 5. 6 7 8 9 ID. - / _ _ _ _ ___ __3 -' _ __ TABLE I. LOGARITHMS OF NUMBERS. 3 N. 0 1 2 3 4 56 6 8 9 D. I60 204I20 4391 4663 4934 5204 5475 5746 60 6286 666556 271 i6i 6826 7096 7365 7634 7'04 8173 8441 8710 8979 9247 269 162 *9515 9783 o051 03I9 0586. 0853 1121 I388 1654 1921 267 I63 21 2188 2454 2720 2986 3252 35i8 3783 4049 4314 4579 266 i64 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 i65 7484 7747 8010 8273. 8536 8798 9060 9323 9585 9846 262 i66 22 0108 0370 0631 0892 1153 I414 i675 936 29621 2456 261 I67 2716 2976.3236 3496 3755 4015 4274 4533 4792 5051 259 i68 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 258 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 +193.256 170 23 0449 0704 0960 1215 1470 1724'979 2234 2488 2742 254 171 2996 3250 3504 3757 4011 4264 4M17 4770 5023 5276 253 172 5528 578I 6033 6285 6537 6789 7041 7292 7544 7795 252 173 *8046 8297 8548 8799 9049 9299 9550 9800 +o5o o300 250 174 240549 0799 Io48 1297 1546 1795 2044 2293 2541 2790 249 175 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 248 176 55i3 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 177 *7973 8219 8464 8709 8954 9198 9443 9687 9932 +176 245 178 25 0420 0664 0908 1151 1395 I638 I881 2125 2368 2610 243 I79 2853 3096 3338 3580 3822 4064 43o6 4548 4790 503i 242 I80 5273 55I4 5755 5996 6237 6477 6718 6058 7198 7439 24i i8i 7679 7918 8i58 8398 8637 8877 91I6 9355 9594 9833 239 182 260071 o3io 0548 0787 1025 1263 i50o 1739 1976 2214 238 i83 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 184 1I8 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 I85 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 i86 *9513 9746 9980 +2i3 0446 0679 0912 1144 1377 1609 233 187 27 1842 2074 2306 2538 2770 3o00 3233 3464 3696 3927 232 I88 4158 4389 4620 4850 508i 5311 5542 5772 6002 6232 230 189 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 229.190 * 8754 8982 9211 9439 9667 9895 +123 o35i 0578 0806 228 IQ1 28 1033 1261 1488 1715 1942 2169 2396 2622 2849 3075 227 192 33o0 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 193 5557 5782 6007 6232 6456 668I 6905 7130 7354 7578 225 194 7802. 8026 8249 8473 8696 8920 9143 9366 9589 9812 223 195 29 0035 0257 o480 0702 0925 1147 1369 1591 i813 2034 222 196 2256 2478 2699 2920 314i 3363 3584 3804 4025 4246 221 I97 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 198 6665 6884 7104 7323 7542 7761 7979 8I98 8416 8635 219 199 8853 9071 9289 9507 9725 9943 +i61 0378 o595 o813 218 200 30 o30 1247 1464 1681 1898 2114 2331 2547 2764 2980 217 201 3196 3412 3628 3844 4059 4275 449I 4706 4921 5136 216 202 535i 5566 5781 5996 6211 6425 663Q 6854 7068 7282 215 203 7496 7710 7924 8137 835i 8564 8778 8991 9204 9417 213 204 *9630 9843 o056 0268 0481 0693 0906 1118 1330 1542 212 205 31 754 1966 2177 2389 2600 2812 3023 3234 3445 3656 211 206 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 207 5970 6i8o 6390 6599 6809 7018 7227 7436 7646 7854 209 208 8063 8272 8481 8689 8898 9 I6 9314 9522 9730 9938 208 209 320146 o354 o562 0769 0977 i184 1391 1598 i805 2012 207 21O.2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 212 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 213 *838o 8583 8787 8991 9194 9398 9601 9805 +oo8 0211 203 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 215 2438 2640 2842 3044 3246 3447 3649 3850 40o5 4253, 202 216 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 217 6460 6660 6860 7060 7260 7459 7659 7858 858 8257 200 218 * 8456 8656 8855 9054 9253 9451 9650 9849 -047 0246 199 219 34 0444 0642 0841 1039 1237 1435 1632 i830 2028 2225 198 N. 0 1 2 3 4 5 6: 8 9 D. 4 LOGARITHMS OF NUMBERS. TABLE L N. 0 1 2 3 4 5 6 8 9 D. 220 34 2423 2620 2817 30I4 3212 3409 3606 3802 3999 4196 197 221 4392 4589 478 4981 517 5374 5570 5766 5962 6 7 196 222 6353 6549 6744 6939 735 7330 7525 7720 7915 8I i 195 223 *83o5 85oo 8694 8889 9o83 9278 9472 9666 9860 *o54 194 224 350248 0442 o636 0829 Io23 1216 1410 i603 1796 I989 I93 225 2183 2375 2568 276I 2954 3I47 3339 3532 3724 3916 193 226 41o8 43oi 4493 4685 4876 5068 5260 5452 5643 5834 192 227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 228 7935 8I25 83i6 85o6 8696 8886 9076 9266 9456 9646 190 229 *9835 +025 0215 0404 0593 0783 0972 I I6 1350 1539 189 230 36 1728 1917 21i5 2294 2482 2671 2859 3048 3236 3424 88 231 3612 38oo 3988 4176 4363 455I 4739 4926 5113 53o0 I88 232 5488 5675 5862 6049 6236 6423 66I0 6796 6983 7I69 187 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 i86 234 - 9216 9401 9587 9772 9958 O143 0328 o5i3 0698 0883 I85 235 37 I068 1253 1437 1622 806' 1991 2I75 2360 2544 2728 184 236 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 237 4748 4932 5115 5298 548I 5664 5846 6029 6212 6394 183 238 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 I82 239 *8398 8580 8761 8943 9124 9306 9487. 9668 9849 o3o0 18 240 38 0211 0392 0573 0754 0934 1115 1296 1476 i656 1837 I8I 241 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 180 242 3815 3995 4174 4353 4533 4712 4891 5070 5249 5428 179 243 5606 5785 5964 6I42 6321 6499 6677 6856 7034 7212 178 244 7390 7568 7746 7923 81oi 8279 8456 8634 88I1 889 178 245 * 9166 9343 9520 9698 9875 0o5 0228 04o5 0582 0759 177 246 390935 1112 I1288 1464 1641 1817 1993 2I69 2345 252I 176 247 2697 2873 3048 3224 3400 3575 375 3926 41 4 4277 76 248 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 I75 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 7940 8114 8287 8461 8634 8808 8981 9154 9328 9501 173 251 * 9674 9847 +020 0192 o365 o538 0711 o883 1056 1228 173 252 40 1401 1573 1745 1917 2089 2261 2433 2605 2777 2949 I72 253 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 171 254 4834 5005 5176 5346 5517 5688 5858 6029 6199 6370 171 255 6540 6710 688I 7051 7221 7391 7561 7731 7901 8070 170 256 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 I169 257 *9933 +0i2 0271 0440 0609 0777 0946 III4 I283 I45I 169 258 41 1620 1788 1956 2124 2293 2461 2629 2796 2964 3132 I68 259 3300 3467 3635 3803 3970 4137 4305 4472 4639 4806 167 260 4973 5I40 5307 5474 564I 5808 5974 6141 6308 6474 167 26I 6641 6807 6973 7I39 7306 7472 7638 7804 797o 8i35 I66 262 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 i65 263 * 9956 I+21 0286 04S1 o6I6 0781 0945 1110 1275 1439 i65 264 42 1604 1768 1933 2097 2261 2426 2590 2754 2918 3082 I64 265 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 64 266 4882 5045 5208 5371 5534 5697 5860 6023 6186 6349 I63 267 65II 6674 6836 6999 7I6I 7324 7486 7648 7811 7973 162 268 8i35 8297 8459 8621 8783 8944 9106 9268 9429 9 91 162 269 9752 9914 o075 0236 0398 0559 0720 088i 1042 1203 I6i 270 43 1364 1525 i685 1846 2007 2167 2328 2488 2649 2809 i6 271 2969.3130 3290 3450 3610 3770 3930 4090 4249 4409 I6o 272 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 159 273 6i63 6322 648I 6640 6799 6957 7116 7275 7433 7592 159 274 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 275 * 9333 9491 9648 Jqo6 9964 +122 0279 0437 0594 0752 158 276 44 909 o166 1224 1381 1538 1695 1852 2009 2166 2323 157 277 2480 2637' 2793 2950 3io6 3263 3419 3576 3732 3889 I57 278 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 156 279 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 155. 0 1 2 3 4 6 6 7 8 9 D.~~~~~~~~69 6 88,03J5 TABLE I. LOGARITHMS OF NUMBERS. 5 N. 0 1 2 3 4 5 6 7 8 9 D. 280 447158 7313 7468 7623 7778 7933 8088 8242 8397 8552 155 281 * 8706 8861 9015 9170 9324 9478 i633 9787 9941 +095 154 282 45 0249 0403 0557 07 1 0865 1018 II72 1326 1479 i633 I54 283 1786 1940 2093 22471 2400 2553 2706 2859 3012 3i65 153 284 3318.3471 3624 3777 3930 4082 4235 4387 4540 4692 153 285 4845 4997 550o 5302 5454 5606 5758 5910 6062 6214 152 286 6366 65i8 6670 6821 6973. 7125 7276 7428 7579 7731 152 287 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 151 288 * 9392 9543 9694 9845 9995 Oi46 0296 0447 0597 0748 15i 289 46 898 1048 1198 1348 1499 1649 1799 1948 2098 2248 i50 290 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 150 291 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 292 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 49 293 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 294 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 295 * 9822 9969 + 16 0263 04Id 0557 0704 0851 0998 1145 147 296 47 I292 1438 I585 1732 1878 2025 2171 2318 2464 26Io 146 297 2756 2903 3049 39 34T 3 4 8 7 3633 3779 3925 4071 I46 298 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 146 299 5671 58i6 5962 6107 6252 6397 6542 6687 6832 6976 145 300 7121 7266 7411 7555 7700 7844 7989 8I33 8278 8422 145 30o 8566 8711 8855 8999 9143 9287 943 9575 9719 9863 I44 302 480007 015I 0294 0438 0582 0725 0869 1012 1156 1299 144 303 I443 586 1729 I872 2016 2159 2302 2445 2588 2731 143 304.2874 3016 3159 3302 3445 3587 3730 3872 4015 4157 143 305 4300 4442 4585 4727 4869 501I'5153 5295 5437 5579 142 306 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997 142 307 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 141 308 855I 8692 8833 8974 9114 9255 9396 9537 9677 9818 I41 309 *9958 *099 0239 0380 0520 066i o8oi 0941 108i 1222 140 310 49 1362 1502 1642 1782 1922 2062 2201 2341 2481 2621 140 311 2760 2900 3040 3179 3319 3458 3597 3737 3876 40I5 139 312 4I55 4294 4433 4572 47 1 4850 4989 5128 5267 5406 I39 313 5544 5683 5822 5960 609 6238 6376 6515 6653. 6791 139 314 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 315 8311 8448 8586 8724 8862 8999 9137 9275 9412 9550 I38 316 9687 9824 9962 0*99 0236 0374 0511 0648 0785 0922 137 317 50o 059 i196 i333 1470 I607 I744 i880 2017 2154 2291 I37 318 2427 2564 2700 2837 2973' 309g 3246 3382 3518 3655 I36 319 3791 3927 4o63 4199 4335 4471 4607 4743 4878 5014 i36 320 5150 5286 5421 5557 5693 5828 5964 6099 6234 6370 136 321 6505 6640 6776 6911 7046 7181 7316 7451 7586 772I i35 322 7856 7991 8126 8260 8395 8530 8664 8799 8934 9068 135 323 *9203 9337 9471 9606 9740 9874 +009 0143 0277 0411 I34 324 51 o545 o679 08I3 0947 Io8I 1215 1349 I482 1616 1750 134 325 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 I33 326 3218 3351 3484 3617 3750 3883 4016 4149 4282 4414 133 327 4548 468I 4813 4946 5079 5211 5344 5476 5609 574I 133 328 5874 6006 6139 6271 6403. 6535 6668 6800 6932 7064 132 329 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 132 330 8514 8646 8777 8909 9040 9171 9303 9434 9566 9697 131 331 *9828 9959 o090 0221 0353.0484 0615 0745 0876 1007 131 332 52 1138 1269 1400 i530 i66I 1792 1922 2053 2183 2314 131 333 2444 2575 2705 2835 2966 3096 3226 3356 3486 36i6 i30 334 3746 3876 4006 4(36 4266 4396 4526 4656 4785 4915 130 335 5045 5174 5304 5434 5563 5693 5822 595I 6o81 6210 129 336 6339 6469 6598 6727 6856 6985 7114 7243 7372 750I 129 337 7630 7759 7888 8o06 8145 8274 8402 853I 866o 8788 129 338 *8917 9045 9174 9302 9430 9559 9687 98i5 9943 ~072 I28 339 53 0200 0328 456 o584 07I2 o840 0968 1096 1223 i35I 128 N. 0 1 2 3 4 6 6 7 8 9'D. '6 LOGARITHMS OF NUMBERS. TABLE I. N. 0 1 2 3 4 5 6 8 9 D.. 340 531479 1607 1734 1862 1990 2117 2245 2372 2500 2627 128 341 2754 2882 3009 3136 3264 3391 3s. 8 3645 3772 3899 127 342 4026 4153 4280 4407 4534 4661 4787 4914 504I 5167 127 343 5294 5421 5547 5674 58o00 5927 6053 6I80 6306 6432 126 344 6558 6685 6811 6937 7063 7189 735 7441 7567 7693 126 345 7819 7945 807I 8197 8322 8448 8574 8699 8825 895 126 346 * 9076 9202 9327 9452 9578 9703 9829 9954 +079 0204 125 347 54 329 0455 o580 0705 o830 0955 1080.1205 I330'I454 125 348 1579 1704 14829 1953 2078 2203 2327 2452 2576 270I 125 349 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 124 350 4068 4192 4316 4440 4564 4688 4812 4936 5060 5183 124 351 5307 543I 5555 5678 5802 5925 6049 6172 6296 6419 124 352 6543 6666 6789 6913 7036 7I59 7282 7405 7529 7652 123 353 7775 7898 8o21 8144 8267 8389 852 8635 8758 888I 123 354 * 9003 9126 9249 937I 9494 96I6 9739 9861 9984 +io6 123 355 550228 0351 0473 0595 0717 o840 0962 1084 1206 1328 122 356 I450 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 357 2668 2790 2911 3033 3155 3276 3398 3519 3640 3762 12I 358 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 359 5094 5215 5336, 5457 5578 5699 5820 5940. 606I 6182 121 360 6303 6423 6544 6664 6785 6905 7026 7146 7267 7387 120 36i 7507 7627 7748 7868 7988 8io8 8228 8349 8469 8589 120 362 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 363 9907 +026 o046 0265 o385 o504 0624 0743 o863 0982 119 364 56 110 1221 340 1459 1578 1698 I817 1936 2055 2174 119 365 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 -1I9 366 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 119 367 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 118 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 118 369 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 8202 8319 8436 8554 8671 8788 8905 9023 9140 9257 117 371 *9374 9491 9608 9725 9842 9959 +076 0193 o309 0426 117 372 57 0543 o66o 0776 0893 ioIo 126 1243 1359 1476 1592 117 373 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 116 374 2872 2988 3104 3220 3336 3452 3568 3684 3800 3915 II6 375 4031 4147 4263 4379 4494 46o1 4726 4841 4957 5072 116 376 5188 5303 5419 5534 565o 5765 5880 5996 6111 6226 115 377 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 115 378 7492 7607 7722 7836 795I 8066 8I8I- 8295 84 10 8525 J15 379 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 380 9784 9898 +012 0126 0241 o355 0469 o583 o697 0811 I 14 38i 58 0925 1039 11 53 1267 1381 1495 i608 1722 I836 1950 114 382 2063 2177 2291 2404 2518 2631 2745 2858 2972 3085 114 383 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 113 384 433I 4444 4557 4670 4783 4896 5009 5122 5235 5348 113 385 5461 5574 5686 5799 5912 6024 6137 6250 6362 6475 ii3 386 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 112 387 77' 7823 7935 8047 8I60 8272 8384 8496 86o8 8720 112 388 8832 8944 9056 9167 9279 9391 9503 9615 9726 9838 112 389 9950 *o6i 0173 0284 0396 0507 0619 o730 0842 0953 112 390 59 I065 11I76 1287 1399 1510 1621 1732 I843 I955 2066 iII 39g 2177 2288 2399 25Io 2621 2732 2843 2954 3064 3175 i I 392 3286 3397 3508 3618 3729 3840 3950 4o06 4I7I 4282 III 393 4393 45o3 4614 4724 4834 4945 5055 5i65 5276 5386 Iio 394 5496 5606 5717 5827 5937 6047 6157 6267 6377 6487 110 395 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 Iio 396 7695 78o5 7914 8024 8I34 8243 8353 8462 8572 8681 iio 397 879I 8900 9009 9119 9228 9337 9446 9556 9665 9774 Io9 398 * 9883 9992 +IOI 0210 0319 0428 o537 0646 0755 o864 109 399 60 0973 I182 1191 1299 1408 I517 I625 1734 I843 195I 109 N. 0 1 2 3 4 5 6 7 8 9 D. TABLE I. LOGARITHMS OF NUMBERS. 7 N. 0 1 2 3 4 5 6 -7 8 9 D. 400 60 2060 2169 2277 2386 2494 2603 2711 2819 2928 3036 Io8 40o 3144 3253 336I 3469 3577 3686 3794 3902 4010 4II8 1o8 402 4226 4334 44 42 4550 4658 4766 4874 4982 5089 5197 108 403 5305 5413 5521 5628 5736 5844 595 60o59 6i66 6274 Io8 404 6381 6489 6596 6704 681 6919 7026 7133 7241 7348 107 405 7455 7562 7669 7777 7884 / 799 8098 8205 8312 84I9 107 406 8526 8633 8740 8847 8954 9061 9167 9274 938I 9488 107 407 9594 9701 9808 9914 4021 0128 0234 0341 0447 0554 107 408:61 o66o 0767 0873 0979 o186 II92 1298 1405 151I I617 Io6 409 I723 1829 1936 2042 2148 2254 2360 2466 2572 2678 io6 410 2784 2890 2996 3102 3207 3313 3419 3525 3630 3736 1o6 4r1 3842 3947 4053 4159 4264 4370 4475 458 4686 4792 o06 412 4897 5oo3 51o8 5213 5319 5424 5529 5634 5740 5845 io5 413 5950 6055 616o 6265 6370 6476 658 6686 6790 6895 1o5 414 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 io5 415 8048 8i53 8257 8362 8466 8571 8676 8780 8884 8989 o5 416 9093 9198 9302 9406 95 1 9615. 9719 9824 9928 0o32 104 417 62 oI36 0240 0344 0448 0552 o656 0760 o864 o968 1072 104 418 J176 1280 1384- 1488 1592 I695 1799 g903 2007 2110 104 419g 2214 2318 2421 2525 2628 2732 2835 2939 3042 3 46 104 420 3249 3353- 3456 3559 3663 3766 3869 3973 4076 4179 I03 421 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 0o3 422 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 1o3 423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 0o3 424 7366 7468 7571 7673 7775 7878 7980 8082 8i85 8287 102 425 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308 I02 426 - 941O 9512 9613 9715 9817 9919 021 0123 0224 0326 102 427 630428 o53o o63i 0733 o835 0936 1o38 1139 1241 1342 I02 428 1444 1545 1647 1748 -849 1951 2052 2I53 2255 2356 IoI 429 2457 2559 2660 2761 2862 2963 3064 3I65 3266 3367 101 430 3468 3569 3670 377I 3872 3973 4074 4175 4276 4376 Io0 431 4477 4578 4679 4779 4880 498I 508I 5182 5283 5383 ioo 432 5484 5584 5685 5785 5886 5986 6087 6I87 6287 6388 Ioo 433 6488 6588 6688 6789 6889 6989 7089 7I89 7290 7390 ioo 434 7490 7590 7690 7790 7890 7990 8090 8g19 8290 8389 99 435 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 99 436 * 9486 9586 9686 9785 9885 9984 *o84 oi83 0283 0382 99 437 640481 0581 o680 0779 0879 0978 1077 I177 1276 I375 99 438 I474 I573 1672 177I 1871 1970 2069 2168 2267 2366 99 439 2465 2563 2662 2761 2860 2959 3058 3i56 3255 3354 99 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 98 441 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 98 442 5422 5521 5619 5717 5815 5913 6o0i 6Iio 6208 63o6 98 443 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 98 444 7383 7481 7579 7676 7774 7872 7969 8067 8i65 8262 98 445 8360 8455 8555 8653 0750 8848 8945 9043 9140 9237 97 446 * 9335 9432 9530 9627 9724 9821 9919 oi6 oii3 0210 97 447 65o308 0405 0502 o599 0696 0793 o89o 0987 o084 1181 97 448 1278 2 375 1472 1569 1666 1762 1859 1956 2053 2150 97 449 2246 2343 2.440 2536 2633 2730 2826 2923 309 3 i6 97 450 3213 3309 3405 3502 3598 3695 379I 3888 3984 4080 96 451 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042.96 452 5i38 5235 533I 5427 5523 5619 5715 58io 5906 6002 96 453 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 96 454 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 455 80I I 8107 8202 8298 8393 8488 8584 8679 8774 8870 95 456 8965 9060 9155 9250 9346 944I 9536 9631 9726 982I 95 457 *9916 o0I oio6 0201 0296 039og 0486 058I 0676 0771 95 458 660865 0960 o155 i 50 1245 1339 I434 I529 I623 I7I8 95 459 I813 1907 2002 2096 2191 2286 2380 2475 2569 2663 95 N. 0 1 2 3 4 5 6 7 8 9. D. 11 8 LOGARITHMS OF NUMBERS. TABLE I. N. 0 1 2 3 4 5 6 l 8 9 D. 460 66 2758 2852 2947 304I 3135. 3230 3324 3418 3512 3607 94 461 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 94 462 4642 4736 4830 4924 5018 5112 5206. 5299 5393 5487 94 463 558i 5675 5769 5862 5956 6050 6143 6237 633I 6424 94 464 6518. 66I2 6705 6799 6892 6986 7079 7173 7266 7360 94 465 7453 7546 7640 7733 7826 7920 8oi3 8io6 8199 8293 93 466 8386 8479 8572 8665 8759 8852 8945 9038 9131 9224 93 467 *9317 9410 9503 9596 9689 9782 9875 9967 o6o 0153 93 468 67 0246 0339 0431 0524 0617 0710 o802 0895 0988 1080 93 469 I173 1265 i358 145I I543 I636 1728 I821 I913 2005 93 470 2098 2190 2283 2375 2467 2560 2652 2744 2836 2292 92 471 321 3113 325 3297 3390 3482 3574 3666 3758 385 92 472 3942 4 034 4126 4218 4310 4402 4494 4586 4677 4769 92 473 4861 4953 5045 5137 5228 5320 5412 5503 5595 5687 92 474 5778 5870 5962 6053 6145 6236 6328 6419 65II 6602 92 475 6694 6785 6876 6968' 7059 715I 724 7333 7424 75I6 9I 476 7607 7698 7789 7881 972 8063 8i54 8245 8336 8427 91 477 8518 8609 700 879I 8882 8973 9064 9155 9246 9337 91 478 *9428 9519 9610 9 900 791 9882 9973'o63 o154 o245 91 479 68 0336 0426 0517 0607 0698 0789 o879 0970 I06o 1151 91 480 124I 1332 I422 I1513 i603 I693 1784 I874 1964 2055 90 48I 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 482 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 90 483 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 484 4845 4935 5025 5114 5204 5294 5383 5473 5563 5652 90 485 5742 583 5921 6oio 6100 6189 6279 6368 6458 6547 89 486 6636 5726 681 6904 6994 7083 7172 7261 735I 7440 89 487 75.29 768 7707 7796 7886 7975 8o64 853 8242 833I 89 488 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 89 489 *9309 9398 9486 9575 9664 9753 9841 9930 *019 0107 89 490 690196 0285 0373 0462 o550 0639 0728 o816 0905 o993 89 491 1081 1170 1258 1347 1435 1524 1612 1700 1789 1877 88 492 IQ65 2053 2142 2230 2318 2406 2494 2583 2671 2759 88 493 2247 2935 3023 3111 3199. 3287 3375 3463 3551 3639 88 494 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 88 495 4605 4693 4781 4868 4956 5044 5131 5219 5307 5394 88 496 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 87 497 6356 6444 653I 66I8 6706 6793 6880 6968 7055 7142 87 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 80I4 87 499 8101 8.88 8275 8362 8449 8535 8622 8709 8796 8883 87 500 8970 9057 9144 9231 9317 9404 9491 9578 9664 9751 87 50i *9838 9924 *oiI o098 OI84 0271 o358 0444 o53I 0617 87 502 70 0704 0790 0877 o063 io50 II36 I222 I309 3 135 482 86 503 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 86 504 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 86 505 3291 3377 3463 3549 3635 3721 3807 3895 3979 4065 86 506 410i 4236 4322 4408 4494 4579 4665 4751 4837 4922 86 507 5008 5094 5179 5265 5350 5436 5522 5607 5693 5778 86 508 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 85 509 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 85 5Io 7570 7655 7740 7826 7911 79968o8 8i66 8251 8336 85 511 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 85 512 *9270 9355 9440 9524 9609 9694 9779 9863 9948.o33 85 513 710117 0202 0287 o371 0456 0540 062 0710 0794 0879 85 514 0963 o148 1132 1217 130o I385 1470 1554 1639 1723 84 515 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 84 516: 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 84 517 3491 3575 3650 3742 3826 39o 3994 4078 4162 4246 84 518 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 84 519 5167 5251 5335 5418 5502 5586 5669 5753 5836 5920 84 N. 0 1 2 3 4 5 6 7 8 9 D. TABLE I. LOGARITHMS OF NUMBERS. 9 N. 0 1 2. 3 4 5 6 7 8 9 D. 520 716003 6087 6170 6254 6337 6421 6504 6588 6671 6754 83 521 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 83 522 7671 7754 7837 792o 8003 8086 8169 8253 8336 8419 83 523 8502 -8585 8668 8751 8834 8917 9000 9083 9165. 9248 83 524 *9331 9414 9497 9580 9663 9745 9828 9911 9994 *+77 83 525 720159 0242 0325 0407 0490 0573 o655 0738 0821 ogo3 83 526 o986 io68 115I 1233 I316 I398 1481 I563 I646 1728 82 527 111 1893 1975 2058 2140 2222 2305 2387 2469 2552 82 528 2634 2716 2798 2881 2963.3045 3127 3209 3291 3374 82 529 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 82 530 4276 4358 4440.4522 4604 4685 4767 4849 4931 5oi3 82 531 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 82 532 5912 5993 6075 6i56 6238 6320 640I 6483 6564 6646 82 533 6727 6809 6890 6972 7053 7134 7216 7297 7379 746 8I 534 754I 7623 7704 7785 7866 7948 8029 8I1o 8191 8273 81 535 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 8i 536 9165 9246 9327 9408 9489 9570 9651 9732 9813 9893 81 537 *9974 *o55 oi36 0217 0298 0378 0459 0540 0621 0702 81 538 73 0782 0863 0944 1024 1105 x186 1266 1347 I428 1508 81 539 i589 1669 1750 18.30 1911 1991 2072 2152 2233 2313 81 540 2394 2474 2555 2635 2715 2796 2876 2956 3037 3117 80 541 3197 3278 3358 3438 3518 3598 3679 3759 3839 3919 80 542 3999 4079 460o 4240 4320 4400 4480 4560 4640 4720 80 543 48oo 4880 4960 5040 5120 5200 5279 5359 5439 5519 80 544 5599 5679 5759 5838 5918 5998 6078 6157 6237 6317 80 545 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 546 7193 7272 7352 7431 751 7590 7670 7749 7829 7908 79 547 797 8 67 0 8 I46 8225 8305 8384 8463 8543 8622 8701 79 548 8781 8860 8939 90o8 9097 9177 9256 9335 9414 9493 79 549 *9572 965I 9731 9810 9889 9968 +047 0126 0205 0284 079 550 74 0363 0442 0521 o6o00 678 0757 o836 0915 0994 1073 79 551 152 I230 1309 I388 1467 I546 I624 i703 1782 i860 79 552 I939 2018 2096 2175 2254 2332 2411 2489 2568 2646 7 553 2723 2804 2882 296I 3039 3118 3196 3275 3353 3431 78 554 351o 3588 3667 3745 3823 3902 3980 4058 4i36 4215 78 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 78 557 5855 5933 60oi 6089 6167 6245 6323 640I 6479 6556 78 558 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 78 559 7412 7489 7567 7645 7722 7800 7878 79 8033 8110 78 560 8188 8266 8343 8421 8498 8576 8653 8731 8808 8885 77 561 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 562 *9736 9814 9891 9968 *o45 0123 0200 0277 o354 o431 77 563 75 0508 o586 0663 0740 0817 0894 0971 1048 I125 1202 77 564 1279 I356 1433 151o 1587 1664 1741 i818 1895 I972 77 565 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 77 566 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 77 567 3583. 3660 3736 3813 3889 3966 4042 4119 4195 4272 77 568 4348 4425 45o0 4578 4654 4730 4807 4883 4960 5o36 76 569 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 570 5875 5951 6027 6103 6180 6256 6332 6408 6484 6560 76 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 572 1396 7472 7548 7624 7700 7775 785 7927 8003 8079 76 573 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 76 574 8912 8988 9063 9139 9214 9290 9366 9441 9517 9592 76 575 *9668 9743 9819 9894 9970 +045 0121 0196 0272 0347 75 576 76 0422 048 0573 0649 0724. 0799 0875 09o0 1025 I101 75 577 1176 1251 1326 1402 1477 I552 I627 1702 1778 i853 75 57 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 75 579 2679 2754 2829 2904 2978 3053 3128 3203 3278 3353 75 N. 0 1 2 3 4 5 6 1 8 9 D. 10 LOGARITHMS OF NUMBERS. TABLE I. N. 0 1 2 3 4 5 66 8 9 D. *580 76 3428 3503 3578 3653 3727 3802 3877 3q52 4027 4101 75 581 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 75 582 4923 4998 5072 5147 522I 5296 5370 5445 5520 5594 75 583 5669 5743 5818 5892 5966 6o4I 6115 6190 6264 6338 74. 584 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 74 585 7156 7230 7304 7379 7453 7527 76i0 7675 7749 7 77823 4 586 7898 7972 8046 8120'8194 8268 8342 8416 8490 8564 74 587 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 74 588 9377 9451 9525 9599 9673 9746 9820 9894 9968 *o42 74 589 77 O115 0189 0263 o336 0410 0484 o557 o63I 0705 0778 74 590 0852 0926 0999 I073 1146 1220 I293 1367 1440 1514 74 591 1587 1661 1734 I808 1i88i 1955 2028 2102 2175 2248 73 592 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 73 593 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 73 594 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 595 4517 4590 4663 4736 4809 4882 4955 5028 5100 5173 73 596 5246 5319 5392 5465 5538 561o 5683 5756 5829 5902 73 597 5974 6047 6120 6193 6265 6338 6411 6483 6556 6629 73 598 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 599 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079 72 600 815i 8224 8296 8368 8441 8513 8585 8658 8730 8802 72 60o 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 602 *9596 9669 9741 9813 9885 9957 +029 OIOI 0173 0245 72 603 78 0317 0389 0461 0533 0605 0677 074 0821 893 0965 72 604 I037 119 181 1253 i324 I396 i468 I540 I612 I684 72 605 I755 1827 I899 197I 2042 2114 2186 2258 2329 2401 72 606 2473 2544 2616 2688 275 283I 2902 2974 3046 3II7 71'607 3189 3260 3332 3403 3475 3546 3618 3689 376I 3832 71 608. 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 609 46 7 4689 4760 483I 4902 4974 5045 5116 5187 5259 71 610 5330 54oi 5472 5543.5615 5686 5757 5828 5899 5970 71 611 604I 6112 6183 6254 6325 6396 6467 6538 6609 6680 71 612 6751 6822 6893 6964 7035 70o6 7177 7248 73I9 7390 71 613 7460 7531 7602 7673'7744 7815 7885 7956 8027 8098 71 614 8 68 8239 83io1 838i 845I 8522 8593 8663 8734 8804 71 615 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 71 616 *958I 965I 9722 9792 9863 9933 oo04 0074 0144 0215 70 617 79 0285 0356 0426 0496 0567 0637 0707 0778 o848 0918 70 6i8 0988 1059 1129 1199 1269 I34o 1410 1480 I55o I620 70 619 1691 1761 i831 1901 1971 2041 2111 2181 2252 2322 70 620 2392 2462 2532 2602 2672 2742 2812 2882 2952 3022 70 621 3092 3162 3231 33o0 3371 344I 3 3835 358 365 3721 70 622 3790 3860 3930 4000 4070 4139- 4209 4279 4349 44I8 70 623 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 70 624 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 70 625 5880 5949 6o09 6088 6I58 6227 6297 6366 6436 6505 69 626 6574 6644 6713 782 6852 6921 6990 7060 7129 7198 69 627 7268 7337 7406 7475 7545 7614 7683 7752 782I 7890 69 628 7960 8029 8098 8167 8236 8305 8374 8443 8513. 8582 69 629 865i 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 630 934I 9409 9478 9547 96I6 9685 9754 9823 9892 9961 69 63I 8o 0029 0098 o167 0236 o305 0373 0442 051I o58o 0648 69 632 0717 0786 0854 0923 0992 o106 1129 118 1266 I335 69 633 1404 I472 1541 I609 1678 1747 I81 I 884 1952 202I 69 634 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 69 635 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 68 636 3457 3525 3594 -3662 3730 3798 3867 3935 4003 4071 68 637 4139 4208 4276 4344 4412 448o 4548 4616 4685 4753 68 638 482I 4889 4957 5025 5093 516i 5229 5297 5365 5433 68 639 55o0 5569 5637 5705 5773 584i 5908 5976 6044 6112 68 N. 0 1 2 3 4 5 6 i 8 9 D. TABLE I. LOGARITHMS OF NUMBERS. II N. 0 1 2 3 4 5 6 7 8 9 D. 640 806180 6248 63i6 6384 6451 6519 6587 6655 6723 6790 68 641 6858 6926 6994 7061 7129 7197 7264 7332 7400 7467 68 642' 7535 7603 7670 7738 7806 7873 794I 8008 8076 8143 68 643 8211 8279 8346 8414 8481 8549 8616 8684 8751 88i8 67 644 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 645 *9560 9627 9694 9762 9829 9896 9964 *o3 0098 oi65 67 646 81 0233 0300 o367 0434 0501 0569 0636 0703 0770 0837 67 647 0904 0971 1039 1106 1173 1240 I307 I374 I441 i508 67 648 I575 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 649 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847 67 650 2913 2980 3047 3114 3181 3247 3314 338i 3448 3514 67 65I 358i 3648 3714 378I 3848 3914 3981 4048 4114 418I 67 652 4248 4314 438I 4447 4514 4581 4647 4714 4780 4847 67 653 4913 4980 5046 5Ii3 5179 5246 5312 5378 5445 5511 66 654 5578 5644, 5711 5777 5843 5910 5976 6042 6g09 6175 66 655 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 66 656 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 66 657 7565 763I 7698 7764 7830 7896 7962 8028 804 8i60 66 658 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 66 659 8885 8951 9017 9083 9i49 9215 9281 9346 9412 9478 66 660 9544 96o0 9676 974I 9807 9873 9939 +004 0070 0o36 66 66i 82 0201 0267 333 0399 0464 0530 066 727 792 66 662 0858 0924 0989 o55 1120 i 86 I251 I317 I382 1448 66 663 I514 I579 I645 1710 1775 1841 1906 1972 2037 2103 65 664 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 65 665 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 666 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 65 667 4126 4191 4256 4321 4386 4451 4516 458I 4646 47II 65 668 4776 4841 4906 4971 5036 5io1 5i66 5231 5296 5361 65 669 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 65 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 65 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 65 672 7369 7434 7499 7563 7628 7692 7757 782I 7886 795i 65 673 8015 8080 8144 82082 273 8338 8402 8467 853 8595 64 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 92J9 64 675 9304 9368 9432 9497 9561 9625 96go 9754 9818 9882 64 676 *9947 o+II 0075 0139 0204 0268 0332 0396 0460 0525 64 677 830589 o653 0717 0781 0845 o909 0973 1037 1102 i 66 64 678 I230 1294 i358 1422 I486 1550 i614 I678 1742 I806 64 679 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 2509 2573 2637 2700 2764 2828 2892 2956 3020 3083 64 68I 3147 3211 3275 3338 3402 3466 3530 3593 365.7 3721 64 682 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 683 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 684 5056 5120 5i83 5247 53io 5373 5437. 5500 5564 5627 63 685 5691 5754 5817 588i 5944 6007 607 6134 6197 6261 63 686 6324 6387 6451 6514 6577 664 6 74 6767 6830 6894 63 687 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 63 688 7588 7652 7715 7778 7841 7904 7967 8030 8093 8I56 63 689 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 8849 8912 8975 9038 910o 9g64 9227 9289 9352 9415 63 691 *9478 9541 9604 9667 9729 9792 9855 9918 9981 +043 63 692 84 0106 0169 o232 0294 0357 0420 0482 o545 o608 o67 63 693 0733 0796 0859 0921 0984 o146 1109 1172 234 1297 63 694 1359 I422 1485 1547 1610 1672 1735 1797 i86o 1922 63 695 1985 2047 2110 2172 2235 2297 2360 2422 2484 2547 62 696 2609 2672 2734 2796 2859 2921 2983 3046 3io8 3170 62 697 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 62 6 3855 3918 3980 4042 4104 4I66 4229 4291 4353 4415 62 699 4477 4539 4601 4664 4726 4788 4850 4912 4974 5036 62 N. 0 1 2 3 4 5 6 7 8 9 D. 12 LOGARITHMS OF NUMBERS. TABLE I. N. O 1 2 3 4 5 6 - 8 9 D. 700 84 5098 5i6 5222 5284 5346 5408 5470 5532 5594 5656 62 701 5718 5780 5842 50o4 5966 6028 6090 615I 6213 6275 62 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 703 6955 7017 7079 714I 7202 7264 7326 7388 7449 7511 62 704 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 705 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 706 88o5 8866 8028 8989 9051 9112 9174 9235 9397 9358 61 707 9419 9481 9542 9604 9665 9726 9788 9849 991 9972 6I 70 85oo33 o0095 o56 0217 0279 o340 0401 462 0524 0585 6 709 0646 0707 0769 o830 0891 0952 1014 1075 136 1197 61 710 1258 1320 I38I 1442 1503 I564 1625 1686 I747 I809 61 71I I870 i93I 1992 2053 2114 2175 2236 2297 2358 2419 61 712 2480 2341 2602 2663 2724 2785 2846 2907 2968 3029 6I 713 3090 3150 3211 3272- 3333 3394 3455 35i6 35'77 3637 61 714 3698 3759 3820 388i 394I 4002 4063 4124 4185 4245 6I 715 4306 4367 4428 4488 4549 461o 4670 4731 4792 4852 6I 716 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 6i 717 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 6I 718 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 719 6729 6789 6850 6910 6970 703I 7091 7152 7212 7272 60 720 7332 7393 7453 7513 7574 7634 7694 7755 7815 7875 60 721 7935 7995 8056 8ii6 8176 8236 8297 8357 8417 8477 60 722 8537 7 87 8657 8718 8778 8838 8898 8958 90o8 9078 60 723 9138 9198 9258 9318 9379 9439 949 9559 9619 9679 60 724 * 9739 9799 9859 9918 9978 +o38 oo98 0158 021 027 60 725 86o338 o398 o458 o5i8 0578 0637 0697 0757 0817 0877 60 726 0937 0996 Io56 1116 1176 1236 I295 i355 14i5 1475 60 727 I 534 1594 654 1714 1773 i833 1893 1952 20o2 2072 60 728 2131 219 2251 23IO 2370 2430 2489 2549 2608 2668 60 729 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 3323 3382 3442 350o 3561 3620 3680 3739 3799 3858 59 731 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 732 41 4570 4630 4689 4748 4808 4867 4926 4985 5045 50 733 5Io4 5I63 5222 5282 534I 5400 5459 55I9 5578 5637 59 734 5696 5755 5814 5874 5933 5992 6o5x 6iio 6669 6228 59 735 6287 6346 6405 6465 6524 6583 6642 670I 6760 6819 59 736 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 737 7467 7526 785 7644 7703 7762 782I 7880 7 793 59 738 8056 8115 8174 8233 8292 835o 8409 8468 8527 88 59 739 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 59 741 *9818 9877 9935 9994 *o53 oiIi 0170 0228 0287 o345 59 742 87 0404 0462 o52I 0 079 o638 0696 0755 o8i3 o872 0930 58 743 0989 1047 Iio6 1164 1223 1281 1339 1398 1456 i515 58 744 1573 i63I 1690 1748 i806 i865 I923 1981 2040 2098 58 745 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 746 2739 2797 2855 2913 2 72 3030 3088 3146 3204 3262 58 747 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 748 3902 3960 40o8 4076 4134 4192 4250 4308 4366 4424 58 749 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 58 750 5o6i 5ii9 5177 5235 5293 5351 5409 5466 5524 5582 58 75i 5640 5698 5756 5813 587I 5929 5987 6045 6102 6i60 58 752 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 58 753 6795 6853 69o1 6968 7026 7083 7141 7199 7256 7314 58 754 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 755 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 57 756 8522 8579 8637 8694 8752 88 8809 8866 8924 8 939 57 757 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 57 758 *9669 9726 9784 9841 9898 9956 *oi3 0070 0127 oI85 57 759 880242 0299 0356 04I3 0471 0528 o585 0642 0699 0756 57 N. 1 2 3 4 5 6 7 8 9 D. TABLE I. LOGARITHMS OF NUMBERS. 18 N. 0 1 2 3 4 5 6 7 8 9 D. 760 88 81 4 o872 0928 0985 I042 1099 i56 1213 1271 1328 57 761 1385 1442 1499 I556 1613 1670 1727 1784 I84I 1898 57 762 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 57 763 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 57 764 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 57 765 366I 3718 3775 3832 3888 3945 4002 4059 4115 4172 57 766 4229 4285 4342 4399 4455 45I2 4569 4625 4682 4739 57 767 4795 4852 4909 4965 5022 5078 5i35 5192 5248 5305 57 768 536I 5418 5474 5531 5587 5644 5700 5757 58I3 5870 57 769 5926 5983 6039 6096 6152 6209 6265 632I 6378 6434 56 770 6491 6547 6604 6660 6716 6773 6829 6885 6942 6998 56 771 7054 7111 7167 7223 7280 7336 7392 7449 75o5 7561 56 772 7617 7674 7730 7786 7842 7898 7955 8oi 8067 8123 56 773 8179 8236 8292 8348 8404 8460 85 6 8573 8629 8685 56 774 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 56 775 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 56 776 *9862 9918 9974 *o30 oo86 0141 0197 0253 o309 o365 56 777 89 0421 0477 o533 0589 0645 0700 0706 0812 o868 0924 56 778 0980 1035 1091 1147 1203 2259 I314 1370 1426 1482 56 779 I537 1593 I649 1705 1760 1816 I872 1928 1983 2039 56 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2595 56 781 2651 2707 2762 2818 2873 2929 2985 3040 3096 313i 56 782'3207 3262 33t8 3373 3429 3484 3540 3595 3651 3706 56 783 3762 3817 3873 3928 3984 4039 4094 41 0 4205 4261 55 784 4316 4371 4427 4482 4538 4593 4648 4704 4759 48I4 55 785 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 55 786 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 55 787 5975 6030 6085 6140 6195 6251 6306 636i 6416 6471 55 788 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 55 789 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 55 790 7627 7682 7737 7792 7847 7902 7957 8012 8067 8122 55 791 8I76 8231 8286 8341 8396 845I 8506 856I 8615 8670 55 792 8725 8780 8835 88o9 8944 8999 9054 9109 9164 9218 55 793 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 55 794 *9821 9875 9930 9985 o039 0094 0149 0203 0258 0312 55 795 90 0367 0422 0476 o53I 0586 0640 0695 0749 0804 o859 55 796 o913 o068 I022 1077 1131 1186 1240 1295 1349 1404 55 7967 1458 1513 1567 I622 I676 173I 1785 i840 1894 1948 54 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 54 799 2547 260I 2655 2710 2764 2818 2873 2927 298I 3036 54 800 3090 3144 3199 3253 3307 336I 3416 3470 3524 3578 54 80O 3633 3687 3741. 3795 3849 3904 3958 4012 4066 4120 54 802 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 54 803 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 54 804 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 54 805 5796 5850 5904 5958 6012 6066 6iI9 6I73 6227 628I 54 806 6335 6389 6443 6497 655I 6604 6658 6712 6766 6820 54 807 6874 6927 6981, 7035 7089 7143 7196 7250 7364 7358 54 808 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 54 809 7949 8002 8056 8iio 8i63 8217 8270 8324 8378 843 54 810 8485 8539 8592 8646 8699 8753 8807 8860 8914 8967 54 8i1 9021 9074 9128 9181 925 9289 9342 9396 9449 93 54 812 *9556 9610 9663 9716 9770 9823 9877 9930 9984 *o37 53 813 91 0091 0144 017 0251 0304 0358 041r1 0464 o5i8 0571 53 814 0624 0678 0731 0784 0838 o891 0944 0998 io5i 1104 53 815 1158 1211 1264 1317 1371 1424'477 i530 1584 1637 53 8i6 -690 I743 1797 1850 I903 I956 2009 2063 2116 2I69 53 817 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 53 818 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 53 819 3284 3337 3390 3443 3496 3549 3602 3655 3708 376I 53 N. 0 1 2 3 4 6 8 9 D. 14 LOGARITHMS OF NUMBERS. TABLE L N. 0 1 2 3 4 5 6 7 8 9 D. 820 91 3814 3867 3920 3973 4026 4079 4132 4184 4237 4290 53 821 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 53 822 4872 4925 4977 5030 5083 5136 5189 5241 5294 5347 53 823 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 53 824 5927 5980 6033 6085 6138 6191 6243 6296 6349 64o0 53 825 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 53 826 6980 7033 7085 7138 790 7243 7295 7348 7400 7453 53 827 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 52 828 8o3o 8083 8135 8188 8240 8293 8345 8397 8450 8502 52 829 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 52 830 9078 9130 9183 9235 9287 9340 9392 9444 9496 9549 52 831 *9601 9653 9706 9758 9810 9862 9914 9967 +019 0071 52 832 92 0123 0176 0228 0280 0332 0384 0436 0489 0541 0593 52 833 0645 0697 0749 080oi o853 0906 0958 IOIO 1062 1114 52 834 1166 1218 1270 1322 1374 1426 1478 I530 1582 1634 52 835 1686 1738 1790 1842 1894 I946 1998 2050 2102 2154 52 836 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 52 837 2725 2777 2829 2881 2933 2985 3037 3089 3140 3I92 52 838 3244 3296 3348 3399 345* 3503 3555 3607 3658 3710 52 839 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 52 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 52 841 4796 4848 4899 4951 5003 5054 5io6 5157 5209 5261 52 842 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 52 843 5828 5879 5931 5982 6034 6085 6137 6188 6240 6291 51 844 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 51 845 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 51 846 7370 7422 7473 7524 7576 7627 7678 7730 778i 7832 51 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 5I 848 8396 8447 8498 8549 86o0 8652 8703 8754 8805 8857 51 849 8908 8959goi 6 9102 o963 9215 9266 9317 9368 51 850 9419 9470 952I 9572 9623 9674.9725 9776 9827 9879 51 851 9930 9981 032 o083 0134. oI85 0236 0287 o338 o389 51 852 93 0440 0491 0542 0592 0643 0694 0745 0796 0847 0898 5I 853 0949 I000 1051 1102 1153 1204 1254 i305 i356 1407 51 854. 1458 1509 1560 16io 166i 1712 1763 1814 i865 1915 51 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 5i 856 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 51 857 298I 303i 3082 3133 3183 3234 3285 3335 3386 3437 5i 858 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 51 859 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 51 860 4498 4549 4599 4650 4700 4751 480o 4852 4902 4953 5o 861 5003 5054 51o4 5154 5205 5255 5306 5356 5406 5457 50 862 5507 5558 5608 5658 5709 5759 5809 5860 591o 5960 50 863 6oii 6o6I 6ii 1 6162 6212 6262 63i 6363 6413 6463 50 864 6514 6564 6614 6665 6715 6765 68i5 6865 6916 6966 5o 865 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 50 866 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 50 867 8019 8 8119 86 8 869 89 69 8320 8370 8420 8470 5 868 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 50 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 5o 870 9519 9569 96i 9769 9819 9869 989 9869998 9968 5 871 940oo1 00o016 o oi68 02 0267 o317 0367 047 0467 50 872 o516 o0566 o616 o666 0716 0765 o815 o865 o915 0964 5o 873 1014 1064 I114 1163 1213 1263 I313 1362 1412 1462 50 874 i511 I561 I611 i660 1710 1760 1809 1859 1909 1958 50 875 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 50 876 2504 2554 2603 2653 2702 2752 2801 2851 2901 2950 50 877 3000 3049 3099 348 3198 3247 3297 3346 3396 3445 49 878 3495 3544 3593 3643 3692 3742 379I 384I 3890 3939 49 879 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 49 N. 0 1 2 3 4 6' 8 9 D. _ - i i i - - i m i TABLE I. LOGARITHMS OF NUMBERS. 15 N. 0 1 2 3 4 5 6 7 1 8 9 D. 880 94 4483 4532 4581 4631 4680 4729 4779 4828 4877 4927 49 88I 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 49 882 5469 5518 5567 56I6 5665 5715 5764 58i3 5862 5912 49 883 5961 6o00 6059 6o08 6157 6207 6256 6305 6354 6403 49 884 6452 6501 655I 6600 6649 6698 6747 6796 6845 6894 49 885 6943 6992 7041 7090 7,40 7I89 7238 7287 7336 7385 49 886 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 887 7924 7973 8b22 8070 8129 8i68 8217 8266 83I5 8364 49 888 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 49 889 8902 8951 8999 9o48 9097 9146 9195 9244 9292 9341 49 890 9390 9439 9488 9536 9585 9634 9683 9731 9780 9829 49 891 *9878 9926 9975 *024 0073 0121 0170 0219 0267 0316 49 892 95 o365 0414 0462 0511 o560 o608 0657 0706 0754 o803 49 893 o85 0900 0949 0997 1046 1095 1143 1192 1240 1289 49 894 i338 I386 I435 1483 1532 i580 1629 1677 1726 1775 49 895 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 48 896 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 48 897 2792 2841 2889 2938 2986 3034 3083 313 3180 3228 48 898 3276 3325 3373 3421 3470 35i8 3566 36i5 3663 3711 48 899 3760 3808 3856 3905 3953 4001 4049 4098 4146 4194 48 900 4243 4291 4339 4387 4435 4484 4532 458o 4628 4677 48 901 4725 4773 4821 4869 4918 4966 50o4 5062 51io 5158 48 902 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 48 903 5688 5736 5784 5832 5880 5928 5976 60o4 6072 6120 48 904 6i68 6216 6265 6313 6361 6409 6457 6505 6553 66o0 48 905 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 48 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 48 907 7607 7655 7703 7751 7799 7847 7894 7792 7990 838 48 908 8086 8i34 818I 8229 8277 8325 8373 8421 8468 85i6 48 909 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 910 9041 9089 9137 9185 9232 9280 9328 9375 9423 9471 48 911 9518 9566 9614 9661 97Q9 9757 9804 9852 9900 9947 48 912 * 9995 +042 0090 0138 oi85 0233 0280 0328 0376 0423 48 913 960471 o058 o566 0613 066i 0709 0756 0804 o851 0899 48 914 0946 0994 1041 1089 1136 1184 123I 1279 1326 1374 47 915 1421 1469 1516 i563 161 I 658 1706 1753 1801 1848 47 916 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 47 917 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795' 47 918 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 47 919 3316 3363 340i 3457 3504 3552 3599 3646 3693 3741 47 920 3788 3835 3882 3929 3977 4024 4071 4118 4I65 4212 47 921 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 47 922 4731 4778 4825 4872 4919 4966 5013 5061 5108 5155 47 923 5202 5249 5296 5343 5390 5437 5484 553i 5578 5625 47 924 5672 5719 5766 5813 5860 5907 5954 6001 6048 6095 47 925 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 47 926 66I 6658 6705 6752 6799 6845 6892 6939 6986 7033 47 927 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 47 928. 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 47 929 80o6 8062 8109 8I56 8203 8249 8296 8343 8390 8436 47 930 8483 853o 8576 8623 8670 8716 8763 88Io 8856 8903 47 93 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 47 932 9416 9463 9509 9556 9602 9649 9693 9742 9789 9835 47 933 * 9882 9928 9975 *02I oo68 0114 0161 0207 0254 o300 47 934 970347 0393 0440 0486 0533 0579 0626 0672 0719 0765 46 935' 0812 o858 0904 0951 0997 1044 1090 o 37 ii83 1229' 46 936 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 46 937 1740 1786 1832 1879 1925 1971 2018 2064 2110 2I57 46 938 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 46 939 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 46 N. 0 1 2 3 4 6 6 I 8 9 D. 16 LOGARITHMS OF NUMBERS. TABLE I. N. 0 1 2 3 4 5 6 8 9 D. 940 97 3128 3174 3220 3266 3313 3359 3405 345i 3497 3543 46 94I 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 46 942 4o5I 4097 4143 4189 4235 4281 4327 4374 4420 4466 46 943 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 46 944 4972 5018 5064 510o. 5156 5202 5248 5294 5340 5386 46 945 5432 5478 5524 5570 566 5662 5707 5753 5799 5845 46 946 5891 5937 5983 6029 6075 612I 6167 6212 6258 6304 46 947 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 46 948 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 949 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 950 7724 7769 7815 7861 7906 7952 7998'8043 8089 8135 46 951 818I 8226 8272 8317 8363 8409 8454 8500 8546 8591 46 952 8637 8683 8728 8774 88 9 8865 8911 8956 9002 9047 46 953 9093 9138 9184 9230 9275 932I 9366 9412 9457 9503 46 954 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 955 980003 0049 0094 0140 oi85 0231 0276 0322 o367 0412 45 956 0458 0503 o549 0594 o640 0685 0730 0776 o821 0867 45 957 0912 0957 Ioo3 1048 I093 1139 1 84 1229 1275 1320 45 958 i366 1411 I456 5oi I1547 I592 1637 i683 1728 1773 45 959 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 45 960 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 45 961 2723 2769 2814 2859 2904 2949 2994 3040 3085 3i30 45 962 3175 3220 3265 33IO 3356 3401 3446 3491 3536 3581 45 963 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 45 964 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 965 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 966 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 45 967 5426 5471 5516 556I 5606 565i 5696 5741 5786 5830 45 968 5875 5920 5965 6010 6055 6100 6144 6I89 6234 6279 45 969 6324 6369 6413 6458 6503 6548 6593 6637 668e 6727 45 970 6772 6817 686i 6o96 6951 6996 7040 7085 71 30 7175 45 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 45 972 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 45 973 81i3 8157 8202 8247 8291 8336 838i 8425 8470 8514 45 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 975 9005 9049 9004 9138 9183 9227 9272 9316 9361 9405 45 976 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 977 *9895 9939 9983 *o28 0072 0117 o016 0206 0250 0294 44 978 99 0339 o83 0428 0472 0o56 o56i o605 o650 o694 0738 44 979 0783 0827 o871 0916 0960 1004 1049 1093 1137 1182 44 980 1226 1270 I315 1359 i403 1448 1492 i536 i580 1625 44 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 44 983 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 987 4317 436x 4405 4449 4493 4537 4581 4625 4669 4713 44 988 4757 4801 4845 4889 4933 4977 5021 5065 5 o8 5152 44 989 5196 5240 5284 5328 5372 54i6 5460 5504 5547 5591 44 990 5635 5679 5723 5767' 5811 5854 5898 5942 5986 6030 44 991 6074 6117 6I61 6205 6249 6293 6337 6380 6424 6468 44 992 6512 6555 6599 6643 6687 6731 6774 68i8 6862 6906 44 993'6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 44 994 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779 44 995. 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 44 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 44 997 8693 8739 8782 8826 8869 8913 8956 90g 9043 9087 44 998 9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 44 999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 N. 0 1 2 3 4 5 6 8 9 D. TAB LTE II. LOGARITHMIC SINES AND TANGENTS, FOR EVERY DEGREE AND MINUTE OF THE QUADRANT. If the logarithms of. the values in Table III. be each increased by io, the results will be the values of this table. The logarithmic Secants and Cosecants are not given. They may be readily obtained, as follows:-Subtract the logarithmic Cosine from 20, and the remainder will be the logarithmic Secant; subtract the logarithmic Sine from 20, and the remainder will be the logarithmic Coseeant. 18 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 00 179~ / Sine. D. Cosine. D. Tang. D. Cotang. 0 Inf. Neg. IO000000 [nf. Neg. Infinite. 6o I 6.463726 5017I7 oooooo oo 6.463726 501717 I3'536274 59 2 764756 293485 oooooo oo 764756 293483 235244 58 3 940847 208231 000000 00 940847 208231 o59i53 57 4 7'o65786 I615i7 000000 00 7.065786 I615I7 I2.934214 56 5 I62696 I31968 000o00 o0 162696 I31969 837304 55 6 241877 111575 9'999999 0I 241878 111578 758122 54 7 308824 96653 999999 oI 308825 99653 69II75 53 & 366816 85254 999999 oi 366817 85254 633183 52 9 417968 76263 999999 01 417970 76263 582030 51 0 463726 68988 999998 o l 463727 68988 536273 50 11 7-505II8 6298I 9.999998 o0 7. 505120 6298I 12.494880 49 12 542906 57936 999997 oI 542909 57933 457091 48 I3 577668 5364I 999997 OI 577672 53642 422328 47 14 53 43 99996 609853 49938 99996 60987 4993 390o43 46 15 639816 46714 999996 oI 639820 46715 36oi80 45 i6 667845 43881 999995 oI 667849 43882 332151 44 17 694173 41372 999995 oI 694179 41373 305821 43 18 718997 39I35 999994 oi 719003 39136 280997 42 19 742478 37127 999993 oi 742484 37I28 257516 41 20 764754 35325 999993 oI 76476I 35I36 235239 40 21 7.785943 33672 9.999992 oI 7.78595I -33673 12.214049 39 22 806146 32175 999991 oI 8o6i55 32176 I93845 38 23 825451 3o8o5 999990 oI 825460 30806 174540 37 24 843934 29547 999989 02 843944 29549 156056 36 25 861662 28388 999989 02 861674 28390 138326 35 26 878695 27317 999988 02 878708 27318 I21292 34 27 895085 26323 999987 02 895099 26325 104901 33 28 90o879 25399 999986 02 9Io894 25401 0o8906 32 29 926119'24538 999985 02 926134 24540 073866 31 30 940842 23733 999983 02 940858 23735 059142 30 31 7.955082 22980 9.999982 02 7-955100.22981 12.044900 2 32 968870 22273 99998I 02 968889 22275 03iiii 28 33 982233 21608 999980 02 982253 216Io oI7747 27 34 995198 20og8 999979 02 995219 20983 004781 26 35 8.007787 20390 999977 02 8o00780 20392 II9921I1 25 36 020021 I9831 999976 02 020044 I9833 9799 24 37 031919 19302 999975 02 03I945 193o5 968055 23 38 o435oI I880o 999973 02 043 27 I88o3 956473 22 39 054781 1I8325 999972 02 054809 18327 945191 21 40 065776 17872 999971 02 065806 17874 934194 20 41 8.076500 17441 9.999969 02 8.07653I 17444 11.923469 19 42 086965 17031 999968 02 086997 17034 913003 8 43 097183 I6639 999966 02 097217 I6642 902783 17 44 107167 I6265 999964 03 107203 6268 892797 i6 45 116926 I5908 999963 03 3 663 15910 883037 15 46 126471 I 5566 999961 o3 126510 15568 873490 14 47 I358io I5238 999959 03 I3585I 15241 864149 13 48 144953 14924 999958 03 144996 14927 855004 12 49 I53907 14622 999956 03 153952 14627 846048 II 50. 162681 I4333 999954 03 162727 I4336 837273 o1 51 8.171280 I4054 9.999952 03 8.171328 14057 11828672 52 1797I3 I3786 999950 03 I79763 I3790 820237 53 187985 13529 999948 03 I88o36 I3532 8iI964 7 54 196102 13280 999946 03 196156 13284 803844 6 55 204070 I304I 999944 03 204I26 i3044 795874 5 56 2zI895 128o1 999942 04 211953 12814 788047 4.57 219581 12587 999940 04 219641 12590 780359 3 58 227134 12372 999938 04 227195 12376 772805 2 59 234557 126I4 999936 04 234621 12168 765379 I 60 241855 11963 999934 04 241921 11967 758079 o Cosine. D. Sine. D. Cotang. D. Tang. 900 890 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 19 10 1780 / Sine. D. Cosine. D. Tang. D. Cotang. / o 8-241855 II963 9.999934 04 8-241921 11967 1 758079 6o0 I 249033 I768 999932 04 249I02 I1772 750898 5 2 256094 / 580 999929 04 256165 I584 743835 58 3 263042 11398 999927 04 263115 11402 736885 57 4 269881 I 1122' 999925 04 269956 11225 730044 56 5 276614 11050 999922 04 276691 11054 723309 55 6 283243 Io883 999920 04 283323 I0887 716677 54 7 289773 10721 999918 04 289856 10726 710I44 53 8 296207 Io565 9999I 5 04 296292 0570 703708 52 9 302546 Io413 999913 04 302634 10418 697366 45 10 308794 10266 999910 04 308884 10270 691116 50 11 8-314954 10122 99999907 04 8.315046 10126 ii~684954 4 12 321027 9982 999905 o4 321122 9987 678878 4 13 327016 9847 999902 04 327II4 9851 672886 47 14 332924 9714 999899 05 333025 9719 666975 46 I5 338753 9586 999897 05 338856 9590 661144 45 16 344504 9460 999894 05 344610 9465 655390 44 17 350181 9338 999891 05 350289 9343 649711 43 I8 355783 9219 999888 05 355895 9224 644105 42 19 361315 1o03 999885 05 361430 Io8 638570 4I 20 366777 8990 999882 o5 366895 8995 633105 40 21 8-372171 8880 9-999879 05 8.372292 8885 1 1627708 39 22 377499 8772 999876 05 377622 8777 622378 38 23 3-82762 8667 999873 05 382889 8672 617111 37 24 387962 8564 999870 05 388092 8570 611908 36 25 39310I 8464 999867 05 393234 8470 606766 35 26 398179 8366 999864 05 398315 8371 6oi685 34 27 403199 827I 99986I 05 403338 8276 596662 33 28 40816 8177 999858 05 408304 8182 591696 32 29 413068 8086 999854 05 413213 809i 586787 31 30 417919 7996 99985I 06 418068 8002 581932 30 31 8-4227I7 7909 9-999848 06 8.422869 7914 11-577131 29 32 427462 7823 999844 06 427618 7830 572382 28 33 432156 7740 99984I 06 432315 7745 567685 27 34 436800 7657 999838 06 436962 7663 563038 26 35 441394 7577 999834 o6 441560 7583 558440 25 36 445941 7499 999831 o6 446Io 7505 553890 24 37 450440 7422 999827 o6 45o613 7428 549387 23 38 454893 7346 999824 06 455070 7352 544930 22 39 4593o0 7273 999820 o6 459481 7279 540519 2 40 463665 7200 9998I6 06 463849 7206 536I5o 20 41 8-467985 7129 9.999813 06 8.468172 7135 11.531828 1 42 472263 7060 999809 06 472454 7066 527546 I 43 476498 6991 999805 06 476693 6998 523307 I7 44 480693 6924 999801 06 480892 6931 5I9108 1 45 484848 6859 999797 07 485030 6865 514950 15 40 488963 6794 999794 07 489170 68o0 5io83o 14 47 493040 673 99979~ 07 493250 6738 506750 3 48 497078 6669 999786 07 497293 6676 502707 12 49 501o80 66o8 999782 07 50o298 66I5 498702 1 50 505045 6548 999778 07 505267 6555 494733 io 51 8-508974 6489 9 999774 07 8509200 6496 1I.490800 9 52 512867 643 999769 07 513098 6439 486902 53 516726 6375 999765 07 51696i 6382 483039 7 54 520551 6319 999761 07 520790 6326 47920I 6 55 524343 6264 999757 07 524586 6272 475414 5 56 528102 621I 999753 07 528349 6218 471651 4 57 531828 6158 999748 07 532080 6165 467920 3 58 535523 6106 999744 07 535779 6113 464221 2 59 539I86 6055 99740 07 539447 6062 460553 I 60 542819 6004 999735 07 543084 6012 4569I6 o Cosine. D. Sine D. Cotang. D. Tang. 910 88~ 20 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 20 117________ __ _____0 S/ ine. D. Cosine. D. Tang. D. Cotang. / o 8-5428I9 6o04 9 999735 07 8.543o84 60o2 II-456916 60 I 546422 5955 999731 07 54669I 5962 453309 59 2 549995. 5906 999726 07 550268 59I4 449732 58 3 553539 5858 999722 08 553817 5866 446 I83 57 4 557054 5811 999717 08 557336 5819 442664 56 5 560540 5765 9997I3 08 560828 5773 439172 55 6 563999 5719 999708 8 564291 5727 435709 54 7 56743I 5674 999704 08 567727 5682 432273 53 8 570836 5630 999699 o8 571137 5638 428863 52 9 574214 5587 9996 4 08 574520 5595 425480 51 io 577566 5544 999689 08 577877 5552 42223 50 ii 8-580892 5502 9.999685 08 858I208 55I0 11418792 49 12 584I93 5460 999680 08 5845I4 5468 45486 4 3 587469 54I9 999675 08 587795 5427 4I2205 47 14 590721 5379 999670 08 591 0I 5387 408949 4 15 593948 5339 999665 o8 594283 5347 405717 45 i6 597152 5300 999660 o8 597492 5308 402508 44 17 600332 5261 999655 o8 600677 5270 399323 43 I8 603489 5223 999650 08 603839 5232 396161 42 19 606623 5186 999645 09 606978 5194 393022 41 20 609734 5149 999640 09 6I0094 5I18 389906 40 21 8-612823 5I12 9.999635 09 8.6I3I89 5121 II-3868ii 39 22 615891 5076 999629 09 616262 5085 383738 38 23 618937 504I 999624 09 619313 5050 380687 37 24 62I962 5006 999619 09 622343 5015 377657 36 25 624965 4972 9996I4 09 625352 4581 374648 35 26 627948 4938 999608 09 628340 4947 371660 34 27 6309 I 4904 999603 09 63i308 4913 368692 33 28 633854 4871 999597 09 634256 4880 36574 32 29 636776 4839 999592 09 637184 4848 362816 31 30 63968 99956 09 6493 4806 64009359907 3 31 8-642563 4775 9.999581 09 8-642982 4784 I -3570I8 29 32 645428 4743 999575 09 645853 4753 354147 28 33 648274 47I2 999570 09 648704 4722 351296 27 34 65I102 4682 999564 09 651537 469I 348463 26 35 653911 4652 999558 io 654352 4661 345648 25 36 656702 4622 999553 0o 657I4 463 34285 24 37 659475 4592 999547 0 659928 4602 340072 23 38 662230 4563 999541 Io 662689 4573 33731 22 39 664968 4535 999535 io 665433 4544 334567 21 40 667689 4506 999529 Io 668160 4526 331840 20 4I 8.670393 4479. 9.999524 Io 8670870 4488 1 329130 1 42 673080 445 999518 Io 673563 446 326437 8 43 675751 4424 999512 10 676239 4434 323761 17 44 678405 4397 999506 10 678900 4417 321100 16 45 680o43 4370 999500 Io 68I 44 4380 38456 x5 46 683665 4344 999493 o1 684172 4354 3I5828 14 47 686272 4318 999487 10 686784 4328 313216 I3 48 688863 4292 999481 10 68938I 4303 310619 12 49 691438 4267 999475 io 691963 4277 308037 II 50 693998 4242 999469.0o 694529 4252 3054 51 8-696543 4217 9.999463 ii 8-69708I 4228 1 3029 9 52 699073 4192 999456 ii 699617 4203 3383 8 53 70I589 4I68 999450 II 702139 4I79 29786I 7 54 704090 4144 999443 1 704646 4155 295354 6 55 706577 4121 999437 II 707I40 4132 292860 5 56 709049 4097 999431 II 7096I8 4I08 290382 4 57 711507 4074 999424 I 712083 4085 287917 3 58 713952 4051 999418 11 714534 4062 285466 2 59 716383 4029 999411 II 716972 4040 283028 I 60 718800 4006 999404 11 719396 4017 280604 o Cosine. D. Sine. D. Cotang. D. Tang. 920 8:8 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 21 30 1760 / Sine. D. Cosine. D. Tang. D. Cotang. I o 8.718800 4006 9999404 II 8.719396 4017 II.280604 60 I 721204 3984 999398 II 721806 3995 278194 59 2 723595 3962 999391 II 724204 3974 275796 58 3 725972 3941 999384 II 726588 3952 273412 57 4 728337 399 999378 I 728959 3930 271041 56 5 730688 3898 99937I 11 731317 3909 268683 55 6 733027 3877 999364 12 733663 3889 266337 54 7 733534 3857 999357 12 735996 3868 264004 53 8 737667 3836 999350 12 738317 3848 261683 52 9 739969 3816 999343 12 740626 3827 259374 51 10 742259 3796 999336 12 742922 3807 257078 50 I 8-744536 3776 9.999329 12 8.745207 3787 11254793 49 12 746802 3756 999322 12 747479 3768 252521 48 13 749055 3737 999315 12 749740 3749 250260 47 14 751297 3717 999308 I2 751989 3729 248011 46 I5 753528 3698 99930I 12 754227 3710 245773 45 i6 755747 3679 999294 12 756453 3692 243547 44 17 757955 366I 999287 12 758668 3673 24I332 43 I8 76015 3642 999279 I2 760872 3655 239128 42 I9 762337 3624 999272 I2 763065 3636 236935 41 20 7645 1 3606 999265 12 765246 3618 234754 40 2I 8.766675 3588 9'999257 I2 8.7674 7 3600 11232583 39 22 768828 3570 999250 I3 769578 3583 230422 38 23 770970 3553 999242 I3 771727 3565 228273 37 24 773101 3535 999235 13 773866 3548 226134 36 25 775223 3518 999227 I3 775995 3531 224005 35 26 777333 350o 999220 i3 778114 3514 221886 34 27 779434 3484 999212 I3 780222 3497 2I9778 33 28 781524 3467 999205 13 782320 3480 217680 32 29 783605 3451 999197 13 784408 3464 215592 31 30 785675 343 I 999189 13 786486 3447 213514 30 31 8.787736 3418 9 999181 13 8. 788554 343I 2 I 1446 29 32 789787 3402 999I74 I3 790613 3414 209387 28 33 791828 3386 999166 I3 792662 3399 207338 27 34 793859 3370 999158 I3 794701 3383 205299 26 35 79588I 3354 999150 3 79673I 3368 203269 25 36 797894 3339 999142 I3 798752 3352 20I248 24 37 799897 3323 999134 13 800763 3337 199237 23 38 801892 3308 999126 I3 802765 3322 197235 22 39 803876 3293 999118 I3 804758 3307 I95242 2I 40 805852 3278 999110 13 806742 3292 193258 20 4I 8-807819 3263 9.999102 i3 8-8087I7 3278 11191283 I9 42 809777 3249 999094 4 80o683 3262 I89317 I8 43 811726 3234 999086 4 812641 3248 187359 17 44 813667 3219 999077 14 8I4589 3233 I85411 16 45 815599 3205 999069 I4 816529 3219 I83471 I5 46 817522 3191 99906I 14 8I846I 3205 I81539 14 47 819436 3177 999053 14 820384 319I i79616 13 48 821343 3163 999044 14 822298 3177 77702 2 49 823240 3I49 999036 I4 824205 3163 175795 I 50 825130 3135 999027 14 826103 3150 I73897 1 5I 8-8270oI 3122 9-999o09 I4 8.827992 3136 12-I72008 52 828884 3108 999010 I4 829874 3123 70126 8 53 830749 3095 999002 14 831748 3Iio 168252 7 54 832607 3082 998993 14 8336I3 3096 I66387 6 55 834456 3069 998984 14 835471 3083 164529 5 56 836297 3056 998976 14 837321 3070 62679 4 57 838i30 3043 998967 15 839163 3057 160837 3 58 839956 3o30 998958 5 840998 3045 I59002 2 59 841774 3017 998950 I5 842825 3032 157I75 I 60 843585 3000 998941 15 844644 3019 i55356 O! Cosine. D. Sine. D. Cotang. D. Tang. _93_____0"____________ ~ 860 22 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 40 175__ / Sine. D. Cosine. |D. Tang. D. Cotang. / o 8.843585 3005 9 99894I 15 8 844644 3019 11.i55356 60 I 845387 2992 998932 I5 846455 3007 I53545 5c 2 847183 2980 998923 15 848260 2995 5174 58 3 848971 2967 998914 i5 850057 2982 4gg943 57 4 850751 2955 998905 15 85I846 2970 I48154 56 5 852525 2943 998896 15 853628 2958 146372 55 6 854291 2931 998887 15 855403 2946 144597 54 7 856049 2919 998878 15 857171 2935 142829 53 8 857801 2907 998869 15 858932 2923 I4I068 52 9 859546 2896 998860 15 860686 2911 139314 51 o0 861283 2884 998851 15 862433 2900 I37567 50 ii 8.863014 2873 9-99884I I5 8.864I73 2888 11.135827 49 12 864738 286I 998832 15 865906 2877 134094 48 13 866455 2850 998823 16 867632 2866 132368 47 14 868165 2839 998813 i6 869351 2854 130649 46 15 869868 2828 998804 i6 871064 2843 128936 45 16 871565 2817 998795 16 872770 2832 127230 44 I7 873255 2806 998785 i6 874469 2821 125531 43 I8 874938 2795 998776 I6 876162 2811 123838 42 19 876615 2786 998766 I6 877849 2800 122151 41 20 878285 2773 998757 i6 879529 2789 120471 40 21 8-879949 2763 9-998747 16 8-88I202 2779 11I18798 3 22 881607 2752 998738 I6 882869 2768 11713 38 23 883258 2742 998728 i6 884530 2758 15470 37 24 884903 2731 998718 16 886185 2747 ii3815 36 25 886542 2721 998708 i6 887833 2737 112167 35 26 888174 2711 998699 I6 889476 2727 110524 34 27 889801 2700 998689 I6 891112 27I7 o18888 33 28 891421 2690 998679 I6 892742 2707 107258 32 29 893035 2680 998669 17 894366 2697 Io5634 31 30 894643 2670 998659 I7 895984 2687 I040o6 30 31 8.896246 2660 9.998649 17 8-897596 2677 11- I 2404 2 32 897842 2651 998639 17 899203 2667 797 28 33 899432 264: 998629 17 900803 2658 099197 27 34 901017 2631 998619 17 902398 2648 097602 26 35 902596 2622 998609 17 903987 2638 og6013 25 36 904169 2612 998599 17 905070 2629 094430 24 37 905736 2603 99858 7 907147 2620 092853 23 38 907297 2593 998578 17 908719 2610 09128 22 39 908853 2584 998568 17 910285 2601 o897i5 2I 40 910404 2575 998558 17 911846 2592 088154 20 41 8-9II949 2566 9.998548 I7 8-913401o 2583 z.o86599 1 42 913488 2556 998537 17 914951 2574 085049 I 43 915022 2547 998527 17 916495 2565 o83505 17 44 916550 2538 998516 18 918034 2556 o81966 16 45 918073 2529 998506 18 919568 2547 080432 I5 46 919591 2520 998495 18 921096 2538 078904 14 47 92II03 2512 998485 I8 922619 2530 07738I 13 48 922610 2503 998474 18 924136 252I 075864 12 49 924112 2494 998464 18 925649 2512 07435I i 5 o 925609 2486 998453 18 927156 2503 072844 Io 5! 8'927I00 2477 9.998442 I8 8-928658 2495 11-0o71342 52 928587 2469 998431 18 930o55 2486 069845 53 930068 2460 998421 18 931647 2478 o68353 7 54 931544 2452 998410 18 933134 2470 o66866 6 55.33015 2443 998399 i8 934616 2461 065384 5 56 934481 2435 998388 18 936093 2453 063907 4 57 935942 2427 998377 I8 937565 2445 062435 3 58. 937398 24I9 998366 I8 939032 2437 060968 2 59 938850 2411 998355 18 940494 2430 o59506 I 6o 940296 2403 998344 I8 94-19 2 242I 058048 / | Cosine. D. Sine. D. Cotang. D. Tang. I 94_ 850 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 23 50 __1740 t Sine. D. Cosine. D. Tan. D. Cotang. o 8-940296 2403 9.998344 19 8.941952 242I I o58o48 60 I 941738 2394 998333 19 943404 243 056596 5 2 943174 2387 998322 19 944852 2405 o55i48 58 3 944606 2379 998311 19 946295 2397 053705 57 4 946034 2371 998300 19 947734 2390 052266 56 5 947456 2363 998289 19 949168 2382 050832 55 6 948874 2355 998277 19 950597 2374 049403 54 7 950287 2348 998266 19 95202I 2366 047979 53 8 951696 2340 998255 19 953441 2360 o46559 52 9 953100 2332 998243 19 954856 235I 045I44 5I o0 954499 2325 998232 19 956267 2344 043733 50 ii 8.955894 2317 9-998220 19 8957674 2337 II-042326 49 I2 957284 2310 998209 19 959075 2329 040925 48 13 958670 2302 998197 19 960473 2323 o3g27 47 I4 960052 2295 998 86 19 96I866 2314 o3 834 46 15 961429 2288 998174 19 963255 2307 036745 45 i6 962801 2280 998163 19 964639 2300 o3536i 44 7 9641 70 2273 998151 i9 966019 2293 033981 43 I8 965534 2266 998139 20 967394 2286 032606 42 I9 966893 2259 998I28 20 968766 2279 031234 41 20 968249 2252 99816 20 970133 2271 029867 40 21 8.969600 2244 9.998104 20 8971496 2265 11-028504 39 22 970947 2238 998092 20 972855 2257 027I45 38 23 972289 223I 998080 20 974209 225I 025791 37 24 973628 2224 998068 20 975560 2244 024440 36 25 974962 2217 998056 20 976906 2237 023094 35 26 976293 2210 998044 20 978248 2230 02I752 34 27 977619 2203 998032 20 979586 2223 0204I4 33 28 978941 2197 998020 20 980921 2217 0I9079 32 29 980259 219o 998008 20 982251 2210 o07749 3I 30 981573 2183 997996 20 983577 2204 o06423 30 3I 8-982883 2177 9.997984 20 8-984899 2197 11015101 129 32 984189 2170 979720 2 986217 2191 3783 28 33 985491 2163 997959 20 987532 2 84 012468 27 34 986789 2157 997947 20 988842 2178 OI0158 26 35 988083 2150 997935 2 I 990149 217 00985I 25 36 989374 2144 997922 21 99145 2165 oo8549 24 37 990660 2138 997910 21 992750 2158 007250 23 38 919i43 213[ 997897 2I 994045 2152 005955 22 39 993222 2I25 997885 21 995337 2146 004663 21 40 994497 2119 997872 21 996624 2I40 003376 20 4I 8.995768 2112 9-997860 21 8-997908 2134 I11002092 9 42 997036 2IO6 997847 21 999I88 2127 0008I2 8 43 998299 299 997835 2I 9.000465 212I 10999535 I7 44 999560 2094 997822 21 001738 2115 998262 6 45 9ooo0816 2087 997809 21 003007 2109 996993 15 46 002069 2082 997797 21 004272 2103 995728 14 47 oo33i8 2076 997784 21 005534 2097 994466 13 48 004563 2070 997771 21 006792 2091 993208 12 49 005805 2064 997758 21 008047 2085 99i953 ii 50 007044 2058.997745 2I 009298 2080 990702 IO 51 9-008278 2052 9.997732 2 I 9oio546 2074 I 989454 52 009510 2046 997719 21 0oI790 2068 988210 8 53 010737 2040 997706 21 O30o31 2062 986969 7 54 011962 2034 997693 22 014268 2056 985732 6 55 o13182 2029 997680 22 015502 205I 984498 5 56 OI4400 2023 997667 22 016732 2045 983268 4 57 015613 2017 997654 22 017959 2040 98204I 3 58 016824 2012 997641 22 019183 2033 980817 2 59 018031 2006 997628 22 020403 2028 979597 I 60 019235 2000 9976I4 22 021620 2023 978380 o Cosine. Sine. D. Cotang. D. Tang. I 950________________ 840"______ 24 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 60 1730 / Sine. D. Cosine. D. Tang. D. Cotang, | o 9.019235 2000 9.997614 22 9.021620 2023 IO-978380 60 I 020435 19I5 99760o 22 022834 20I7 977166 5 2 021632 1989 997588 22 024044 2011 975956 58 3 022825 1984 997574 22 025251 2006 974749 57 4 024016 1978 997561 22 026455 2000 973545 56 5 025203 1973 3 997547 22 027655 i995 972345 55 6 026386 1967 997534 23 028852 1990 971148 54 7 027567 I962 997520 23 030046 1985 969954 53 028744 1957 997507 23 031237 1979 968763 52 9 029918 195I 997493 23 032425 1974 967575 51 10 031089 1947 997480 23 033609 2969 966391 50 I[ 9-032257 1941 9 997466 23 9.034791 1964 I0o965209 4 I2 033421 1936 997452 23 035069 1958 96403o 48 I3 034582 I930 997439 23 037144 1953 962856 47 14 035741 1925 997425 23 038316 I948 961684 46 I5 o36896 1920 997411 23 039485 I943 960515 45 i6 038048 1915 997397 23 o40651 1938 959349 44 I7 039197 1910 997383 23 04181 933 3 958187 43 18 040342 I905 997369 23 042973 1928 957027 42 19 04i485 1899 997355 23 044130 1923 955870 41 20 042625 1894 997341 23 045284 1918 95476 40 21 9.043762 1889 9'997327 24 9.o46434 I913 Io-953566 39 22 044895 1884 997313 24 o47582 1908 9524I8 38 23 046026 1879 997299 24 048727 iO93 95I273 37 24 o47i54 I875 9972.9 24 049869 1898 950o13 36 25 048279 I870 997271 24 05 008 1893 948992 35 26 049400 1865 997257 24 052144 1889 947856 34 27 050519 i86 997242 24 053277 I884 946723 33 2 051635 i855 997228 24 054407 1879 945593 32 29 952749 I850 9972I4 24 055535 1874 944465 31 30 o53859 1845 997199 24 o56659 1870 943341 30 31 9-054966 1841 9.997185 24 9-05778i 1865 10.942219 29 32 056071 I836 997170 24 o58900 1869 941100 28 33 o57172 i83i 997156 24 0600I6 I855 939984 27 34 o58271 I827 997141 24 o6ii3o I85I 938870 26 35 059367 1822 997127 24 062240 1846 937760 25 36 060460 1817 997112 24 063348 1842 936652 24 37 06i551 I813 997098 24 064453 i837 935547 23 38 062639 I8o8 997083 25 065556 i833 934444 22 39 o63724 1804 997068 25 o66655 1828 933345 21 40 o64806 1799 997053 25 o67752 1824 932248 20 4I 9-065885 1794 9.997039 25 9.068846 1819 Io0931154 I 42 066962 I790 997024 25 069938 1813 930062 I8 43 o68036 1786 997009 25 071027 I8Io 92873 17 44 069107 I78I 996994 25 072 I3 1806 927887 45 070176 I777 996979 25 073I97 1802 926803 15 46 071242 I772 996964 25 074278 I797 925722 14 47 072306 1768 996949 25 075356 1793 924644 13 4 073366 1763 996934 25 076432 1789 923568 I2 49 o74424 1759 996919 25 077505 I784 922495 ii 50 075480 1753 996904 25 078576 1780 92I424 IO 51 9-076533 i750 9-996889 25 9-079644 1776 IO0920356 52 077583 1746 996874 25 080710 1772 919290 8 53 07863I I742 996858 25 08I773 1767 918227 7 54 079676 I738 996843 25 082833 1763 917I67 6 55 080719 1733 996828 25 083891 1759 91 109 5 56 081759 1729 9968I2 26 084947 1755 915053 4 57 082797 1725 996797 26 086000 I751 914000 3 58 083832 172I 996782 26 087050 1747 9I295 2 59 084864 1717 996766 26 088098 1743 911902 I 60 085894 1713 996751 26 089,44 1738 910856 o - Cosine. D. Sine. D. Cotang. D. Tang. 1 r — ioi —------------- TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 25 170 1 72~ _ Sine. D. Cosine. D. Tang. D. Cotang. o 9.085894 17I3 9.996751 26 9-08g144 1738 0ogI90856 60 J 086922 I709 996735 26 090187 1734 909813 59 2 087947 I704 996720 26 091228 1730.908772 58 3 088970 1700 996704 26 092266 1727 907734 57 4 089990 1696 996688 26 093302 1722 906698 56 5 091008 1692 996673 26 094336 1719 905664 55 6 092024 1688 996657 26 095367 I7I2 904633 54 7 093037 1684 996641 26 096395 1711 903605 53 8 094047 I680 996625 26 097422 I707 902578 52 9 095056 1676 996610 26 098446 1703 901554 51 10 096062 1673 996594 26 099468 1699 900532 50 II 9-097065 1668 9.996578 27 9-100487 I695 io 899513 49 12 098066 i665 996562 27 101504 1691 898496 48 13 099065 I661 996546 27 102519 1687 897481 47 14 100062 1657 996530 27 103532 1684 896468 46 I5 101056 I653 996514 27 104542 i68o 895458 45 16 102048 1649 996498 27 o15550 1676 894450 44 17 103037 i645 996482 27 106556 1672 893444 43 I8 104025 I64I 996465 27 I07559 1669 892441 t42 19 i050o10 638 996449 27 o08560 i665 891440 41 20 105992 I634 996433 27 o09559 I66 890441 40 21 9-106973 I630 9.996417 27 9.110556 I658 Io.88944 39 22 I0795I 1627 996400. 27 111551 1654 888449 38 23 108927 1623 996384 27 112543 1650 887457 37 24 109901 16I9 996368 27 ii3533 1646 886467 36 25 110873 I6I6 99635I 27 114521 I643 885479 35 26 111842 1612 996335 27 115507 1639 884493 34 27 112809 i6o8 996318 27 116491 i636 883509 33 28 113774 i605 996302 28 I17472 1632 882528 32 29 114737 I60o 996285 28 II 8452 1629 881548 31 30 II5698 i597 996269 28 119429 I625 880571 30 31 9-II6656 1594 9-996252 28 9.120404 1622 IO0879596 2 32 117613 1590 996235 28 121377 I618 878623 28 33 118567 1587 996219 28 122348 16I5 877652 27 34 19519 583 996202 28 123317 1611 876683 26 35' 120469 i58o 996185 28 124284 1607 875716 25 36 121417 1576 996168 28 125249 I604 874751 24 37 122362 I573 996i5I 28 126211 1601 873789 23 38 123306 1569 996134 28 127172 1597 872828 22 39 124248 1566 996117 28 128130 1594 871870 21 40 125187 1562 996100 28 129087 1591 870913 20 41 9-126125 i 559 9996083 29 9.130041 1587 0o.869959 9 42 127060 1556 996066 29 130994 1584 869006 I 43 127993 552 996049 29 131944 58 868056 17 44 128925 1549 996032 29 132893 I577 867107 16 45 129854 I545 996015 29 133839 1574 866161 15 46 130781 1542 995998 29 134784 1571 865216 14 47 I31706 I539 9998 29 135726 1567 864274 13 48 132630 1535 995963 29 136667 i564 863333 12 49 133551 1532 995946 29 137605 1561 862395 ii 5o 134470 1529 995928 29 I38542 i558 861458 io 51 9-135387 1525 9.995911 29 9.139476 1555 10o860524 52 i36303 1522 995894 29 140409 I55I 85959I 53 137216 15&9 995876 29 141340 I548 858660 7 54 138128 I 56 995859 29 142269 1545 857731 6 55 139037 51I2. 995841 29 143196 1542 856804 5 56 139944 i509 995823 29 144121 1539 855879 4 57 140850 1506 995806 29 i45044 535 854956 3 58 141754 1503 995788 29 145966 1532 854034 2 59 142655 5oo0 995771 29 146885 1529 853115 I 60 143555 1496 995753 29 147803 1526 852197 0 / Cosine. D. SineD.D. Cotang. D. Tang. / J __ _._.. _.970 _ 821 so VV1uVV * 9 9 / 0 Sine. D. Cosine. D. Tang. D. Cotang. I o 9'143555 1496 9-995753 30 9' 147803 1526 o1852107 60 I 144453 1493 995735 30 148718 1523 85I282 5 2 145349 1,90 995717 3o 149632 1520 850368 58 3 146243 1487 995699 30 50544 1517 849-456 57 4 147136 1484 995681 30 I51454 I5I4 848546 56 5 148026 1481 995664 30 152363 5I i 847637 55 6 148915 1478 995646 30 I53269. I 50 846731 54 I49802 1475 995628 30 154174 I505 845826 53 50686 1472 995610 30 155077 1502 844923 52 9 I51569 I469 995591 30 I 5978 I499 844022 51 o1 152451 1466 995573 30 156877 I496 843123 50 II 9- 53330 I463 9.995555 30 9.157775 I493 10842225 49 I 2 54208 I460 995537 30 158671 1490 84I329 48 I3 I55083 1457 9955I9 30 I59565 487 84435 4 14 155957 1454 995501 31 160457 1484 839543 46 I5 i56830 I45I 995482 3I 161347 I481 838653 45 i6 I57700 I448 995464 3i. 62236 1479 837764 44 17 158569 i445 995446 3i 163123 1476 836877 43 I8 I59435 I442 995427 3I 164008 1473 835992 42 19 i6030oI 439 995409 3I 164892 1470 8350o8 41 20 161164 1436 995390 31 165774 1467 834226 40 21 9-162025 I433 9'995372 3I 9-166654 1464 io.833346 3 22 I62885 i430 995353 31 167532 i46I 832468 38 23 163743 1427 995334 3I I68409 i458 83159I 37 24 164600 1424 995316 31 I69284 I455 8307I6 36 25 I65454 1422 995297 3i 170157 1453 829843 35 26 166307 1419 995278 31 171029 1450 82 97I 34 27 I67159 I416 995260 3I I71899 I447 828IOI 33 28 i68008 1413 99524I 32 172767 1444 827233 32 29 i68856 I410 995222 32 173634 1442 826366- 31 30 169702 1407 995203 32 I74499 I439 825501 30 31 9-I70547 I4o5 9.995I84 32 9. 75362 I436 10-824638 29 32 171389 1402 995165 32 176224 I433 823776 28 33 172230 1399 995I46 32 177084 1431 822916 27 34 173070 1396 995127 32 I7794 1428 822058 26 35 173908 1394 995o18 32 178799 1425 82I20I 25 36 174744 139 995089 32 179655 I423 820345 24 37 175578 1388 995070 32 180508 1420 819492 23 38 176411 I386 995051 32 I81360 I417 818640 22 39 177242 1383 995032 32 182211 I145 817789 21 40 178072 1380 995013 32 183059 1412 8I694I 20 4I 9-178900 1377 9.994993 32 9-I83907 1409 108I6093 19 42 179726 1374 994974 32 184752 1407 815248 I8 43 180551 1372 994955 32 i85597 I404 814403 17 44 18I374 I369 994935 32 186439 I402 8I356i i6 45 182196 I366 9949 6 33 187280 399 812720 15 46 i830I6 1364 994896 33 188120 1396 8i88o0 14 47 183834 136I 994877 33 I88958 1393 811i42 13 48 18465I I359 994857 33 189794 I39I 8I0206 12 49 185466 i356 994838 33 190629 1389 809371 II 50 186280 i353 994818 33 191462 i386 808538 io 5I 9-I87092 135I 9.994798 33 9-I92294 I384 10o807706 9 52 187903 1348 994779 33 I93124 I38I 806876 53 I887I2 I346 994759 33 I93953 1379 806047 7 54 189519 I343 994739 33 194780 1376 805220 6 55 190325 1341 994720 33 i95606 1374 804394 5 56 191130 I338 994700 33 I96430 1371 803570 4 57 I9I933 I336 994680 33 197253 I369 802747 3 58 192734 I333 994660 33 198074 1366 801926 2 59 i93534 1330 994640 33 198894 I364 801106 I 60 194332 1328 994620 33 I99713 i36i 800287 0 I Cosine. D. Sine. D. Cotang. D. Tang. I 980 810 TABLE II.. LOGARITHMIC SINES, TANGENTS, ETC.' 90 _170! Sine. D. Cosine. D. Tang. D. Cotang. _ o 9.I94332 1328 9.994620 33 9- I997I3 I361 10800287 60 I 195129 1326 994600 33 200529 1359 79947I 5 2 195925 1323 994580 33 201345 1356 798655 5 3 196719 1321 994560 34 202159 I354 79784I 57 4 197511 1318 994540 34 202971 1352 797029 56 5 198302 i316 994519 34 203782 1349 796218 55 6 199091 1313 994499 34 204592 1347 795408 54 7 199879 311 994479 34 205400 1345 794600 53 8 200666 I308 994459 34 206207 1342 793793 52 9 201451 i306 994438 34 207013 i340 792987 51 Io 202234 I304 994418 34 207817 i338 792183 50 II 9-203017 I3oi 9.994398 34 9208619 1335 I0791381 4 12 203797 1299 994377 34 209420 1333 790580 48 I3 204577 1296 994357 34 2I0220 I331 789780 47 I4 205354 1294 994336 34 2I018 I328 788982 46 i5 20613I 1292 994316 34 211815 1326 788185 45 i6 206906 1289 994295 34 212611 1324 787389 44 17.207679 1287 994274 35 213405 1321 786595 43 18 208452 I285 994254 35 214I98 13I9 785802 42 19 209222 1282 994233 35 2I4989 1317 785011 41 20 209992 1280 994212 35 215780 1315 784220 40 21 9-210760 1278 9.994191 35 9-216568 1312 I0-783432 39 22 211526 1275 994171 35 217356 1310 782644 38 23'212291 1273 994150 35 218142 1308 781858 37 24 2I3055 I271 g99412 35 218926 1305 781074 36 25 213818 1268 99410I 35 2197I0 i303 780290 35 26 214579 1266 994087 35 220492 I30I 779508 34 27 2I533 1264 994066 35 221272 1299 778728 33 2 26097 126I 994045 35 222052 1297 777948 32 29 216854 1259 994024 35 222830 1294 777170 31 30 217609 1257 994003 35 223607 1292 776393 30 31 9-218363 1255 9.993982 35 9-224382 I290 10 775618 2 32 219116 1253 993960 35 225156 1288 774844 2 33 219868 1250 993939 35 225929 1286 774071 27 34 2206i8 1248 993918 35 226700 1284 773300 26 35 221367 I246 993897 36 227471 1281 772529 25 36 222115 1244 993875 36 228239 1279 771761 24 37 222861 1242 993854 36 229007 1277 770993 23 38 223606 1239 993832 36 229773 1275 770227 22 39 224349 1237 993811 36 230539 1273 769461 21 40 225092 1235 993789 36 231302 1271 768698 20 41 9-225833 1233 9.993768 36 9-232065 I269 10.767935 Ig 42 226573 1231 993746 36 232826 1267 767174 I 43 2273II 1228 993725 36 233586 I265 766414 17 44 228048 1226 993703 36 234345 1262 765655 16 45 228784 1224 993681 36 235103 1260 764897 I5 46 229518 1222 993660 36 235859 1258 764141 14 47 230252 1220 993638 36 236614 1256 763386 13 48 230984 1218 993616 36 237368 I254 762632 12 49 231715 1216 993594 37 238120 1252 761880 ii 50 232444 1214 993572 37 238872 1250 761128 Io 5I 9.233172 1212 9.993550 37 9.239622 1248 10i760378 52 233899 1209 993528 37 240371 1246 759629 53 234625 1207 993506 37 241118 1244 758882 7 54 235349 1205 993484 37 241865 1242 758135 6 55 236073 1203 993462 37 242610 1240 757390 5 56 236795 1201 993440 37 243354 1238 756646 4 57 237515 1199 993418 37 244097 1236 755903 3 58 238235 1197 993396 37.244839 1234 755161 2 59 238953 IIg9 993374 37 245579 1232 754421 I 60 239670 1193 993351 37 246319 1230 753681 0 i Cosine. - D.- Sine. D. Cotang. D. Tang. I a99~ __ __ 80~ 28'LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 10o _______1690 / Sine.. Cosine. D. Tang. D. Cotang. o 9'239670 1193 9.993351 37 9.2463 19 1230 10753681 60 I 240386 1191 993329 37 247057 1228 752943 59 2 241101 I189 993307 37 247794 I226 752206 58 3 241814 1187 993284 37 248530 1224 5JI470 57 4 242526 II85 993262 37 249264 1222 J5o736 56 5 243237 ii83 993240 37 249998 1220 750002 55 6 243947 II8I 993217 38 250730 1218 749270 54 7 244656 11 79 99395 38 25I46I I217 748539 53 8 245363 1177 993172 38 25219I 1215 747809 52 9 246069 1175 993149 38 252920 1213 747080 51 io 246775 1173 993127 38 253648 1211 746352 50 II 9.247478 1171 9.993ro 4 38 9.254374 1209 10745626 49 1 24818I II69 993081 - 38 255100 1207 744900 48 3 248883 1167 993059 38 255824 1205 744176 47 14 249583 1165 993036 38 256547 1203 743453 46 15 250282 1163 993013 38 257269 1201 742731 45 I6 250980 116I 992990 38 257990 1200 742010 44 I7 251677 1159 992967 38 258710 II98 741290 43 8 252373 1158 992944 38 259429 I96 740571 42 19 253067.I56 992921 38 260146 1194 739854 41 20 25376I 1154 992898 38 260863 1192 739137 40 21 9-254453 1152 9.992875 38 9-261578 1190 IO738422 39 22 255144 1i50 992852 38 262292 1189 737708 38 23 255834 1148 992829 39 263005 I187 736995 37 24 256523 1146 992806 39 263717 I 18 736283 36 25 257211 1144 992783 39 264428 1183 735572 35 26 257898 1142 992759 39 265138 1181 734862 34 27 258583 114I 992736 39 265847 1I79 734153 33 28 259268 1139 992713 39 266555 1178 733445 32 29 259951 I137 992690 39 267261 1176 732739 31 30 260633 1135 992666 39 267967 I174 732033 30 31 9-261314 ii33 9-992643 39 9.268671 1172 10-731329 2 32 261994 I131 9926I9 39 269375 II 730625 2 3 26 673 1130 99259 39 270077 119 729923 27 34 263351 1128 992072 39 270779 1167 729221 26 35 264027 1126 992549 39 271479 165 728521 25 36 264703 1124 992525 39 272178 1164 727822 24 37 265377 I122 992501 39 272876 1162 727124 23 38 266051 1120 992478 40 273573 1160 726427 22 39 266723 1119 992454 40 274269 Ii58 725731 21 40 267395 1117 992430 40 274964 1157 725036 20 41 9g268065 1115 9.992406 40 9.275658 1155 10o724342 I9 42 268734 1113 992382 40 276351 153 723649 i8 43 269402 1111 992359 40 277043 115I 722957 17 44 270069 1110 992335 40 277734 II50 722266 16 45 270735 1108 992311 40 278424 II48 721576 15 46 271400 1106 992287 40 279113 1147 720887 14 47 272064 1105 992263 40 279801 145 720I99 13 48 272726 1103 992239 40 280488 1143 719512 I2 49 273388 11io 992214 40 281174 14I 718826 II 50 274049 I099 992190 40 281858 1140 718142 10.5 9.274708 1098 9-992166 40 9.282542 138 10717458 9 52 275367 1096 992142 40 283225 1136 716775 53 276025 I094 992118 41 283907 II35 716093 7 54 27668I 1092 992093 41 284588 1133 715412 55 277337 I091 992069 41 285268 1131 714732 5 56 277991 I89 992044 41 285947 1130 714053 4 7 278645 I 087 992020 41 28624 I128 713376 3 58 279297 1086 99I996 41 287301 1126 712699 2 59 279948 1084 991971 41 287977 1125 712023 I 60 280599 1082 991947 4I 288652 1123 711348 O / Cosine. D. Sine. D. C otang. I D. Tang.' 100"lo ~'90 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 29 110 __________ 1680 Sine. D. Cosine. D. Tang. D. Cotang. 6 o 9.280599 1082 999I947 4I 9-288652 1123 10-711348 60 I 281248 1081 991.922 41 289326 122 710674 59 2 281897 1079 991897 41 289999 1120 710001 58 3 282544 I077 991873 41 290671 1118 709329 57 4 283190 1076 991848,41 291342 117 708658 56 5 283836 1074 99823 41 292013 1115 707987 55 6 284480 1072 99I799 41 292682 1114 707318 54 7 285124 1071 991774 42 293350 1112 706650 53 8 285766 Io6g 991749 42 294017 111 705983 52 9 286408 1067 991724 42 294684 109 705316 51 o1 287048 Io66 991699 42 295349 1107 70465I 50 I I 9*287688 1064 9-99I674 42 9-296013 II16 10-703987 4 12 288326 o163 991649 42 296677 1104 703323 48 13 288964 o06I 991624 42 297339 io03 702661 47 14 289600 1059 991599 42 298001 111 70I999 46 15 290236 058 99I574 42 298662 I100 701338 45 i6 290870 io56 991549 42 299322 1098 700678 44 17'291504 054 99I 524 42 299980 196 700020 43 i8 292137 io53 991498 42 300638 1095 699362 42 19 292768 1051 991473 42 301295 1093. 698705 41 20 293399 1o50 991448 42 30I95 1 092 698049 40 21 9-294029 I48 9.991422 42 9.302607 i90 o1697393 3a 22 294658 o1046 991397 42 30326i 1089 696739 3 23 295286 1045 991372 43 303914 1087 696086 37 24 295913 o043 991346 43 304567 1086 695433 36 25 296539 1042 991321 43 305218 1084 694782 35 26 297164 1040 991295 43 305869 Io83 694131 34 27 297788 1039 991270 43 3065i 9 Io8 69348I 33 28 298412 I037 991244 43 307168 i080 692832 32 29 299034 1036 99I218 43 3078I6 1078 692184 31 30 299655 io34 991193 43 308463 1077 691537 30 13 2 93002991167 43 9.309109 IO75 o1069g891 29 32 300895 i03i 991141 43 309754 1074 690246 28 33 30514 1029 991115 43 310399 1073 689601 27 34 302I32 1028 991090 43 311042 I07I 688958 26 35 302748 1026 991064 43 311685 1070 6883I5 25 36 303364 1025 991038 43 3 312327 io68 687673 24 37 303979 1023 991 12 43 312968 o067 687032 23 38 304593 I022 990986 43 336o8 Io65 686392 22 39 305207 1020 990960 43 314247 o164 685753 21 40 305819 I019 990934 44 314885 1062 6851I5 20 41 9.306430 IOI7 9.990908 44 9-3I5523 Io6I Io0684477 1 42 307041 ioi6 990882 44 316159 Io60 683841 43 307650 1014 990855 44 316795 o58 683205 17 44 308259 ioi3 990829 44 317430 1057 682570 i 45 308867 Io I 990803 44 318064 I o5 681936 15 46 309474 Ioo1 990777 44 318697 o054 681303 14 47 31oo080 008 990750 44 319330 io53 680670 13 48 3io685 1007 990724 44 4 319961 I05 680039 12 49 311289 io15 990697 44 320592 Io50 679408 II 50 311893 1004 990671 44 321222 1048 678778 IO 5i 9.312.495 oo003 9990645' 44 9-32I851 I047 10-678I49 9 52 313097 100oo 990618 44 322 479 1045 677521 8 53 313698 Iooo 990591 44 323Io6 1044 676894 7 54 314297 998 990565 44 323733 1043 676267 6 55 314897 997 990538 44 324358 1041 675642 5 56 3I5495 996 990511 45 324983 1040 6750I7 4 57 3 6092 994 990485 45 325607 I039 674393 3 58 316689 993 990458 45 326231 Io37 673769 2 59 317284 99 990431 45 326853 o36 673147 I 60 317879 990 990404 45 327475 1035 672525 o / C.i. D. Sin. D. Cot3a. D. 6Tan. 63 / osine. D. Sine. D. Cotang. D. Tang. / 7 10 S0. 80 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 120 _167 f Sine. D. Cosine. D. Tang. D. Cotang. 0 9-317879 990 9.990404 45 9.327475 Io35 10-672525 60 I 3I8473 988 990378 45 328095 io33 671905 59 2 319066 987 990351 45 328715 1032 671285 58 3 319658 986 990324 45 329334 o030 670666 57 4 320249 984 990297 45 329953 1029 670047 56 5 320840 983 990270 45 330570 1028 669430 55 6 321430 982 990243 45 331187 1026 668813 54 322019 980 990215 45 331803 1025 668I97 53 322607 979 990188 45 332418 1024 667582 52 9 323194 977 990161 45 333033 1023 666967 5i Io 323780 976 990134 45 333646 102I1 666354 50 I 9.324366 975 9-990107 46 9.334259 I020 1066574I 49 12 324950 973.990079 46 334871 1019 665129 48 I3 325534 972 990052 46 335482 10I7 66458 47 14 326117 970 990025 46 336093 io16 663907 46 15 326700 969 989997 46 336702 I015 663298 45 16 327281 968 989970 46 337311 o113 662689 44 17 327862 966 989942 46 337919 1012 662081 43 I8 328442 965 9899 5 46 338227 111 661473 42 19 329021 964 989887 46 339133 1010 660867 41 20 329599 962 989860 46 339739 ioo8 660261 40 2I 9.330176 961 9.989832 46 9-340344 1007 10 659656 39 22 330753 960 989804 46 340948 o006 659052 38 23 331329 958 989777 46 34I552 I004 658448 37 24 331903 957 989749 47 342155 oo003 657845 36 25 332478 956 989721 47 342757 1002 657243 35 26 33305I 954 989693 47 343358 1000 656642 34 27 333624 953 989665 47 343958 999 656042 33 28 334I95 952 989637 47 344558 998 655442 32 29 334767 950 989610 47 345157 997 654843 31 30 335337 949 989582 47 345755 996 654245 30 31 98335906 948 9.989553 47 9.346353 994 10o653647 29 32 336475 946 989525 47 346949 993 65305I 28 33 337043 945 989497 47 347545 992 652455 27 34 337610 944 989469 47 348141 991 65i859 26 35 338176 943 989441 47 348735 990 6526 25 36 338742 94' 989413 ~47 349329 988 650671 24 37 339307 940 989385 47 349922 987 650078 23 38 339871 939'989356 47 350514 986 649486 22 39 340434 937 989328 47 351106 985 648894 21 40 340996 936. 989300 47 351697 983 648303 20 41 9.34I558 935 9 989271 47 9.352287 982 0 64773 19 42 342119 934 989243 47 352876 981 647124 8 43 342679 932 989214 47 353465 980 646535 17 44 343239 931 989186 47 354053 979 645947 16 45 343797 930 989157 47 354640 977 645360 15 46 344355 929 989128 48 355227 976 644773 14 47 344912 927 989100 48 3558i3 975 644187 13 48 345469 926 989071 48 356398 974 643602 12 49 346024 925 989042 48 356982 973 6430I8 ii 50 346579 924 989014 48 357566 971 642434 io 5I 9.347134 922 9-988985 48 9 358I49 970 Io-64I851 52 347687 921 988956 48 358731 969 641269 8 53 348240 920 988927 48 359313 968 640687 7 54 348792 9I9 988898 48 359893 967 640107 6 55 349343 917 988869 48 360474 966 63526 5 56 349893 916 988840 48 36o053 965 638947 4 57 350443 915 988811 49 361632 963 638368 3 58 350992 914 988782 49 362210 962 637790 2 59 35 140 9I3 988753 49 362787 961 637213 I 60 352088 91 988724 49 363364 960 636636 o t Cosine. D, Sine. D. Cotang. D. Tang. / 1020 717 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 31 130 1660 Sine. D. Cosine. D. Tan g. D. Cotang. I 0 9.352088 g 9.98872 44 4 9.363364 960 Io636636 6o I 352635 910 988695 49 363940 959 636060 59 2 353I8I 909 988666 49 364515 058 635485 58 3 353726 908 988636 49 36500o 957 6349o0 57 4 354271 907 988607 49 365664 955 634336 56 5 3548i5 905 988578 49 366237 954 633763 55 6 355358 904 988548 49 3668io 953 633190 54 7 355901 903 988519 49 367382 952 6326I8 53 8 356443 902 988489 49 367953 951 632047 52 9 356984 901 988460 49 368524 950 631476 5 10 357024 899 988430 49 369094 949 630906 50 II 9.358064 898 9'98840I 49 9.369663 948 Io 630337 4 12 3586o3 897 988371 49 370232 946 629768 4 13 359141 896 988342 49 370799 945 629201 47 14 359678 895 988312 50 371367 944 628633 46 15 360215 893 988282 50 37I933 943 628067 45 i6 360752 892 988252 50 372499 942 627501 44 17 36I287 89I 988223 50 373064 941 626936 43 i8 361822 890 988193 50 373629 940 626371 42 I9 362356 889 988I63 50 374I93 939 625807 41' 20 362889 88 988I33 50 374756 938 625244 40 21 9-363422 887 9.988I03 50 9-375319 937 10 62468i 39 22 363954 885 988073 50 375881 935 624119 38 23 364485 884 988043 50 376442 934 623558 37 24 365o06 883 988013 50 377003 933 622997 36 25 365546 882 987983 50 377563 932 622437 35 26 366075 88I 987953 50 378122 931 62I878 34 27 366604 880 987922 50 378681 930 621319 33 28 36713I 879 987892 50 379239 929 62076I 32 29 367659 877 987862 50 379797 928 620203 31 3 368I8 876 987832 5i 380354 927 619646 30 31 9.3687 I 875 9.987801 5I 9-380910 926 10 619090 29 32 369236 874 987771 51 381466 925 618534 28 33 369761 873 987740. 5 382020 924 617980 27 34 370285 872 987710 5I 382575 923 617425 26 35 370808 87I 987679 5i 383129 922 616871 25 36 371330 870 987649 51 383682 921 6I6318 24 37 371852 869 987618 51 384234 920 6I5766 23 38 372373 867 987588 51 384786 919 615214 22 39 372894 866 987557 5I 385337 918 614663 21 40 373414 865 987526 51 385888 9I7 614112 20 41 9-373933 864 9.987496 51 9-386438 915 10-613562 19 42 374452 863 987465 51 386987 914 6i30o3 18 43 374970 862 987434 51 387536 913 6I2464 17 44 375487 86i 987403 52 388084 9I2 611916 16 45 376003 860 987372 52 388631 911 611369 15.46 37659 859 98734I 52 389I78 910 610822 14 47 377035 85 987310 52 389724 90 6I0276 13 48 377549 857 987279 52 390270 90 609730 12 49 378063 856 987248 52 390815 907 609185 ii 50 378577 854 9872I7 52 391360 906 608640 10 5I 9-379089 853 9-987i86 52 9.391903 90o5 Io608097 9 52 379601 852 987I55 52 392447 904 607553 53 380113 851 987124 52 392989 903 607011 7 54 380624 850 987092 52 393531 902 606469 6 55 38II34 849 98706I 52 394073 90o 605927 5 56 38i643 848 987030 52 394614 900 605386 4 57 382152 847 986998 52 395I54 899 604846 3 58 382661 846 986967 52 395694 898 604306 2 59 383168 845 986936 52 396233 897 603767 I 60 383675 844 986904 52 396771 896 603229 0 t Cosine. D. Sine. D1. Cotang. D. Tang. I 10 _~___________________________'1 32 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 1.40 1650 t Sine. D. Cosine. D. Tang, D. Cotang. 0 9.383675 844 9.986904 52 9.39677I 896 10-603229 6c I 384182 843 986873 53 397309 896 60269 5 2 384687 842 986841 53 397846 895 60214 58 3 385192 84 986809 53 398383 894 601617 57 4 385697 840 986778 53 398919 893 60oo8I 56 5 386201 839 986746 53 399455 892 600545 55 6 386704 838 986714 53 399990 89 6000oo 54 7,387207 837 986683 53 400524 890 599476 53 8 387709 836 986651 53 401058 889 598942 52 9 3882 1 835 986619 53 401591 888 598409 5I Io 388711 834 986587 53 40224 887 597876 50 II 9.389211 833 9.986555 53 9.402656 886 10.597344 40 12 38971I 832 986523 53 403187 885 59683 48 13 3902IO 83i 986491 53 403718 884 596282 47 I4 390708 830 986459 53 404249 883 595751 46 I5 391206 828 986427 53 404778 882 595222 45 16 39I703 827 986395 53 405308 88I 594692 44 I7 392199 826 986363 54 405836 880 594164 43 i8 392695 825 986331 54 4o6364 879 593636 42 I9 393IOI 824 986299 54 406892 878 593I08 4I 20 393685 823 986266 54 4074I9 877 59258I 40 2 9-394179 822 9986234 54 9 407945 876 I 0592055 3 22 394673 82I 986202 54 408471 875 591529 3 23 395I66 820 986169 54 408996 874 591004 37 21 395658 89 986I37 54 409521 874 590479 36 25 396150 818 986104 54 410045 873 589955 35 26 396641 817 986072 54 410569 872 589431 34 27 397132 817 986039 54 411092 87I 588908 33 28 307621 816 986007 54 4 1615 870 588385 32 29 88III 815 985974 54 412137 869 587863 31 30 398600 8I4 985942 54 412658 868 587342 30 31 9.399088 803 9985909 55 9.413179 867 o10586821 2 32 399575 812 98587 55 413699 866 5863o0 2i 33 400062 81 985843 55 414219 865 585781 27 34 400549 810 985811 55 414738 864 585262 26 35 4oio35 809 985778 55 415257 864 584743 25 36 401520 808 985745 55 415775 863 584225 24 37 402005 807 985712 55 4I6293 862 583707 23 38 402489 806 985679 55 4i681o 86i 583190 22 39 402972 805 985646 55 417326 860 582674 21 40 403455 804 985613 55 417842 859 582158 20 41 9-403938 803 9.985580 55 9.418358 858 Io058I642 J 42 404420 802 985547 55 418873' 857 581127 18 43 404901 8o0 985514 55 419387 856 5806I3 17 44 405382 800 985480 55 41990I 855 580000 I6 45 405862 799 985447 55 4204I5 855 579585 i5 46 40634i 71 ) 985414 56 420927 854 579073 I4 47 406820 797 98538I 56 421440 853 578560 13 48 407299 796 985347 56 421952 852 578048 12 49 407777 795 985314 56 422463 85i 577537 II 50 408254 794 985280 56 422974 850 577026 so 51 9.408731 794 9.985247 56 9-423484 849 0ro576516 52 409207 793 98521I3 56 423993 848 576007 53 409682 792 985180 56 424503 848 575497 7 54 410157 791 985I46 56 425011 847 574989 6 55 410632 790 985113 56 425519 846 57448I 5 56 4Io106 789 985079 56 426027 845 573973 4 57 410579 788 985045 56 426534 844 573466 3 58 4I2052 787 98501I 56 42704I 843 572959 2 59 4I2524 786 984978 56 427547 843 572453 I 60 4I2996 785 984944 56 428052 842 571948 0 Cosine. D. Sine. D. Cotang. D. Tang. 1040 ________ * _ TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 33 150 1640 Sine. D. Cosine. D. Tang. D. Cotang. 0 9.412996 785 9.984944 57 9 -428052 842 10-51948 60 I 413467 784 984910 57 428558 841 571442 59 2 413938 78.3 984876 57 429062 840 570938 58 3 414408 783 984842 57 429566 839 570434 57 4 414878 782 984808 57 430070 838 569930 56 5 415347 781 984.774 57 430573 838 569427 55 6 415815 780 984740 57 431075 837 568925 54 7 416283 779 984706 57 431577 836 568423 53 8 416751 778 984672 57 432079 835 567921 52 9 417217 777 984638 57 432580 834 567420 5I 10 417684 776 984603 57 433080 833 566920 50 11 9.418150 775 9.984569 57 9 433580 832 10.566420 49 12 418615 774 984535 57 434080 832 565920 4 I3 419079 773 984500 57 434579 831 565421 47 14 4I9544 773 984466 57 435078 830 564932 46 i5 420007 772 984432 58 435576 829 564424 45 i6 420470 77I 984397 58 436073 828 563927 44 17 420933 770 984363 58 436570 828 563430 43 18 421395 769 984328 58 437067 827 562933 42 19 421857 768 984294 58 437563 826 562437 41 20 422318 767 984259 58 438059 825 56I941 40 21 9.422778 767 9.984224 58 9-438554 824 10.561446 39 22 423238 766 984190 58 439048 823 560952 38 23 423697 765 984155 58 439543 823 560457 37 24 424156 764 984120 58 440036 822 559964 36 25.424615 763 984085 58 440529 821 559471 35 26 425073 762 984050 58 44I022 820 558978 34 27 425530 761 984015 58 4415I4 819 558486 33 28 425987 760 983981 58 442006 8I 557994 32 29 426443 760 983946 58 442497 8 8 557503 31 30 426899 759 983911 58 442988 817 557012 30 31 9-427354 758 9.983875 58 9-443479 816 10.556521 29 32 427809 757 983840 59 443968 816 556032 28 33 428263 756 983805 59 44458 815 555542 27 34 428717 755 983770 59 444947 8I4 555053 26 35 429170 754 983735 59 445435 813 554565 25 36 429623 753 983700 59 445923 812 554077 24 37 430075 752 983664 59 446411 812 553589 23 38 430527 752 983629 59 446898 811 553I02 22 39 430978 75I 983594 59 447384 8io 5526I6 21 40 431429 750 983558 59 447870 809 552130 20 4I 9.431879 749 9-983523 59 9.448356 809 o.55I644 I9 42 432329 749 983487 59 44884I 808 551159 18 43 43277 74 983452 59 449326 807 55674 17 44 433226 747 9834I6 59 449810 806 550190 16 45 433675 746. 983381 59 450294 806 549706 I5 46 434122 745 983345 59 450777 805 549223 4 47 434569 744 983309 59 451260 804 548740 13 48 4350I6 744 983273 60 45I743 803 548257 12 49 435462 743 983238 60 452225 802 547775 11 50 435908 742 983202 60 452706 802 547294 io 51 9.436353 741 9 983166 60 9-453187 80o I0.5468I3 52 436798 740 983130 60 453668 800 546332 53 437242 740 983094 60 454148 799 545852 7 54 437686 739 983058 60 454628 799 545372 6 55 438129 738 983022 60 455107 798 544893 5 56 438572 737 982986 60 455586 797 544414 4 57 439014 736 982950 60 456064 796 543936 3 58 439456 736 982914 60 456542 796 543458 2 59 439897 735 982878 60 457019 795 54298I I 60 440338 734 982842 6o 457496 794 542504 0 Cosine, D. Sine. D. Cotang. D. Tang. / l1050 40 34 LOGARITHMIC SINES, TANGENTS, ETO. TABLE II. 16~ 1630 / Sine. D. Cosine. D. Tang. D. Cotang. o 9.440338 734 9-982842 60 9-457496.794 10O542504 60 I 440778 733 982805 60 457973 793' 542027 59 2 4412 I8 732 982769 6i 458449 793 54i55i 58 3 441658 73I 982733 6I 458925 792 541075 57 4 442096 731 982696 6i 459400 791 540600 56 5 442535 730 982660 6 459875 790,540I25 55 6 442973 729 982624 61 460349 79 53965I 54 7 4434Io 728 982587 6 460823 79 539177 53 8 443847 727 982551 6 461297 788 538703 52 9 444284 727 982514 61 461770 788 538230 51 io 444720 726 982477 61 462242 787 537758 50 ii 9.445155 725 9.982441 6i 9.462715 786 o10537285 4 12 445590 724 982404 6i 463186 785 5368I4 48 13 446025. 723 982367 6i 463658 785 536342 47 I4 446459 723 982331 61 464128 784 535872 46 I5 446893 722 982294 6i 464599 783 53540 45 16 447326 721 982257 61 465069 783 534931 44 17 447759 720 982220 62 465539 782 53446I 43 I8 448I9I 720 982183 62 466008 781 533992 42 19 448623 71 982146 62 466477 780 533523 41 20 449054 718 982109 62 466945 780 533055 40 21 9 449485 717 9982072 62 9 467413 779 o10532587 3 22 449915 716 982035 62 467880 778 532120 3 23 450345 716 981998 62 468347 778 531653 37 24 450775 715 98 1961 62 468814 777 53ii86 36 25 45I204 7I4 981924 62 469280 776 530720 35 26 451632 713 981886 62 469746 775 530254 34 27 452060 713 981849 62 470211 775 529789 33 28 452488 712 981812 62 470676 774 529324 32 29 452915 711 981774 62 471141 773 528859 31 30 453342 710 981737 62 471605 773 528395 30 31 9 453768 710 9-98I700 63 9 472069 772 10-527931 29 32 454194 709 981662 63 472532 771 527468 28 33 454619 708 981625 63 472995 771 527005 27 34 455044 707 981587 63 473457 770 526543 26 35 455469 707 981549 63 473919 769 526081 25 36 455893 706 981512 63 47438I 769 52569 24 37 4563I6 705 981474 63 474842 76 525 58 23 38 456739 704 981436 63 475303 767 524697 22 39 457I62 704 981399 63 475763 767 524237 21 40 457584 703 981361 63 476223 766 523777 20 41 9 45800o 6 702 9.98323 63 9-476683 765 o105233I7 19 42 458427 701 981285 63 477142 765 522858 I 43 458848 701 981247 63 477601 764 522399 17 44 459268 700 981209 63 478059 763 521941 i6 45. 459688 699 981171 63 478517 763 521483 I 46 460I08 698 981133 64 478975 762 521025 I4 47 460527 698 981095 64 479432 76I 520568 I3 48 460946 697 981i57 64 479889 76I 52011I 12 49 46I364 696 981019 64 480345 760 59655 ii 50 461782 695 980981 64 480801 759 519i99 10 5I 9-462199 695 9980942 64 9-481257 759 I0-58743 9 52 462616 694 980904 64 48I712 758 5I8288 53 463032 693 980866 64 482167 757 5I7833 7 54 463448 693 980827 64 48262I 757 517379 6 55 463864 692 980789 64 483075 756. 5925 5 56 464279 691 980750 64 483529 755 51647I 4 57 464694 690 980712 64 483982 755 5I6018 3 58 465108 690 980673 64 484435 754 515565 2 59 465522 689 980635 64 484887 753 515113 I 60 465935 688 980596 64 485339 753 51466I 0 Cosine. D. Sine. D. Cotang. D. Tang. / 106~0 - 3 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 85 17~0 - 1620 / Sine. D. Cosine. D. Tang. I D. I Cotang. _ o 9-465935 688 9 980o56. 64 9.485339 755 io051466I 60 466348 688 980558 64 485791 752 514209 5 2 466761 687 980519 65 486242 75I 51375 58 3 467173 686 980480 65 486693 751 513307 57 4 467585 685 980442 65 487I43 750 512857 56 5 467996 685 980403 65 487593 749 5I2407 55 6 468407 684 980364 65 488043 749 51957 54 7 468817 683 980325' 65 488492 748 5115o8 53 8 469227 683 980286 65 488941 747 511059 52 9 469637 682 980247 65 489390 747 51o06I 5i Io 470046 681 980208 65 489838 746 510162 50 Ii 9.470455 5 68 98 65 9490286 746 10-50974 49 12 470863 68o 980130 65 490733 745 509267 48 13 47127I 679 980091 65 491180 744 508820 47 14 471679 678 980052 65 491627 744 508373 46 I5 472086 678 980012 65 492073 743 507927 45 I6 472492 677 979973 65 49259 743 50748I' 44 I7 472898 676 97934 66 492965 742 507035 43 18 473304 676 979895 66 493410 741 506590 42 19 473710 675 979835 66 493854 740 506146 4I 20 474115 674 979816 66 494299 740 505701 40 21 9.474519 674 9.97977666 9,494743 740 IO 505257 39 22 474923 673 979737 66 495186 739.504814 38 23 475327 672 979697 66 495630 738 504370 37 24 475730 672 979658 66 496073 737 503927 36 25 476133 671 979618 66 496515 737 503485 35 26 476536 670 979579 66 496957 736 503043 34 27 476938 669 979539 66.'97399 736 502601 33 28 47734 669 979499 66 497841 735 502 I5 32 29 47774I 66 9794 9 66 498282 734 501718 3I 30 478142 667 979420 66 498722 734 501278 30 31'9-478542 667 9 979380 -66 9-499163 733 o10500837 2 32 478942 666 979340 66 499603 733 500397 28 33 479342 665 979300 67 500042 732 49998 27 34 47974I 665 979260 67 500481 731 499519 26 35 480140 664 979220 67 500920 731 499080 25 36 480539 663 979180 67 501359 730 49864I 24 37 480937 663 979140 67 501797 730 498203 23 38 481334 662 979100 67 502235 729 497765 22 39 481731 66I 979059 67 502672 728 497328 21 40 482128 66I 979019 67 503109 728 496891 20 41 9.482525 660 9-978979 67. 9-503546 727 10-496454 Ig 42 482921 659 978939 67 503982 727 496018 I8 43 4833I6 659 978898 67 5044I8 726 495582 17 44 483712 658 978858 67 504854 725 495146 i6 45 484107 657 978817 67 505289 725 4947II 15 46 48450o 657 978777 67 505724 724 494276 14 47 484895 656 978737 67 506I59 724 49384 13 48 485289 655 978696 68 506593 723 493407 12 49 485682 655 978655 68 507027 722 49273 II 50 486075 654 978615 68 |507460 722 492540 10 51 9.486467 653 9'978574 68 9.507893 721 10-492107 9 52 486860 653 978533 68 508326 721 491674 53 487251 652 978493 68 508759 720 491241 7 54 487643 65I 978452 68 50o99 719 490809 6 55 488034 65I 978411 68 509622 7 49o378 5 56 488424 650 978370 68 5Ioo54 71 489946 4 57 488814 650 978329 68 5I0485 718 4895I5 3 5 489204 649 978288 6 51096 717 489084 2 59 489593 648 978247 68 5ii346 716 488654 i 6 489982 648 978206 68 511776 716 488224 0 I | Cosine. D; Sine. D. C ot Tng. D. Tng. 1070 720 86 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 180 1610 I Sine. D. Cosine. D. Tang. D. Cotang. I 0 9.-489982 648 9-978206 68 9-51I776 716 10-488224 60 I 490371 648 978165 68 512206 716 487794 50 2 490759 647 978124 68 512635 715 487365 58 3 49I147 646 978083 69 5I3064 714 486936 57 4 491535 646 978042 69 513493 714 486507 56 5 491922 645 978001 69 513921 713 48679 55 6 492308 644 977959 69 5I4349 73 48565I 54 7 492695 644 977918 69 514777 712 485223 53 8 493081 643 977877 69 515204 712 484796 52 9 493466 642 977835 69 51563I 711 484369 51 io 493851 642 977794 69 516057 710 483943 50 II 9-494236 64I 9-977752 69 9.5I6484 710 io04835I6 4 12 494621 64i 977711 69 516910 709 483090 4 I3 495005 640 977669 69 017335 709 482665 47 I4 495388 639 977628 69 51776 708 482239 46 I5 495772 639 977586 69 518186 708 481814 45 I6 496154 638 977544 70 5r86ro 707 481390 44 I7 496537 637 977503 70 519034 706 480966 43 i8 496919 637 977461 70 519458. 706 480542 42 I9 497301 636 977419 70 519882 705 480118 41 20 497682 636 977377 70 520305 705 479695 4o 21 9-498o64 635 9.977335 70 9.520728 704 o10479272 3 22 498444 634 977293 70 521151 703 478849 38 23 498825 634 9772 I 70 521573 703 478427 37 24 499204 633 977209 70 521995 703 478005 36 25 499584 632 977i67 70 522417 702 477583 35 26 499963 632 977125 70 522838 702 477162 34 27 500342 63i 977083 70 523259 701 476742 33 28 500721 63i 977041 70 523680 70I 476320 32 29 50I099 630 976999 70 524100oo 700 475900 3I 30 501476 629 976957 70 524520 699 475480 30 31 9.501854 629 9.976914 70o 9.524940 699 10-475060 29 32 502231 628 976872 71 52535 9 474641 28 33 502607 628 976830 71 525778 698 474222 27 34 502984 627 976787 71 526197 697 473803 26 35 503360 626 976745 71 526615 697 473385 25 36 503735 626 976702 71 527033 696 472967 24 37 504Iio 625 976660 71 527451 696 472549 23 38 504485 625 976617 71 527868 695 472132 22 39 504860 624 976574 71 528285 695 471715 21 40 505234 623 976532 71 528702 694 471298 20 4I 950o56o8 623 9.976489 71 9.529119 693 10o47088I 1 42 505981 622 976446 71 529535 693 470465 18 43 506354 622 976404 71 529951 693 470049 17 44 506727 621 976361 71 530366 692 469634 I6 45 607099 620 9763I8 71 530781 69 469219 15 46 507471 620 976275 71 531196 691 468804 14 47 507843 619 976232 72 53I6II 690 468389 13 48'508214 619 976189 72 532025 690 467975 12 49 508585 68 976146 72 532439 689 467561 II 50 508956 618 976103 72 532853 689 467I47 10 5I 9.509326 617 99 6 72 976060533266 688 466734 52 509696 66 976017 72 533679 688 466321 53 510065 616 975974 72 534092 687 465908 7 54 510434 615 975930 72 534504 687 465496 6 55 51o803 6I5 975887 72 534916 686 465084 5 56 511172 6I4 975844 72 535328 686 464672 4 57 51i540 6I3 975800 72 535739 685 46426I 3 58 511907 6I3 975757 72 536I50 685 463850 2 59 5I2275 6I2 975714 72 53656i 684 463439 I 60 512642 612 975670 72 536972 684 463028 o Cosine. D. Sine. D. Cotang. D. Tang, / 1 08~ _11-~ ~os. TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 37 190 _ _______ _______ _____ 1600 190 1600 _ Sine. D. Cosine. D. Tang. D. Cotang. / 0 9.-512642 6I2 9-975670 73 9.536972 684 I0o463028 60 513009'6II 975627 73 537382 683 462618 50 2 513375 6II 975583 73 537792 683 462208 58 3 51374I 60 975539 73 538202 682 461798 57 4 514107 609 975496 73 53861I 682 4613g9 5 5 5I4472 609 975452 73 5390o0 68I 460980 55 6 514837 6o 975408 73 539429 68 I 460571 5 7 5I.5202 608 975365 73 539837 680 460o63 53 8 55566 607 975321 73 540245 680 459755 52 9 515930 607 975277 73 540653 679 459347 5I 10 516294 606 975233 73 54IO6I 679 458939 5o ii 9.516657 65 9-975I89 73 94541468 678 10I458532 49 12 517020 605 975145 73 541875 678 458I25 48 13 517382 604 975IoI 73 542281 677 457719 47 I4 517745 604 975057 73 542688 677 4573I2 46 15 518107 603 9750o3 73 543094 676 456906 45 I6 5I8468 603 974969 74 543499 676 456501 44 I7 5I8829 602 974925 74 543905 675 456095 43 I8 519190J 601 974880 74 5443 I 675 455690 42 19 51955I 6o0 974836 74 544715 674 455285 4I 20 5199iI 600 974792 74 545I19 674 454881 40 21 9.520271 600 9.974748 74 9.545524 673 10.454476 39 22 520631 599 974703 74 545928 673 454072 38 23 520990 59o 974659 74 546331 672 453669 37 24 521349 59 97464 74 546735 672 453265 36 25 521707 598 974570 74 547138 671 452862 35 26 522066 597 974525 74 547540 671 452460 34 27 522424 596 974481 74 547943 670 452057 33 28 522781 596 974436 74 548345 670 451655 32 29 523138 595 974391 74 548747 669 451253 31 30 523495 595 974347 75 549149 669 45085I 30 3I 9.523852 594 9.974302 75 9.549550 668 io0450450 29 32 524208 594 974257 75 549951 668 450049 28 33 524564 593 974212 75 550352 667 449648 27 34 524920 593 974167 75 550752 667 449248 26 35 525275 592 974122 75 5515 3 666 448847 25 36 525630 591 974077 75 55I552 666 448448 24 525984 59I 974032 75 55I952 665 448048 23 526339 590 973987 75 552351 665 447649 22 39 526693 590 973942 75 552750 665 447250 21 40 527046 589 973897 75 553149 664 446851 20 4I 9-527400 589 9.973852 75 9-553548 664 1 o446452 19 42 527753 588 973807 75 553946 663 446054 43 528105 588 973761 75 554344 663 445656 17 44 528458 587 9737I6 76 554741 662 445259 16 45 528810 587 973671 76 555139 662 444861 15 46 529161 586 973625 76 555536 661 444464 14 47 529513 586 97358o 76 555933 661 444067 13 48 529864 585 973535 76 556329 660 443671 12 49 530215 585 973489 76 55672 660 443275 ii 50 530565 584 973444 76 557121 659 442879 o1 5I 9.5309I5 584 9-973398 76 9-5575I7 659 I044.2483 52 531265 583 973352 76 5579I3 659 442087 53 53I6I4 582 973307 76 558308 658 44692 54 53i963 582 973261 76 558703 658 441297 55 532312 581 973215 76 559097 657 440903 5 56 532661 58i 973169 76 55949I 657 440509 4 57 533009 580 973124 76 55985 656 4401 5 3 58 533357 580 973078 76 560279 656 439721 2 59 533704 579 973032 77 560673 655 439327 I 6o 534052 578 972986 77 561066 655 438934' Cosine. D. Sine. D. Cotang. D. Tang. t 109 - - IO~ 98 __8 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 20~ 1590 / Sine. D. Cosine. D. Tang. D. Cotang. I 0 9.534052 578 9.972986 77 9.56io66 655 o10438934 60 I 534399 577 972940 77 56 459 654 43854i 59 2 534745 577. 972 94 77 56i85 654 438149 58 3 535092 577 972848 77 562244 653 437756 57 4 535438 576 972802 77 562636 653 437364 56 5 535783 576 972755 77 563028 653 436972 55 6 536129 575 972709 77 563419 652 436581 54 7 536474 574 97266 77 5638iI 652 436189 53 8 5368i8 574 972617 77 564202 65i 435798 52 9 537163 573 972570 77 564593 65 435407 51 io 537507 573 972524 77 564983 65o 4350o7 50 ii 9-53785i 572 9-972478 77 9*565373 650 10o434627 49 12 538194 572 97243 78 565763 649 434237 48 i3 538538 57i 972385 78 566153 649 433847 47 14 538880 571 972338 78 566542 649 433458 46 15 539223 570 972291 78 566932 648 433068 45 16 539565 570 972245 78 567320 648 432680 44 17 539907 569 972198 78 56770 647 432291 43 i8 540249 56 972151 78 56809j 647 431902 42 i9 540590 56 972105 78 568486 646 43i5I4 41 20 540931 568 972058 78 568873 646 431127 40 21 9.541272 567 9-9720II 78 9.569261 645 I0o430739 30 22 54613 567 971964 78 569648 645 430352 3 23 541953 566 971917 78 570035 645 429965 37 24 542293 566 971870 78 570422 644 429578 36 25 542632 565 971823 78 57080 644 429191 35 26 54297I 565 97I776 78 5711 643 428805 34 27 5433o 564 971729 79 571581 643 428419 33 28 543649 564 971682 79 571967 642 428033 32 29 543987 563 971635 79 572352 642 427648 3x 30 544325 563 971588 79 572738 642 427262 30 31 9.544663 562 9-971540 79 9-573I23 641 o10426877 29 32 545000 562 971493 79 573507 64, 426493 28 33 545338 56i 971446 79 573892 640 426108 27 34 545674 56I 971398 79 574276 640 425724, 26 35 5460oI 560.97135I 79 574660 639 425340 25 36 546347 560 971303 79 575044 639 424956 24 37 546683 559 971256 79 575427 639 424573 23 38 547019 55 97'1208 79 57580 638 424190 22 39 547354 558 971161 79 57693 638 423807 21 40 547689 558 971113 79 576576 637 423424 20 41 9.548024 557 9-971066 80 9.576959 637 10-423041 19 42 548359 557 97I018 80 577341 636 422659 18 43 548693 556 970970 80 577723 636 422277 17 44 549027 556 970922 80 578104 636 421896 16 45 549360 555 970874 80 578486 635 421514 15 46 549693 555 970827 80 578867 635 421133 14 47 550026 554 970779 80 579248 634. 420752 x3 48 550359 554 970731 80 579629 634 420371 12 49 550692 553 970683 80 580009 634 419991 II 50 55I024 553 970635 80 580389 633 41961 I 5I 9. 55I356 552 9'970586 80 9-580769 633 0o.419231 I 52 55I687 552 970538 80 581149 632 418851 8 53 552018 552 970490 80 581528 632 418472 7 54 552349 551 970442 80 581907 632 418093 6 55 552680 551 970394 80 582286 63 417714 5 56 553010 550 970345 81 582665 63i 417335 4 57 553341 550 970297 8I 583044 630 41656 3 58 553670 549 970249 8I 583422 630 4I678 2 59 554000 549 970200 81 583800 629 416200 I 60 554329 548 970152 8i 584177 629 415823 0 I Cosine. D. Sine. D. Cotang. D. Tang. I 1100 69~ TABLE II. OG ITHM 969957 8 585686 TA ENTS, ETC.39 210 1580 Sine597I D. Cosine. D. 586o62Tang. D. Cotang.938 o6 9554329 548 997052 8 9586417 629 10i415823 60 7 554626 545 97698I 8I 5864555 626 435445 53 2 556953 544 969762 8I 587493 626 415068 58 3- 55531 547 970006 81 585309 628 414691 57 4 5556428 544 969957 8i 58566 62 412434 56 5 557606 543 969690 8I 587946 625 420935 55 6 5593299 5453 9.969660 82 586839 625 410 6 5684 2 556626 545 969567 82 588698 626 4I1385 53 3 558583 542 969762 82 5879066 624 410934 4 9 557280 544 9697]4 81 587566 625 412434 51 55768909 542 969469 82 589440 625 42056 46 I5 9559232 54I 9969420 82 589814 623 4I-4 868 4S I6 559558 54I 969370 82 588691 623 439809 44 17 559883 540 969351 82 590562 624 409434 47 14 558909 542 969469 82 589440 623 409060 46 15 559234 541 969420 82 589814 623 4io186 45 16 559558 541 969370 82 590188 623 409812 44 17 559883 540 969321 82 5.90562 622 409438 43 18 560207 540 969272 82 590935 622 40go65 42 19 56053i 539 969223 82 59I308 622 408692 4I 20 560855 539 969173 82 591681 621 408319 40 21 9.561178 538 9-969124 82 9.592054 621 I0407946 39 22 56i501 538 969075 82 592426 620 407574 38 23 561824 537 969025 82 592799 620 40720I 37 24 562146 537 968976 82 593171 619 406829 36 25 562468 536 968926 83 593542 6 9 40645 35 26 562790 536 968877 83 593914 68 406086 34 27 56312 536 968827 83 594285 6I8 405715 33 28 563433 535 968777 83 594656.6I8 405344 32 29 563755 535 968728 83 595027 617 404973 31 30 564075 534.968678 83 595398 617 404602 3o 3I 9.564396 534 9.968628 83 9.595768 617 10404232 29 32 564716 533 968578 83 596138 6i6 403862 28 33 565036 533 968528 83 596508 6I6 403492 27 34 565356 532 968479 83 596878 6I6 403122 26 35 565676 532 968429 83 5977247 6 5 402753 25 36 565995 531 968379 83 5976i6 6i5 402384 24 37 566314 531 968329 83 597985 6i5 4020o 5 23 38 566632 53i 968278 83 598354 614 401646 22 39 56695 530 968228 84 598722 614 401278 21 40 567269 530 968178 84 59909i 613 400909 20 4I 9. 567587 529 9.968128 84 9.599459 6i3 Io-400541 1 42 567904 529 968078 84 599827 63 400oo73 I 43 568222 528 968027 84 600194 612 399806 17 44 568539 528 967977 84 600562 6I2 399438 i6 45 568856 528 967927 84 600929 6ii 39907I 15 46 569172 527 967876 84 60I296 611 398704 14 47 569488 527 967826 84 601663 611 398337 13 48 569804 526 967775 84 602029 6io 397971 12 49 570o20 526 967725 84 602395 6io 3976o5 II 50 570435 525 967674 84 602761 6io 397239 Io 51 9.57075i 525 9-967624 84 9-603127 609 10.396873 52 571066 524 967573 84 603493 609 396507 53 57I380 524 967522 85 603858 609 396142 7 54 57i695 523 967471 85 604223 608 395777 6 55 572009 523 967421 85 604588 608 395412 5 56 572323 523 967370 85 604953 607 395047 4 57 572636 522 967319 85 605317 607 394683 3 58 572950 522 967268 85 605682 607 394318 2 59 573263 521 967217 85 606046 606 393954 I 60 573575 521 967166 85 6064i0 6o6 393590 0' Cosine. D. Sine. D. Cotang. D. Tang. / 1110 o ___- 68 o 40 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 22~ 1567 / Sine. D. Cosine, D. Tang. D. Cotang. i o 9.573575 521 9.967I66 85 09606410 606 10-393590 60 I 573888 520 967115 85 60677 6o6 393227 59 2 574200 520 967064 85 607137 605 392863 58 3 574512 5I9 967013 85 607500 605 392500 -57 4 574824.' 519 96696I 85 60863 604 392137 56 5 575I36 5I9 966910 85 608225 604 391775 55 6 575447 58 96685 85 608588 604 391412 54 7 575758 5i8 966808 85 608950 603 391050 53 8 576069 517 966756 86 609312 603 390688 52 9 576379 517 966705 86 609674 603 390326 5I I0 576689 56.966653 86 610036 602 389964 50 II 9.576999 5I6 9.966602 86 9-6I0397 602 Io.389603 4 12 577309 5i6 966550 86 6Io759 62 38924 48 3 577618 515 966499 86 611120 6oi 38888 47 I4 577927 5I5 966447 86 611480 6ox 388520 46 i5 578236 514 966395 86 61184i 601 388159 45 i6 578545 5I4 966344 86 612201 600 387799 44 I7 578853 513 966292 86 612561 600 387439 43 18 579162 513 966240 86 612921 600 387079 42 19 579470 513 966188 86 613281 599 386719 41 20 579777 512 966136 86 6I3641 599 386359 40 2I 9-580085 512 9-966085 87 9-614000 598 0io386000 39 22 580392 511 966033 87 614359 598 385641 38 23 580699 511 965981 87 614718 598 385282 37 24 58I005 511 965928 87 615077 597 384923 36 25 581312 5Io 965876 87 615435 597 384565 35 26 58I618 51o 965824 87 61573 597 384207 34 2 581924 509 965772 87 616161 596 383849 33 582229 509. 965720 87 6I6509 596 383491 32 29 582535 509 965668 87 616867 596 383 33 31 30 582840 508 965615 87 617224 595 382776 30 3i 9.583145 508 9.965563 87 9.617582 595 I0 382418 2 32 583449 507 96551.1 87 617939 595 382062 28 33 583754 507 965458 87 618295 594 38 705 27 34 584058 506 965406 87 6I8652 594 381348 26 35 584361 506 965353 88 619008 594 380992 25 36 584665 506 965301 88 6I9364 593 380636 24 37 584968 505 965248 88 619720 593 380280 23 38 585272 505 965195 88 620076 593 379924 22 39 585574 504 965143 88 620432 592 379568 21 40 58577 504 965090 88 620787 592 379213 20 41 9.586179 503 9.965037 88 9.621142 592 10o378858 I 42 586482 503 964984 88 621497 59I 378503 18 43 586783 503 964931 88 6218 2 591 378148 17 44 587085 502 964879 88 622207 590 377793 6 45 587386 502 964826 88 62256I 590 377439 5 46 587688 50o 964773 88 622915 590 377085 4 47 587989 501 964720 88 623269 589 37673 13 48 588289 501 964666 89 623623 589 376377 12 49 588590 500 964613 89 623976 589 376024 II 50 588890 500 964560 89 624330 588 375670 I0 51 9.589190 499 9-964507 89 9 624683 588 I0-375317 52 589489 499 964454 89 625036 588 374964 8 53 589789 499 964400 89 625388 587 374612 7 54 590088 498 964347 89 625741 587 374259 6 55 590387 498 964294 89 626093 587 3737 5 56.590686 497 964240 89 626445 586 373 55 4 57 590984 497 964187 89 626797 586 373203 3 58 591282 497 964133 89 627149 586 372851 2 59 59I580 496 964080 89 627501 585 372499 I 60 591878 496 964026 89 627852 585 372148 0' Cosine. D. Sine. D. Cotang. D. Tang. 1120 6106 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 41 230 1560 / Sine. D. Cosine. D. Tang. D. Cotang. / o 9.591878 496 9.964026 89 9.62 852 585 10.372148 60 I 592176 495 963972 89 62203 585 37I797 5 2 592473 495 963919 89 628554 585 37I446 3 592770 495 963865 90 628905 584 371095 57 4 593067 494 963811 9o 629255 584 370745 56 5 593363 494 963757 90 629606 583 370394 55 6 593659 493 963704 90 629956 583 370044 54 7 593955 493 963650 90 630306 583 369694 53 8 594251 493 963596 90 630656 583 369344 52 9 594547 492 963542 g 63oo005 582 368995 51 10 594842 492 963488 90 631355 582 368645 50 ii 9-595I37 491 9-963434 90 9.631704 582 10.368296 49 12 595432 49I 963379 90 632053 58i 367947 48 13 595727 491 963325 90 632402 581 36798 47 14 596021 490 963271 90 632750 58I 367250 46 15 596315 490 963217 g9 633099 580 366901 45 6 596609 489 963163 90 633447 580 366553 44 17 596903 489 963108 9 633795 580 366205 4.3 I8 597196 489 963054 91 634I43 579 365857 42 I9 507490 488 902999 91 634490 579 3655io 4I 20 597783 488 962945 91 634838 579 365I62 40 21 9.598075 487 9.962890 91 9.635185 578 o10364815 3 22 598368 487 962836 91 635532 578 364468 38 23 598660 487 962781 91 635879 578 364121 37 24 598952 486 962727 91 636226 577 363774 36 25 599244 486 962672 91 636572 577 363428 35 26 599536 485 962617 91 63691 577 363081 34 27 599827 485 962562 91 63726 577 362735 33 28 60oo08 485 962508 91 637611 576 362389 32 29 600409 484 962453 91 637956 576 362044 31 30 600700 484 962398 92 638302 576 36I698 30 3I 9600o99o 484 9.962343 92 9-638647 575 Io.361353 29 32 601280 483 962288 92 638992 575 36Ioo8 28 33 601570 483 962233 92 639337 575 360663 27 34 6oi860 482 962178 92 639682 574 3603I8 26 35 602150 482 962123 92 640027 574 359973 25 36 602439 482.962067 92 640371 574 359629 24 37 602728 481 962012 92 640716 573 359284 23 38 603017 481 961957 92 64io60 573 358940 22 39 603305 481 961902 92 641404 573 358596 2I 40 603594 480 961846 92 641747 572 358253 20 41 9-603882 480 9-961791 92 9.642091 572 I0o357909 10 42 604170 479 961735 92 642434 572 357 66 I 43 604457 479 961680 92 642777 572 357223 17 44 604745 479 961624 93 643120 571 356880 16 45 605032 478 96I569 93 643463 57I 356537 15 46 605319 478 961513 93 643806 57I 356194 14 47 6o56o06 478 96I458 93 644148 570 355852 13 48 605892 477 961402 93 644490 570 35551I 12 49 606179 477 961346 93 644832 570 355168 ii 50 606465 476 961290 93 645174 569 354826 o1 51 9.60675 476 9.961235 93 9.6455I6 569 Io-354484 52 607036 476 961179 93 645857 569 354i43 53 607322 475 961123 93 646199 569 35380 7 54 607607 475 961067 93 646540 56 35346 6 55 607892 474 961011 93 646881 568 35319 5 56 608177 474 960955 93 647222 568 352778 4 57 608461 474 960899 93 647562 567 352438 3 58 608745 473 960843 94 647903 567 352097 2 59 609029 473 960786 94 648243 567 351757 60 609313 473 960730 94 648583 566 35I417 0 - Cosine. D. Sine. D. Cotang. D. Tang. / 113_____________.66 42 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 240 1655~ I Sine. D. Cosine. D. Tang. D. Cotang. 0 9 609313 473 9-960730 94 9.648583 566 10o351417 60 I 609597 472 960674 94 648923 566 351077 5 2 609880 472 960618 94 649263 566 350737 58 3 6ioi64 472 960561 94 649602 566 350398 57 4 610447 471 96505 94 649942 565 350058 56 5 610729 47 960448 94 650281 565 349719 55 6 611,I2 470 960392 94 650620 565 349380 54 7 611294 470 960335 94 650959 564 349041 53 8 611576 470 960279 94 651297 564 348703 52 9 6ii858 469 960222 94 651636 564 348364 51 Io 612140 469 960165 94 651974 563 348026 50 ii 9-6I242I 469 9.960109 95 9.652312 563 10-347688 49 12 612702 468 960052 95 652650 563 347350 48 13 612983 468 959995 95 652988 563 347012 47 14 6I3264 467 959938 95 653326 562 346674 46 i5 6i3545 467 959882 95 653663 562 346337 45 16 613825 467 95 9 525 9 64000 562 346000 44 17 614105 466 959768 95 654337 56i 345663 43 I8 614385 466 959711 95 654674 56i 345326 42 19 6i4665 466 959654 95 655011 56i 344989 41 20 614944 465 959596 95 655348 56i 344652 40 21 9-6I5223 465 9.959539 95 9.655684 56o 103443I6 3 22 615502 465 959482 95 656020 560 343980 38 23 6157j81 464 959425 95 656356 560 343644 37 24 6I6060 464 959368 95 656692 559 343308 36 25 616338 464 959310 96 657028 559 342972 35 26 6i66i6 463 959253 96 657364 559 342636 34 27 616894 463 959I95 96 657699 559 34230I 33 28 67I72 462 959138 96 658034 558 34966 32 29 6 174-50 462 959080 96 658369 558 34I63I 31 30 6I7727 462 959023 96 658704 558 341296 3o 31 9-618004 461 9-958965 96 9.659039 558 10-34096i 29 32 GI8281 46i 958908 96 659373 557 340627 2 33 6i8558 46i 958850 96 659708 557 340292 27 34 618834 460 958792 96 660042 557 339958 26 35 6i9iio 460 958734 66 660376 557 339624 25 36 619386 460 958677 96 6607IO 556 339290 24 37 6I9662 459 958619 96 661043 556 338957 23 38 619938 459 95856I 96 661377 556 338623 22 39 6202I3 459 958503 97 66I710 555 338290 21 40 620488 458 958445 97 662043 555 337957 2C 4I 9.620763 458 9-958387 97 9.662376 555 10-337624 19 42 62038 457 958329 97 662709 554 337291 43 62I313 457 927 958 7 663042 554 33698 17 44 62I587 457 958213 97 663375 554 336625 16 45 62I86I 456 958 958 7 663707 554 336293 5 46 622135 456 958096 97 664039 553 33596I 14 47 622409 456 958038 97 664371 553 335629 13 48 622682 455 957979 97 664703 553 335297 12 49 622956 4 955 9 2 9 7 665035 553 334965 ii 50 623229 455 957863 97 665366 552 334634 io 51 9-623502 454 9.957804 97 9-665698 552 o10334302 52 623774 454 957746 98 666029 552 33397 8 53 624047 454 957687 98 666360 55I 33364o 7 54 624319 453 957628 98 66669I 551 333309 6 55 624591 453 957570 98 667021 551 332979 5 56 624863 453 9575 5 98 667352 55 332648 4 57 625135 452 957452 98 667682 550 3323I8 3 58 625406 452 957393 98 6680i3 550 331987 2 59 625677 452 957335 98 668343 550 331657 I 60 625948 451 957276 98 668673 550 331327 I Cosine. D. Sine. D. Cotang. D. Tang. I 1 14~ 650 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. _ 43 25 _______ 1540 / Site. D. Cosine. D. Tang. D. Cotang. t o 9^625948 451 9'957276 98 9.668673 550 IO033I327 60 I 626219 45 957217 98 669002 549 330998 59 2 626490 45I 957158 98 669332 549 330668 58 3 626760 450 957099 98 669661 549 330339 57 4 627030 450 957040 98 669991 548 330009 56 5 627300 450 956981 98 670320 548 329680 55 6 627570 449 956921 99 670649 548 329351 54 7 627840 449 956862 99 670977 548 329023 53 8 628109 449 956803 99 671306 547 328694 52 9 628378 448 956744 99 671635 547 328365 51 Io 628647 448 956684 99 671963 547 328037 50 II 9-628916 457 9.956625 99 9672291 547 I10327709 49 12 629185 447 956566 99 672619 546 327381 48 13 629453 447 956506 99 672947 546 327053 47 14 629721 446 956447 99 673274 546 326726 46 15 629989 446 956387 99 673602 546 326398 45 I6 630257 446 956327 99 673929 545 326071 44 17 630524I 446 956268 99 674257 545 325743 43 I8 630792 445 956208 i00 674584 545 325416 42 19 631059 445 956148 10oo 6749I 544 325089 41 20 631326 445 956089 o00 675237 544 324763 40 21 9.631593 444 9.956029 o00 9.675564 544 10-324436 30 22 631859 444 955969 100 675890 544 324110 38 23 632125 444 955909 10o 676217 543 323783 37 24 632392 443 955849 100 676543 543 323457 36 25 632658 443 9557'89 100oo 676869 543 323131 35 26 632923 443 955729 100oo 677194 543 322806 34 27 633189 442 955669 100 677520 542 322480 33 28 633454 442 95569 100. 677846 542 322154 32 29 6337I9 442 955548 I00 67817I 542 321829 31 30 633984 441 955488 100 678496 542 321504 30 3I 9 634249 44I 9.955428 IoI 9.678821 54I I o321179 29 32 6345I14 440 955368 1io 679146 541 320854 28 33 634778 440 955307 o10 67947I 541 320529 27 34 635042 440 955247 10I 679795 541 320205 26 35 635306 439 955186 101 680120 540 -39880 25 36 635570 439 955126 o10 680444 540 319556 24 37. 635834 439 955065 o0I 680768 540 3I9232 23 38 636097 43 955005 10o 681092 540 318908 22 39 63636o 438 954944 ioi 6841i6 539 318584 21 40 636623 438 954883 Ioi 681740 539 3I8260 20 41 9-636886 437 9-954823 101 9.682063 539 10-317937 19 42 637148 437 954762 101 682387 539 317613 I8 43 63741 I 437 95470I IoI 682710 538 317290 17 44 637673 437 954640 o1I 683033 538 316967 i6 45 637935 436 954579 11 683356 538 316644 15 46 638197 436 954518 102 683679 538 3I6321 14 47 638458 436 954457 102 68400o 537 315999 13 48 638720 435 954396 102 684324 537 315676 12 49 638981 435 954335 102 684646 537 315354 II 50 639242 435 954274 102 684968 537 315032 10 5 9.639503 434 9'954213 I2 9.685290 536 I0-3I47IO 52 639764 434 954152 102 685612 536 314388 8 53 640024 434 954090 102 685934 536 314066 7 54 640284 433 954029 102 686255 536 313745 6 55 640544 433 953968 I02 686577 535 313423 5 56 640804 433 953906 102 686898 535 313102 4 57 64o064 432 953845 102 687219 535 312781 3 58 64I324 1432 953783 102 687540 535 312460 2 59 641583 432 953722 Io3 68786I 534 312139 I 60 641842 43I 953660 o103 688182 534 31181 I Cosine. D. ine D. Cotang. D. Tang. 11 o __-___50_ _ 64 44 LOGARITHMIC SINES, TANGENTS, ETC. TABLE I. 26 _1530 Sine. D. Cosine. D. Tang. D. Cotang. o 9.641842 431 9-953660 103 9.688182 534 xo-3ii818 60 I 642IOI 43 r 953599 io3 688502 534 3II498 59 2 642360 43 953537 io3 688823 534 31177 58 3 642618 430 953475 io3 689143 533 310857 57 4 642877 430 953413 Io3 689463 533 310537 56 5 643I35 430 953352 1o3 689783 533 3102I7 55 6 643393 430 953290 103 690103 533 309897 54 7 643690 429 953228 o13 690423 533 3o9577 53 8 643908 429 953166,03 690742 532 309258 52 9 644165 429 953104 Io3 691062 532 308938 5i 10 644423 428 953042 o13 69138I 532 308619 50 II 9.644680 428 9-952980 Io4 9.691700 53i 10-308300 49 12 644936 428 952918 104 692019 531 307981 4 13 645193 427 952855 104 692338 53I 307662 47 14 645450 427 952793 104 692656 531 307344 46 15 645706 427 952731 I04 692975 531 307025 45 16 645962 426 952669 04 693293 530 306707 44 17 6462I8 426 952606 104 693612 530 306388 43 18 646474 426 952544 104 693930 530 306070 42 19 646729 425 95248 I0o4 694248 530 305752 41 20 646984 425 952419 04 694566 529 305434 40 21 9 647240 425 9-952356 104 9.694883 529 10305117 3 22 647494 424 952294 I04 695201 529 304799 3 23 647749 424 95223I 104 6955I8 529 304482 37 24 648004 424 952168 I05 695836 529 304164 36 25 648258 424 952106 105 696153 528 303847 35 26 648512 423 952043 I05 696470 528 303530 34 27 648766 423 951980 I05 696787 528 303213 33 28 64900 423 951917 105 697103 528 302897 32 29 649274 422 951854 I05 697420 527 302580 31 30 649527 422 95179I I05 697736 527 302264 30 3I 9.64978I 422 9.951728 Io5 9.698053 527 o103oI947 2 32 650034 422 951665 o15 698369 527 3oi63I 2 33 650287 421 951602 I05 698685 526 3013I5 27 34 650539 421 951539 105 69g900 526 300999 26 35 650792 421 951476 io5 699316 526 300684 25 36 651044 420 951412 105 699632 526 300368 24 37 651297 420 951349 Io6 699947 526 300053 23 38 651549 420 951286 lO6 700263 525 299737 22 39 65i800 419 951222 106 700578 525 299422 21 40 652052 419 95II59 io6 700893 525 299107 20 4I 9.652304 419 9.951096 I06 9 -70208 524 10298792 1 42 652555 418 951032 Io6 70I523 524 298477 I1 43 652806 418 950968 106 701837 524 298163 17 44 653057 418 950905 o06 702152 524 297848 i6 45 653308 418 950841 io6 702466 524 297534 15 46 653558 417 950778 io6 702781 523 2972I9 14 47 653808 417 950714 o06 703095 523 296905 13 48 654059 417 950650 lo6 703409 523 296591 12 49 6543o0 416 950586 106 703722 523 296278 II 50 654558 4i6 950522 107 704036 522 295964 io 5I 9.654808 4i6 9.950458 107 9.704350 522 10-295650 52 655058 4I6 950394 107 704663 522 295337 8 53 655307 415 950330 107 704976 522 295024 7 54 655556 415 950266 107 705290 522 294710 6 55 655805 415 950202 107 705603 521 294397 5 56 656054 44 14 950138 107 705916 52I 294084 4 57 656302 414 950074 107 706228 521 293772. 3 58 65655I 414 9500oI 107 706541 521 293459 2 59 656799 413 949945 107 706854 521 293146 I 60 657047 413 949881 107 707166 520 292834 o / Cosine.. Sine. D. Coang. t D. Tang. I 116" 63~ TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 45 27~0 1520 Sine. D. Cosine. Tang. D. Cotang. o 9.657047 4I3 9.94988I I07 Q' 707I66 520 10 292834 60 I 657295 413 949816 107 707478 520 292522 59 2 657542 412 949752 I07 707790 520 292210 58 3 657790 412 949688 108 708102 520 291898 57 4 658037 412 949623 I08 708414 519 291586 56 5 658284 412 949558 108 708726 5I9 291274 55 6 658531 4I 949494 io8 709037 519 290963 54 7 658778 41 949429 108 709349 519 290651 53 8 659025 4I 949364 108 709660 519 290340 52 9 659271 410 949300 108 709971 518 290029 51 10 659517 410 949235 108 7I0282 5I8 28978 50 11 9.659763 410 9-9491 70 0o8 9-710593 5I8 10289407 49 12 660009 409 949105 108 710904 518 289096 48 13 660253 409 949040 108 711215 518 288785 47 I4 660501 409 948975 Io8 71525 517 288475 46 15 660746 409 948910 108 711836 517 288164 45 I6 660991 408 948845 108 712146 517 287854 44 17 661236 408 948780 0o9 712456 517 287544 43 18 66I481 408 948715 109 7I2766 516 287234 42 19 661726 407 948650 I09 713076 5i6 286924 4I 20 66I970 407 948584 o19 713386 5i6 286614 40 21 9.662214 407 9-948519 109 9*73696 5i6 10-286304 39 22 662459'407 948454 109 714005 516 285995 38 23 662703 406 948388 0o9 714314 515 285686 37 24 662946 406 948323 0og 714624 515 285376 36 25 663I90 406 948257 109 714933 5I5 285067 35 26 663433 4o5 948I92 109 715242 515 284758 34 27 663677 405 948I26 109 715551 514 284449 33 28 663920 405 948060 09 715860 5,4 284I40 32 29 664I63 405 947995 II 716168 514 283832 31 30 664406 404 947929 Ii0 716477 5i4 283523 30 31 9-664648 404 9.947863 Iio 9.716785 514 0-2832I5 29 32 664891 404 947797 I 1 717093 513 282907 28 33 665i33 403 947731 Io0 717401 513 282599 27 34 665375 403 947665 I1o 717709 513 282291 26 35 665617 403 947600oo I 7I8017 5I3 281983 25 36 665859 402 947533 Io 7I8325 513 281675 24 37 666I00 402 947467 110 718633 512 281367 23 38 666342 402 947401 I o 718940 512 281060 22 39 666583 402 947335 II0 719248 5I2 280752 21 40 666824 401 947269 Iio 719555 512 280445 20 41 9.667065 401 9'947203 II0 971I9862 512 10I280138 19 42 667305 401 947136 III 720169 5ii 27983I 18 43 667546 401 947070 III 720476 51 279524 17 44 667786 400 947004 III 720783 51I 279217 16 45 668027 400 946937 III 72I089 511 278911 15 46 668267 400 946871 III 721396 5I 278604 14 47 668506 399 946804 III 721702 510 278298 13 48 668746 399 946738 II 722009 510 277991 12 49 668986 399 946671 III 722315 510 277685 II 50 669225 399 946604 11 722621 510 277379 10 5i 9.669464 398 9'946538 I II 9722927 510 10-277073 9 52 669703 398 94647I III 723232 509 276768 8 53 669942 398 946404 III 723538 509 276462 7 54 670I81 397 946337 III 723844 509 276156 6 55 670419 397 946270 112 724149 509 275851 5 56 670658 397 946203 112 724454 509 275546 4 57 670896 397 946136 112 724760 50o 275240 3 58 671134 396 946069 112 725065 508 274935 2 59 671372 396 946002 12 725370 508 274630 I 60 671609 396 945935 II2 725674 508 274326 0 / Cosine. D. Sine. D. Cotang. D. Tang. / 1170 62. 46 LOGARITHMIC SINES, TANGENTS, ETC. TABLE IH. 280 _ ____ _ _ ____________1510 I Sine. ID. Cosine. D. Tang. D. Cotang. / o 9671o60o 396 9'94-535 112 9-725674 5o08 10274326 60 I 671847 395 945868 112 725979 5o8 274021 5 2 672084 395 9458oo00 112 726284 507 273716 5 3 672321 395 945733 112 726588 507 273412 57 4 672558 395 945666 112 726892 507 273108 56 5 672795 394 945598 112 727197 507 272803 55 6 673032 394 945531 112 727501 507 272499 54 7 673268 394 945464 113 727805 5o6 27219 53 8 673505 394 945396 113 728109 5o6 27189 52 9 673741 393 945328 113 728412 506 271588 51 io 673977 393 945261 113 728716 5o6 271284 50 II 9-674213 393 9945193 113 9.729020 5o6 10-270980 4 12 674448 392 945125 113 729323 505 270677 4 13' 674684 392 945058 113 729626 505 270374 47 143 674919 392 944-990 1i3 729929 o 270071 46 15 675.155 392 944922 113 730233 5o5 269767 45 16 675390 391 944854 113 730535 505 269465 44 17 675624 391 944786 113 730838 504 269162 43 I8 675859 39I 944718 113. 73114I 504 268859 42 19 676094 391 944650 113 731444 504 268556 41 20 676328 390 944582 114 731746 504 268254 40 21 9.676562 390 9-944514'14 9-732048 5o4, 10267952 3 22 676796 390 944446 114 732351 5o3 267649 3 23 630 390 944377 4 732653 503 267347 3 24 677264 389 944309 114 732955 503 267045 36 25 677498 389 944241' 11/4 733257 5o3 266743 35 26 677731 389 944172 114 733558 5o3 266442 34 27 677964 388 944104 1I4 733860 502 266140 33 28 678197 388 944036 114 734162 502 265838 32 29 678430 388 943967 114 734463 502 265537 3i 3o 67.8663 388.943899 114 734764 502 265236 3o 31 9.678895 387 9.943830 114 9-735066 502 10-264934 2 32 679128 387 943761 114 735367 502 264633 2 33 679360 387 943693 15 735668 50I 264332 27 34 679592 387 943624 115 735969 50o 26403I 26 35 679824 386. 943555 15 736269 501 263731 25 36 68oo56 386 943486 115 736570 501 263430 24 37 680288 386 943417 115 736870 501o 263130. 23 38 680519 385 943348 115 737171 500oo 262829 22 39 680750 385 943279 5 737471 500oo 262529 21 40 680982 385. 943210 115 737771 500 262229 20 41 9.681213 385 9.943141 15 9-738071 5o00 Io-26929 42 681443 384 943072 115 738371 500oo 26629 43 681674 384 943003 115 738671 499. 261329 17 44 681905 384 942934 115 738971 499 261029 i6 45 682135 384 942864 115 739271 499 260729 15 46 682365 383 942795 116 739570 499 260430 14 47 682595 383 942726 116 739870 499 260130 13 48 682825 383 942656 116 740169 499 259831 12 49 683055 383" 942587 116 740468 498 259532 II 50 683284 382 942517 116 740767 498 259233 io 51 9-683514 3872 9-942448 116 9-741366 498 10258934 52 6837493 382 942378 116 7465 498 258635 8 53 683972 382 942308 116 741664 498 258336 7 54 684201 38i 942239 1i6 741962 497 258038 6 55 684430 381 942169 116 742261 497 257739 5 56 684658 38i 942099 116 742559 497 257441 4 57 684887 380 942029 116 742858 497 2571 I2 3 58 685i15 38o 941959 116 743156 497 256844 2 59 685343 38o 941889 117 743454 497 256546 i 60 685571 38o 941819 117 743752 496 25'248 0 CO S118~ ine D 3 ie D C3n / D Tn 610 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 41 29~ 1500 Sine. D. Cosine. D. Tang. D. Cotang. t 0 90685571 380 9.94I81i 117 9743752 496 10-256248 60 i 685799 379 941749 117 744050 496 255950 50 2 686027 379 941679 117 744348 496 255652 58 3 686254 379 94I609 117 744645 496 255355 57 4 686482 379 941539 1I7 744943 496 255057 56 5 686709 37 94I469 117 745240 496 254760 55 6 686936 378 941398 117 745538 495 254462 54 7 687163 378 941328 117 745835 495 254165 53 687389 378 941258 117 74632 495. 253868 52 687616 377 94 187 117 746429 495 253571 5i io 687843 377 941117 117 746726 495 253274 50 ii 9.688069 377 9-941046 118 9.747023 494 10-252977 4 12 688295 377 940975 118 747319 494 25268 48 13 688521 376 940905 118 7476I6 494 252384 47 14 688747 376 940834 118 74793 494 252087 46 15 688972 376 940763 I18 748209 494 251791 45 i6 689I98 376 940693 118 748505 493 251495 44 17 689423 375 940622 118 748801 493 251199 43 I8 689648 375 940551 118 749097 493 250903 42 19 689873 375 940480 18 749393 493 250607 4I 20 6goo98 375 940409 118 749689 493 25031I 40 21 9.690323 374 9'940338 ii8 9 749985 493 10-250015 39 22 690548 374 940267 118 750281 492 24979 38 23 690772 374 940196 118 750576 492 249424 37 24 690996 374 940125 119 750872 492 2491I28 36 25 691220 373 940054 119 751167 492 248833 35 26 69144 4 373 939982 119 751462 492 248538 34 27 69I668 373 939 11 119 751757 492 248243 33 28 69892 373 939840 119 752052 49I 247948 32 29 692115 372 939768. 119 752347 491 247653 31 30 692339 372 939697 119 752642 491 247358 30 31 9.692562 372 9.939625 119 9*752937 49I I10247063 29 32 692785 371 939554 119 753231 49I 246769 28 33 693008 371 939482 119 753526 491 246474 27 34 693231 371 939410' I9 753820 490 246180 26 35 693453 371 939339 119 754115 490 245885 25 36 693676 370 939267 120 754409 490 24559I 24 37 693898 370 939195 120 754703 490 245297 23 38 694120 370 939123 120 754997 490 245003 22 39 694342 370 939052 I20 755291 490 244709 21 40 694564 369 938980 120 755585 489 244415 20 41 9.694786 369 9-938908 I20 9-755878 489 10-244I22 19 42 695007 369 938836 I20 756172 489 243828 18 43 695229 369 938763 120 756465 489 243535 17 44 695450 368 938691 120 756759 489 243241 i6 45 69567I 368 938619 120 757052 489 242948 I5 46 695892 368 938547 I20 757345 488 242655 I4 47 696113 368 938475 120 757638 488 242362 13 48 696334 367 938402 121 75793i 488 242069 12 49 696554 367 938330 121 758224 488 241776 ii 50 696775 367' 938258 12I 758517 488 241483 10 51 9. 966995 367 9-938I85 121 9-758810 488 10241190 9 52 697215 366 938113 121 759102 487 240898 8 53 697435 366 938040 121 759395 487 240605 7 54 697654 366 937967 I21 759687 487 2403I3 6 55 1 697874 366 937895 121 759979 487 240021 5 56 698094 365 937822 121 760272 487 239728 4 57 698313 365 937749 121 760564 487 239436 3 58 698532 365 937676 I21 760856 486 239144 2 59 698751 365 937604 121 761I48 486 238852 I 60o 698970 364 93753i 121 761439 486 23856I 0 / Cosine. D. Sine. D. Cotang. D. Tang. / 698 9 8 7 573 _9'60 48 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 300 _1490 _ Sine. D. Cosine. D. Tang. D. Cotang. / o 9.698970 364 9.937531 121 9*76I439 486 o1023856i 60 I 699189 364 937458 122 761731 486 238269 59 2 699407 364 937385 122 762023 486 237977 58 3 699626 364 937312 I22 762314 486 237686 57 4 699844 363 937238 122 762606 485 237394 56 5 700062 363 93765 122 762897 485 237o03 55 6 700280 363 937092 122 763188 485 236812 54 700498 363 937019 122 763479 485 236521 53 700716 363 936946 122 763770 485 236230 52 9 700933 362 936 72 122 76406 485 235939 5i o1 70115I 362 936799 122 764352 484 235648 50 II 9-701368 362 9.936725 122 9.764643 484 10-235357 49 12 701585 362 936652 123 764933 484 235067 48 13 701802 361 936578 123 765224 484 234776 47 14 702019 36i 936505 123 765514 484 234486 46 i5 702236 36i 93643 I 123 765805 484 234195 45 16 702452 36i 936357 123 766095 484 233905 44 I7 702669 36o 936284 123 766385 483 233615 43 18 702883 360 936210 123 766675 483 233325 42 I9 703o10 360 936136 123 766965 483 233035 41 20 703317 360 936062 123 767255 483 232745 40 21 9.703533 359 9.935988 123 9.767545 483 10.232455 39 22 703749 359 9359I4 I23 767834 483 232166 38 23 703964 359 935840 123 768124 482 231876 37 24 704170 359 935766 124 7684I4 482 231586 36 25 704395 359 935692 124 768703 482 231297 35 26 704610 358 935618 124 768992 482 23o008 34 27 704825 358 935543 124 769281 482 2307I9 33 28 705040 358 935469 124 769571 482 230429 32 29 705254 358 935395 124 769860 481 230140 31 30 705469 357 935320 124 770148 481 229852 30 31 9'705683 357 9.935246 124 9'770437 481 1IO229563 29 32 705898 357 935171 124 770726 481 229274 25 33 706112 357 935097 124 771015 481 228985 27 34 706326 356 935022 124 77I303 48I 228697 26 35 706539 356 934948 124 771592 48 228408 25 36 706753 356 934873 124 771880 480 228120 24 37 706967 356 934798 125 772168 480 227832 23 38 707180 355 934723 125 772457 480 227543 22 39 707393 355 934649 125 772745 480 227255 21 40 707606 355 934574 125 773033 480 226967 20 41 9-707819 355 9.934499 125 9-773321 480 o10226679 19 42 708032 354 934424 125 773608 479 226392 18 43 708245 354 934349 125 773896 479 226104 17 44 708458 354 934274 125 774184 479 2258I6 16 45 708670 354 934199 125 774471 479 225529 15 46 708882 353 934123 I25 774759 479 225247 14 47 709094 353 934048 125 775046 479 224954 13 48 709306 353 933973 125 775333 479 224667 12 49 709518 353 933898 126 775621 478 224379 I 50 709730 353 933822 126 775908 478 224092 10 5I 9-709941 352 9'933747 I26 9.776I95 478 10-223805 9 52 710153 352 93367I 126 776482 478 223518 53 7I0364 352 933596 126 776768 478 223232 7 54 710575 352 933520 126 777055 478 222945 6 55 70o786 351 933445 126 777342 478 222658 5 56 710997 351 933369 126 777628 477 222372 4 57 711208 35i 933293 126 777915 477 222085 3 58 711419 35i 933217 126 778201 477 22I799 2 59 711629 350 933141 126 778488 477 221512 I 6o 711839 350 933066 126 778774 477 221226 o Cosine. D. Sine. D. Cotang. D. Tang. t 1200~.._.o5.. I 690 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 49 810 1480! Sine. D. Cosine. D. Tang. D. Cotang. I o 9'7I-839 350 9-933066 126 9.778774 477 10.22I226 60 I 712050 350 932990 127 77906o 477 220940 59 2 712260 350 932914 127 779346 476 220654 58 3 712469 349 932838 127 779632 476 220368 57 4 712679 349 932762 127 7799I8 476 220082 56 5 712889 349 932685 127 780203 476 219797 55 6 713098 549 932609 127 780489 476 219511 54 713308 349 932533 127 780775 476 219225 53 713517 348 932457 127 78I060 476 218940 52 9 713726 348 932380 127 78134.6 475 218654 51 o1 713935 348 932304 127 781631 475 218369 50 ii 9.714144 348 9.932228 I27 9.781916 475 10-218084 49 12 714352 347 93215I 127 782201 475 217799 4 13 714561 347 932075 128 782486 475 217514 47 14 714769 347 931998 128 782771 475 217229 46 15 714978 347 931921 128 783056 475 216944 45 I6 715186 347 931845 128 783341 475 216659 44 17 715394 346 931768 128 783626 474 216374' 43 i8 715602 346 931691 128 783910 474 216090 42 19 715809 346 931614 128 784195 474 215805 41 20 7 i6017 346 931537 128 784479 474 215521 40 21 9-7I6224 345 9.931460 I28 9-784764 474 10-215236 3 22 716432 345 931383 128 785048 474 214952 38 23 716639 345 931306 128 785332 473 214668 37 24 716846 345 931229 129 785616 473 214384 36 25 717053 345 931152 129 785900 473 214100 35 26 717259 344 931075 I29 786184 473 2138I6 34 27 717466 344 930998 129 786468 473 213532 33 28 717673 344 930921 I29 786752 473 213248 32 29 717879 344 930843 129 787036 473 212964 3I 30 71808 343 930766 129 7873I9 472 2I2681 3o 31 9-718291 343 9.930688 129 9-787603 472 10-212397 29 32 718497 343 930611 129 787886 472 212114 28 33 718703 343 930533 129 788170 472 211830 27 34 718909 343 930456 129 788453 472 211547 26 35 719114 342 930378 129 788736 472 211264 25 36 719320 342 930300 i30 789019 472 210981 24 37 719525 342 930223 130 789302 471 210698 23 38 719730 342 930145 i30 789585 471 210415 22 39 719935 341 930067 130 789868 471 210I32 21 40 720140 34I 929989 i30 790151 471 209849 20 41 9-720345 34I 9.929911 130 9-790434 471 10o209566 1 42 720549 34i 929833 I30 790716 47I 209284 1I 43 720754 340 929755 i30 790999 47I 209001 17 44 720958 340 929677 130 79128I 471 208719 16 45 72I162 340 929599 I30 791563 470 208437 15 46 72I366 340 929521 i30 79I846 470 208154 14 47 721570 340 929442 I30 792128 470 207872 13 48 721774 339 929364 131 79241o 470 207590 12 49 721978 339 929286 131 792692 470 207308 ii 50 722181 339 929207 131 792974 470 207026 IO 51 9-722385 339 9.929129 i31 9.793256 470 o10206744 52 722588 339 929050 13i 793538 469 206462 53 722791 338 928972 131 7938I9 469 206181 7 54 722994 338 928893 131 794101 469 205899 6 55 723197 338 928815 131 794383 469 205617 5 56 723400 338 928736 131 794664 469 205336 4 57 723603 337 928657 131 794946 469 205054 3 58 723805 337 928578 131 795227 469 204773 2 59 724007 337 928499 131 795508 468 204492 I 60 724210 337 928420 131 795789 468. 204211 o' Cosine. D. Sine. D. Cotang. D. Tang. I 1210 _680 50 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 32~0 1410 r Sine. D. Cosine. D. Tang. D. Cotang.' o 9-724210 337 9.928420 132 9.795789 468 10I204211 60 I 724412 337 928342 132 796070 468 203930 5 2 724614 336 028263 I32 79635I 468 203649 58 3 724816 336 928183 132 796632 468 203368 57 4 725017 336 928104 132 796913 468 203087 56 5 725219 336 928025 132 797I94 468 202806 55 6 725420 335 927946 I32 797474 468 202526 54 725622 335 927867 132 797755 468 202245 53 8 725823 335 927787 132 798036'467 201964 52 9 726024 335 927708 132 798316 467 20I684 51 Io 726225 335 927629 132 798596 467 20I404 50 II 9-726426 334 9.927549 I32 9-798877 467 o10201123 49 12 726626 334 927470 133 799157 467 200843 48 13 726827 334 927390 133 799437 467 200563 47 I4 727027 334 92731o 133 7997I7 467 200283 46 I5 727228 334 927231 133 799997 466 200003 45 I6 727428 333 927151 133 800277 466 199723 44 17 727628 333 927071 I33 800557 466 199443 43 i8 727828 333 926991 133 800836 466 I99I64 42 19 728027 333 926911 133 801o16 466 I98884 4, 20 728227 333 926831 133 801396 466 I98604 40 21 9-728427 332 9-92675I 133 9.801675 466 Io.198325 39 22 728626 332 926671 133 801955 466 I98045 38 23 728825 332 926591 133 802234 465 197766 37 24 729024 332 92651I 134 802513 465 197487 36 25 729223 331 926431 I34' 802792 465 197208 35 26 729422 331 92635I 134 803072 465 196928 34 2 729621 331 926270 134 80335I 465 I96649 33 2 729820 33I 926190 134 803630 465 I96370 32 29 73o008 330 926110 34 803909 465 196091 31 30 730217 330 926029 i34 804187 465 195813 30 31 9.73o415 330 9.925949 34 9.804466 464 10oI95534 2 32 7306I3 330 925868 134 804745 464 I95255 28 33 730811 330 925788 134 805023 464 194977 27 34 730oo9 329. 925707 134 805302 464 194698 26 35 731206 329 925626 i34 805580 464 I94420 25 36 731404 329 925545 I35 805859 464 194141 24 37 731602 329 925465 I35 806137 464 I93863 23 38 731799 329 925384 I35 806415 463 193585 22 39 731996 32 925303 135 806693 463 I93307 21 40 73'2193 328 925222 135 806971 463 193029 20 4I 9-732390 328 9.925141 135 9.807249 463 10-19275I 19 42 732587 328 925060 I35 807527 463 192473 18 43 732784 328 924979 135 807805 463 I92195 17 44 732980 327 924897 135 808083 463 191917 16 45 733177 327 924816 135 8o836i 463 I91639 15" 46 733373 327 924735 136 808638 462 191362 14" 47 733569 327 924654 I36 808916 462 I9I084 13 48 733765 327 924572 136 809193 462 I90807 12 49 733961 326 924491 136 809471 462 190529 II 50 734157 326 924409 i36 809748 462 190252 IO 5I 9.734353 326 9-924328 i36 9-810025 462 10.I89975 9 52 734549 326 924246 136 810302 462 I89698 8 53 734744'325 924164 136 810580 462 189420 7 54 734939 325 924083 136 810857 462 189I43 6 55 735I35 325 924001 136 811134 46I1 88866 5 56 735330 325 923919 i36 811410 461 i88590 4 57 735525 325 923837 136 8 1687 461 i883I3 3 58 7357I9 324 923755 137 811964 461 i88036 2 59 7359,4 324 923673 137 812241 461 187759 I 60 736109 324 923591 137 812517 46I 187483 o / Cosine. D. Sine. D. Cotang. D. Tang. t 1220o 6 7 _ o TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 51 830 146~ Sine. I D. Cosine. D. Tang. D. Cotang. o 9'736IO9 324 9.92359I 137 9-812517 46i 10 187483 60 I 736303 324 923509 137 812794 461 187206 5 2 736498 324 923427 I37 813070 46 186930 58 3 736692 323 923345 137 813347 460 186653 57 4 736886 323 923263 137 813623 460 186377 56 5 737080 323 923181 137 813899 460 1860oi 55 6 737274 323 923098 I37 814176 460 185824 54 7 737467 323, 923016 137 814452 460 185548 53 8 737661 322 922933 137 814728 460 185272 52 9 737855 322 92285I I37 8r5004 460 184996 51 o1 738048 322 922768 138 815280 460 184720 50 II 9.738241 322 9-922686 I38 9.815555 459 Io.i84445 4 12 738434 322 922603 138 815831 459 184169 48 13 738627 321 922520 138 816107 459 183893 47 14 I 738820 321 922438 8 8382 459 I83618 46 15 739013 321 922355 I38 816658 459 183342 45 I6 739206 321 922272 138 816933 459 183067 44 17 739398 321 922189 138 817209 459 182791 43 18 739590 320 922106 138 817484 459 182516 42 19 739783 320 922023 138 817759 459 182241 41 20 739975 320 921940 i38 818o35 458 I81965 40 21 9-740167 320 9-921857 139 9.8i83io 458 10181690 39 22 740359 320 921774 I39 8i8585 458 I81415 3 23 740550 319 921691 139 8i8860 458 I81i40 37 24 740742 319 921607 139 819135 458 I80865 36 25 740934 319 921524 139 819410 458 180590 35 26 74I125 319 921441 139 819684 458 180316 34 27 741 /316 319 921357 139 8I9959 458 18004r 33 2 741508 318 921274 139 820234 458 179766 32 29 741699 3i8 921190 139 820508 457 179492 3i 30 741889 318 921107 139 820783 457 179217 30 31 9.742080 3I8 9.921023 130 9.821057'457 1 I178943 29 32 742271 318 920939 140 821332 457 78668 28 33 742462 317 920856 140 821606 457 178394 27 34 742652 317 920772 140 82188c 457 178120 26 35 742842 3I7 920688 140 822154 457 177846 25 36 743033 317 920604 140 822429 457 177571 24 37 743223 317 920520 140 822703 457 177297 23 38 743413 316 920436 140 822977 456 177023 22 39 743602 316 920352 140 823251 456 176749 21 40 743792 3I6 920268 140 823524 456 176476 20 41 9.743982 316 9-920I84 I40 9.823798 456 10.176202 19 42 744171 3i6 920099 I40 824072 456 175928 I 43 744361 315 92001 140 824345 456 I75655 17 44 744550 3i5 919931 141 824619 456 175381 6 45 744730 3I5 919846 141 824893 456 175107 15 46 744928 315 919762 141 825166 456 174834 14 47 745117 315 9I9677 I41 825439 455 I74561 13 48 745306 3 4 919593 141 825713 455 174287 12 49 745494 314 919508 141 825986 455 174014 ii 50 745683 314 9I9424 141 826259 455 17374I io 5I 9-745871 314 9.919339 141 9-826532 455 10-173468 52 746060 314 919254 141 826805 455 173I95 53 746248 3I3 919169 141 827078 455 172922 7 54 746436 313 919085 141 827351. 455 172649 6 55 746624 313 919000 141 827624 455 172376 5 56 746812 3i3 9189 5 142 827897 454 172103 57 746999 313 918830 142 828170 454 171830 3 58 747187 312 918745 142 828442 454 171558 2 59 747374 312 918659 142 828715 454 I71285 I 60 747562 312. 98574 142 828987 454 171013 o' Cosine. D. Sine. D. Cotang. D. Tang. / 12 ________________________________6 52 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 340 145~ _ Sine. D. Cosine. D. Tang. I D. Cotang. / o 9.747562 312 991 8574 142 9-828987 454 io-I7ioI3 6o I 747749 32 918489 142 829260 454 170740 5 2 747936 312 918404 I42 829532 454 170468 58 3 748123 311 918318 142 829805 454 170195 57 4 7483Io 311 918233 142 830077 454 I69923 56 5 748497 311 918147 142 830349 453 169651 55 6 748683 311 918062 142 830621 453 169379 54 7 748870 311 917976 I43 830893 453 I69107 53 8 749056 310 917891 I43 831165 453 I68835 52 9 749243 310 917805 I43 831437 453 168563 51 io 749429 3io 917719 I43 831709 453 16829I 50 II 9.749615 310 9.917634 I43 9.83198 I 453 10168019 49 I2 749801 3io 917548 I43 832253 453 167747 48 13 749987 309 9I7462 I43 832525 453 167475 47 I4 750172 309 917376 I43 832796 453 167204 46 I5 750358 309 917290 143 833068 452 166932 45 I6 750543 309 917204 I43 833339 452 16666I 44 17 750729 309 917I8 144 833611 452 I66389 43 I8 750914 30 917032 144 833882 452 i66ii8 42 19 751099 30 916946 I44 834154 452 165846 4I 20 751284 308 916859 I44 834425 452 165575 40 21 9.751469 308 9-96773 144 9.834696 452 IO1I65304 39 22 751654 308 916687 144 834967 452 i65033 38 23 75I839 308 916600 I44 835238 452 164762 37 24 752023 307 9I6514 144 835509 452 164491 36 25 752208 307 916427 I44 835780 451 164220 35 26 752392 307 916341.144 83605i 451 163949 34 27 752576 307 916254 I44 836322 45I I63678 33 28 752760 307 916167 I45 836593 451 163407 32 29 752944 306 91608I 145 836864 451 16336 31 30 753128 3o6 915994 145 837134 451 162866 30 31 9-753312 306 9-.95907 I45 9'837405 451 o10162595 29 32 753495 306 9I5820 i45 837675 451 162325 2 33 753679 306 915733 145 837946 451 162054 27 34 753862 305 9i5646 145 838216 451 161784 26 35 754046 305 915559 I45 838487 450 I61513 25 36 754229 305 915472 I45 838757 450 I61243 24 37 754412 305 915385 145 839027 450 160973 23 38 754595 305 915297 145 839297 450 I60703 22 39 754778 304 915210 145 839568 450 I60432 21 40 754960 304 915123 I46 839838 450 I60162 20 41 9-755143 304 9-.95035 i46 9.840Io8 450 1o.059892 19 42 755326 304 914948 146 840378 450 i59622 18 43 755508 304 914860 146 840648 450 159352 17 44 755690 304 9I4773 146 840917 449 I59o83 16 45 755872 303 914685 146 841187 449 I588I3 15 46 756054 303 914598 i46 84I457 449 I58543 14 47 756236 303 91451o i46 841727 449 158273 13 48 7564i8 303 914422 146 841996 449 158004 I2 49 756600 303 914334 146 842266 449 I57734 ii 50 756782 302 914246 147 842535 449 157465 io 51 9.756963 302 9-91458 I47 9-842805 449 10-157195 52 757144 302 914070 147 843074 449 156926 53 757326 302 913982 147 843343 449 i56657 7 -54 757507 302 913894'47 843612 449 i56388 6 55 757688 3o0 913806 147 843882 448 15618 5 56 757869 30I 913718 i47 844I5I'448 i55849 4 57 758o050o 3o 913630 147 844420 448 i55580 3 58 758230 301 913541 147 844689 448 1553I1 I 59 7584 I 301 913453 147 844958 448 155042 I 60 758591 301 913365 147 845227 448 154773 Cosine. D. Sine. D. Cotang. D. Tang. 1240 _____ TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. _ 3 860 ____ 1440 f Sine. D. ]Cosine. D. Tang. D. Cotang. 0 9'758591 301 9913365 147 9.845227 448 o10'54773 6o 1 758772 3oo 913276 147 845496 448 154504 50 2 758952 3oo 913187 148 845764 448 154236 58 3 759132 3oo 913099 I48 846033 448 153967 57 4 759312 3oo 913010 148 84.6302 448 153698 56 5 759492 3oo 912922 I48 846570 447 i53430 55 6 759672 299 912833 148 846839 447 i53i6i 54 759852 299, 912744 148 847108 447 152892 53 760031 299 912655 148 847376 447 152624 52 9 760211 299 912566 i48 847644 447 152356 51 10 760390 299 912477 148 847913 447 152087 50 ii 9.760569 298 9.912388 I48 9.848i81 447 io-151819 40 12 760748 298 912299 149 848449 447 151551 48. 13 760927 298 912210 I49 8487I7 447 151283 47 14 76 6 298 912121 149 848986 447 15ioi4 46 15 761285 298 912031 149 849254 447 150746 45 i6 761464 298 911942 149 849522 447 150478 44 17 761642 297 911853 149 849790 446 150210 43 i8 761821 297 911763 149 850057 446 149943 42 19 761999 297 911674 149 850325 446 149675 41 20 762177 297 9i1584 149 85o0593 446 149407 40 21 9.762356 297 9911495 149 9.850861 446 10-1O491 39 22 762534 296 911405 149 851129 446 14887I 38 23 762712 296 9113i5 I50 851396 446 I486o4 37 24 762889 296 911226 150 851664 446 148336 36 25 763067 296 911136 i5o 851931 446 148069 35 26 763245 296 911046 150 o 852199 446 147801 34 27 763422 296 910056 50o 852466 446 147534 33 28 763600 295 910866 150 852733 445 147267 32 29 763777 29 910776 i50 85300oo 445 146999 3i 30 76394 295 910o686 150 853268 445 146732 30 31 9.764131 295 9-9o596 150 9.853535 445 o10.44665 29 32 764308 295 9105o6 i50 853802 445 146198 28 33 764485 294 910415 i50 854o69 445 i45931 27 34 764662 294 91o325 151 854336 445 145664 26 35 764838 294 910235 15I 854603 445 145397 25 36 765015 294 910144 151 854870 445 14510c 24 37 765191 294 910054 151 855137 445 144863 23 38 765367 294 909963 151 855404 445 144596 22 39 765544 293 909873 151 855671 444 144329 21 40 765720 293 909782 151 855938 444 144062 20 4, 9.765896 293 9.909691 151 9-856204 444 10.143796 1 42 766072 293 909601 151 856471 444 143529 18 43 766247' 293 gog909510 151 856737 444 143263 17 44 766423 293 909419 i5i 857004 444 142996 16 45 766598 292 909328 152 857270 444 142730 15 46 766774 292 909237 152 857573 444. I42463 14 47 766949 292 909146 152 8578& 444 142197 13 48 767124 292 909055 152 858069 444 I4193 12 49 767300 292 908964 152 858336 444 141664 11 50 767475 291 908873 152 858602 443 141398 o10 5i 9.767649 291 9-908781 152 9.858868 443 10I141132 52 767824 291 908690 152 859134 443 140866 53 767999 291 908599 152 859400 443 140600 7 54 768173 291 908507 152 859666 443 14o334 6 55 768348 290 908416 153 859932 443 140068 5 56 768522 290 908324 153 860198 443 139802 4 57 768697 290 908233 153 860464 443 139536 3 58 768871 290 908141 I53 860730 443 139270 2 59 769o45 290.908049 i53 860995 443 i3005 60 769219 290 907958 i53 861261 443 138739 Cosine. D D Sine. D. Cotan-. D. Tang. 1250'_640 54 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II 860 1794 oy u-u, UUJ 4 _1430 I Sine. D. Cosine. D. Tang. D. Cotang. I o 9-769213 290 9-907?58 153 9'86l25l 443 IO-138739 60 i 76939 289 907 66 153 861527 443 138473 59 2 769566 289 ~907774 153 861792 442 138208 5 3 769740 289 907682 153 862058 442 137942 57 4 769913 289 907590 153 862323 442 137677 56 5 770087 289 907498 I53 862589 442 13741I 55 6 770260 288 907406 153 862854 442 137146 54 7 770433 288 907314 154 863 19 442 13688i 53 8 770606 288 907222 154 863385 442 i36615 52 9 770779 288 907129 I54 863650 442 136350 51 Io 770952 288 907037 144 863915 442 136085 50 II 9-77II25 288 9-906945 I54 9.864I80 442 Io1035820 4c 12 77I298 287 906852 i54 864445 442 135555 4z 13 771470 287 906760 I54 864710 442 135290 47 14 771643 287 906667 I54 864975 441 135025 46 15 771815 287 906575 I54 865240 441 134760 45 i6 771987 287 906482 154 865505 441 I34495 44 I7 772I59 287 906389 55 865770 441 134230 43 18 772331 286 906296 155 866035 441 133965 42 I9 772503 286 906204 155 866300 441 133700 41 20 772675 286 906111 I 55 866564 441 133436 40 21 9-772847 286 9-906018 155 9.866829 441 10-33171 3 22 7730I8 286 905925 I55 867094 441 132906 3 23 773190 286 905832 155 867358 441 I32642 37 24 77336I 285 905739 155 867623 441 132377 36 25 773533 285 905645 155 867887 441 13213 35 26 773704 285 905552 155 868152 440 131848 34 27 773875 285 905459 I55 868416 440 131584 33 28 774046 285 905366 i56 868680 440 131320 32 29 774217 285 905272 I56 868945 440 i3o055 31 30 774388 284 905179 i56 869209 440 I3079I 30 31 9.774558 284 9.905085 i56 9 869473 440 10o.30527 2 32 774729 284 904992 i56 869737 440 130263 28 33 774899 284 904898 i56 870001 440 129999 27 34 775070 284 904804 156 870265 440.29735 26 35 775240 284 90471I I56 87052 440 12947I 25 36 775410 283 904617 I56 870793 440 129207 24 37 775580 283 904523 i56 871037 440 128943 23 38 775750 283 904429 57 871321 440 128679 22 39 775920 283 904335 157 871585 440 128415 21 40 776090 283 904241 157 871849 439 128151 20 4I 9.776259 283 9.904147 I57 9-872112 439 10127888 1 42 776429 282 904053 157 872376 439 127624 I8 43 776598 282 903959 157 872640 439 127360.7 44 776768 282 903864 157 872903 439 127097 I6 45 776937 282 903770 157 873167 439 126833 15 46 777106 282 903676 157 873430 439 126570 14 47 77275 281 903581 167 873694 439 126306 13 48 777444 281 903487 157 873927 439 126043 12 49 777613 281 903392 i58 874220 439 125780 II 50 777781 28 903298 158 874484 439 125516 10 51 9'777950 281 9'903203 I58 9'874747 439 10-125253 52 778119 281 90318 8 i5 875010 439 24990 8 53 778287 280 9030o4 I58 875273 438 124727 7 54 778455 280 902919 i58 875537 438 124463 6 55 778624 280 902824 158 875800 438 124200 5 56 778792 280 902729 58 876063 438 123937 4 57 778960 280 902634 i58 876326 438 123674 3 58 779128 280 902539 59 876589 438 123411 2 59 779295 279 902444 159 876852 438 123148 I 60 779463 279 902349 59 877114 438 122886 0' Cosine. D. Sine. D. Cotang. D. Tang. / 1260 ~ 30____________ TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 58 3'iO _________________________________________14 2 ~ Sine.. Cosine. D. Tang. D. Cotang. 1 o 9-779463 279 9.902349 159 9.8771J4 438 10I122886 60 I 77963I 279 902253 159 877377 438 122623 5 2 779798 279 902158 159 877640 438 I22360 58 3 779966 279 902063 59 877903 438 122097 57 4 780133 279 901967 I59 878165 438 121835 56 5 780300 278 901872 159 878428 438 121572 55 6 780467 278 90I776 159 878691 438 121309 54 780634 278. 90o68I 159 878953 437 121047 53 8 780801 278 901585 159 879216 437 120784 52 0 780968 278 901490 I59 879478 437 120522.5I Io 781134 278 901394 i60 879741 437 120259 50 11 9.78130I.277 9-901298 60o 9.880003 437 10-II9997 49 12 781468 277 901202 i60 880265 437 119735 48 I3 781634 277 901106 160 880528 437 119472 47 14 781800 277 90OI0 i60 880790 437 119210 46 I5 781966 277 900914 I60 88o 52 437 11 948 45 i6 782132 277 900818 i60 881314 437 II8686 44 17 782298 276 900722 i60 881577 437 118423 43 I8 782464 276 900626 160 88I839 437 ii816I 42 19 782630 276 900529 60 882101 437 17899 41 20 782796 276 900433 I6I 882363 436 117637 40 21 9-78296I 276 9-900337 161 9-882625 436 10 117375 39 22 783127 276 900240 I16 882887 436 117113 38 23 783292. 275 900144 i6I 883148 436 116852 37 24 783458 275 900047 I61 8834 1 436 16590 36 25 783623 275 89995I 16I 883672 436 116328 35 26 783788 275 899854 161 883934 436 ii6o66 34 27 783953 275 899757 161 884196 436 115804 33 28 784118 275 899660 161 884457 436 15543 32 29 784282 274 899564 161 884719 436 115281 31 30 784447 274 899467 162 884980 436 115020 30 3I 9-784612 274 9.899370 I62 9.885242 436 1io114758 2 32 784776 274 899273 162 885504 436 114496 28 33 784941 274 899176 162 885765 436 114235 27 34 785105 274 899078 162 886026 436 13974 26 35 785269 273 89898981 162 886288 436 113712 25 36 785433 273 898884 162 886549 435 II345i 24 37 785597 273 898787 I62 886811 435 113189 23 38 785761 273 898689 I62 887072 435 11 2928 22 39 785925 273 898592 162 887333 435 II2667 21 40 786089 273 898494 i63 887594 435 112406 20 4I 9.786252 272 9-898397 I63. 9-887855 435 IOII2145 I 42 786416 272 898299 i63 888116 435 I 1884 I8 43 786579 272 898202 163 888378 435 I 1622 17 44 786742 272 898104 I63 888639 435 11136i 6 45 786906 272 898006 I63 888900 435 IIIloo 5 46 787069 272 897908 163 889161 435 110839 14 47 787232 271 897810 i63 889421 435 10579 13 48 787395 271 897712 I63 889682 435 io3i8 12 49 787557 271 897614 i63 889943 435 Ii0057 II 50 787720 271 897516 i63 890204 434 109796 io 5i 9-787883 271 9.897418 i64 9 890465 434 io-o09535 9 52 788045 271 897320 164 890725 434 109275 8 53 788208 271 897222 164 890986 434 109014 7 54 788370 270 897I23 I64 89I247 434 108753 6 55 788532 270 897025 i64 891507 434 108493 5 56 788694 270 896926 164 891768 434 I o8232 4 57 788856 270 896828 164 892028 434 107972 3 58 789018 270 896729 I64 892289 434 107711 2 59 789180 270 896631 164 892549 434 o0745 I 60 789342 269 896532 i64 8928o1 434 107190 o / Cosine. D. Sine. D. Cotang. D. Tang. / 1 27~ 562~0 56 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 380 1410 Sire. D. I Cosine. D. Tang. D. Cotang. / o 9.789342 269 9.896532 I64 9.892810 434 10 107190 60 I 789504 269 896433 165 893070 434 106930 59 2 789665 269 896335 165 893331 434 106669 5 3 789827 269 896236 165 89359I 434 o06409 57 4 789988 269 896137 165 89385I 434 I06149 56 5 790149 269 896038 165 894 11 434 io5889 55 6 790310 268 895939 i65 894372 434 o5628 54 7 790471 268 895840 i65 894.632 433 io5368 53 8 790632 268 895741 i65 894892 433 1051o8 52 9 790793 268 895641 i65 895152 433 104848 51 io 790954 268 895542 i65 895412 433 0o4588 50 II 9-791II5 268 9.895443 I66 9.895672 433 10I04328 4 12 791275 267 895343 I66 895932 433 104068 13 791436 267 895244 I66 896192 433 o13808 47 I4 791596 267 895145 i66 896452 433 103548 46 15 791757 267 895045 i66 896712 433 I03288 45 i6 791917 267 894945 I66 896971 433 I03029 44 17 792077 267 894846 I66 897231 433 102769 43 I8 792237 266 894746 i66 897491 433 102509 42 19 792397 266 894646 66 897751 433 102249 41 20 792557 266 894546 i66 898010 433 11990 40 21 9-792716 266 9-894446 167 9.898270 433 10-101730 39 22 792876 266 894346 I67 898530 433 101470 38 23, 793035 266 894246 167 898789 433 101211 37 24 793195 265 894146 167 899049 432 100951 36 25 793354 265 894046 167 899308 432 100692 35 26 793514 265 893946 167 899568 432 100432 34 27 793673 265 893846 167 899827 432 100173 33 28 793832 265 893745 167 900087 432 gg9993 32 29 793991 265 893645 167 900346 432 099654 31 30 794150 264 893544 I67 900605 432 099395 30 31 9-794308 264 9-893444 I68 9.900864 432 0-099136 2 32 794467 264 893343 168 go I24 432 098876 2 33 794626 264 893243 168 90o383 432 o98617 27 34 794784 264 893142 i68 goI642 432 098358 26 35 794942 264 893041 i68 901901 432 098099 25 36 795101 264 892940 i68 902160 432 097840 24 37 795259 263 892839 i68 902420 432 097580 23 38 7954I7 263 892739 i68 902679 432 097321 22 39 795575 263 892638 i68 902938 432 097062 21 40 795733 263 892536 168 903197 431 096803 20 41 9-795891 263 9-892435 169 9.903456 431 o.o096544 19 42 796049 263 892334 169 903714 431 096286 18 43 796206 263 892233 I69 903973 431 096027 17 44 796364 262 892132 169 904232 431 095768 i6 45 796521 262 892030 169 904491 431 095509 i5 46 796679 262 891929 169 904750 43i 095250 14 47 796836 262 891827 169 905008 431 094992 13 48 796993 262 891726 169 905267 431 094733 12 49 797150 261 891624 169 905526 43I 094474 1 50 797307 261 891523 170 905785 43I 094215 10 51 9-797464 26I 9-891421 170 9906043 43I 10o.93957 52 797621 261 891319 170 906302 431 093698 53 797777 26I 891217 I70 906560 43I 093440 7 54 797934 26I 891I 5 70 906819 43I 093181 6 55 798091 261 891013 170 907077 43 092923 5 56 798247 261 890911 170 907336 43I 092664 4 57 798403 260 890809 170 907594 43i 092406 3 58 798560 260 890707 170 907853 431 092147 2 59 798716 260 890605 170 9081II 430 I91889 I 60 798872 260 890503 170 908369 430 091631 o Cosine. D. Sine. D. Cotang. D. Tang. / 128~0 8910Io TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 57 390 140~ _ Sine. D. Cosine. D. Tang. D. Cotang.' 0 9.798872 260 9g89o503 I70 9-908369 430o io*09i63i 60 1 799028 260 890400 171 908628 430 091372 59 2 799184 260 890298 171 908886 430 091114 58 3 799339 259 890o95 171 909144 430 ogo856 57 4 799495 259 890093 171 909402 430 ogo598 56 5 799651 259 889990 171 909660 430 ogo340 55 6 799806 259 889888 171 909918 430 ogoo82 54 7 799962 259 889785 171 91077 430 089823 53 8 800oo 7 259 889682 I71 910435 43o0 o8565 52 9 800272 258 889579 171 910693 430 089307 51 10 800427 258 889477 171 91095I 430 089049 50 II 9.800582 258 9.889374 172 -9911209 430 10-088791 4a 12 800737 258 889271 172 g11467 430 o88533 4 13 800892 258 889168 172 911725 430 088275 47 I4 801047 258 889064 172 911982 430 o880o8 46 15 801201 258 888961 172 912240 430 087760 45 I6 80o356 257 888858 172 912498 430 087502 44 17 8o0151 257 888755 172 912756 430 087244 43 18 801665 257 88865I 172 913014 429 086986 42 19 801819 257 888548 172 913271 429 086729 41 20 8017 3 257 888444 I73 9I3529 429 08647 40 21 9.802128 257 9 888341 173 9'913787 429 IO086213 3 22 802282 256 888237 173 914044 429 085956 38 23 802436 256 888134 173 g14302 429 085698 37 24 802589 256 888030 173 914560 429 085440 36 25 802743 256 887926 173 914817 429 o85183 35 26 802807 256 887822 173 915075 429 084925 34 27 803050 256 887718 173 915332 429 o84668 33 28 803204 256 887614 173 915590 429 08441o 32 29 803357 255 887510 173 915847 429 084153 31 30 803511 255 887406 174 916104 429 083896 30 31 9. 803664 255 9.887302 174 9-916362 429 io.o83638 29 32 803817 255 887198 174 916619 429 o83381 2 33 803970 255 887093 174 916877 429 o83123 27 34 804123 255 886989 174 917I34 429 082866 26 35 804276 254 886885 74 917391 429 082609 25 36 804428 254 886780 174 917648 429 082352 24 37 8o458J 254 886676 I74 917906 42 o082094 23 38 804734 254 886571 174 918163 42 081837 22 39 804886 254 886466 174 918420 428 o8i580 21 40 805039 254 886362 ~75 918677 428 081323 20 41 9.80519I 254 9.886257 175 9-918934 428 Io.o8o166 I 42 805343 253 886152 75 9191911 428 o08809 8 43 805495 253 886047 175 919448 428 080552 17 44 805647 253 885942 175 919705 428 o80295 I6 45 805799 253 885837 175 919962 428 o80038 15 46 80595I 253 885732 175 920219 428 07978. 14 47 806103 253 885627 175 920476 428 079524 13 48 806254 253 885522 175 920733 428 079267 12 49 806406 252 885416 175 920990 428. 079010 II 50 806557 252 885311 176 921247 423 07753 10 51 9.806709 252 9-885205 176 9.921503 428 10-078497 52 80686o 252 8851oo 176 921760 428 078240 53 807011 252 884994 176 922017 428 077983 7 54 807163 252 884889 76 922274 428 077726 6 55 807314 252 884783 176 922530 428 077470 5 56 807465 251 884677 176 922787 428 077213 4 57 807615 251 884572 176 923044 428 076956 3 58 807766 25I 884466 176 923300 428 076700 2 59 807917 251 884360 176 923557 427 076443 I 60 808067 251 884254 177 923814 427 076186 o / Cosine. D. Sine. D. Cotang. D. Tang. 129-'0 ___________i'___, 0 68 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 400 139~ I Sine. D. Cosine. D. Tang. D. Cotang. o 9.808067 251 9.884254 177 9-923814 427 Io0o76186 60 i 808218 251 884148 177 924070 427 075930 59 2 808368 251 884042 177 924327 427 075673 58 3 808519 250 883936 177 924583 427 075417 57 4 808669 250 883829 177 924840 427 075160 56 5 808819 250 883723 177 925096 427 074904 55 6 808969 250 883617 I77 925352 427 074648 54 7 809119 250 8835io 177 925609 427 074391 53 8 809269 250 883404 177 925865 427 074135 52 9 809419 249 883297 178 926122 427 073878 5i o1 809569 249 883191 178 926378 427 073622 50 II 9-809718 249 9.883084 178 9.926634 427 10-073366 4 12 809868 249 882977 178 926890 427 o73110 48 13 810017 249 882871 178 927147 427 072853 47 14 810167 249 882764 178 927403 427 072597 46 15 8I03i6 248 882657 I78 927659 427 o72341 45 I6 810465 248 882550 178 92791I 427 072085 44 I7 8o0614 248 882443 178 928171 427 071829 43 I8 810763 248 882336 179 928427 427 071573 42 19 810912 248 882229 179 928684 427 071316 4I 20 81Io6I 248 882121 I79 928940 427 071060 40 21 9'811210 248 9-8820I4 179 9-929196 427 10o070804 3 22 811358 247 881907 I79 929452 427 070548 38 23 811507 247 881799 179 929708 427 070292 37 24 8II655 247 881692 179 029964 426 070036 36 25 811804 247 881584 179 930220 426 069780 35 26 811952 247 881477 179 930475 426 069525 34 27 812100 247 881369 179 930731 426 069269 33 28 812248 247 881261 I80 930987 426 06q903 32 29 812396 246 881i53 I80 931243 426 o68757 31 30 812544 246 88o146 180 931499 426 068501 30 3I 9812692 246 9-880938 i8o 9-931755 426 Io1068245 29 32 812840 246 880830 180 932010 426 067990 2 33 8I2988 246 880722 I80 932266 426 067734 27 34 8i3135 246 88o063 I80 932522 426 067478 26 35 813283 246 880505 180 932778 426 067222 25 36 8I3430 245 880397 I80 933033 426 066967 24 37 835-78 245 880289 r8i 933289 426 o66711 23 38 813725 245 880180 181 933545 426 066455 22 39 813872 245 880072 181 933800 426 066200 21 40 8140g1 245 879963 181 934056 426 065944 20 4I 9.8I4166 245 9.879855 I8I 9.934311 426 o0-065689 1 42 814313 245 879746 18I 934567 426 065433 18 43 814460 244 879637 i8I 934822 426 065178 17 44 814607 244 879529 18I 935078 426 064922 i6 45 814753 244 879420 181 935333 426 064667 15 46 814900 244 879311 181 935589 426 o64411 14 47 815046 244 879202 182 935844 426 064156 13 48 8i5193 244 879093 182 936I00 426 063900 12 49 815339 244 878984 I82 936355 426 063645 ii 50 815485 243 878875 182 936611 426 063389 Io 51 9.8I563i 243 9-878766 182 9-936866 425 Io-o6334 9 52 815778 243 878656 182 937121 425 062879 8 53 815924 243 878547 182 937377 425 062623 7 54 8I6069 243 878438 182 937632 425 062368 6 55 8I6215 243 878328 182 937887 425 062113 5 56 8i636i 243 878219 i83 938142 425 o61858 4 57 816507 242 878109 i83 938398 425 o61602 3 58 816652 242 877999 183 938653 425 061347 2 59 8I6798 242 877890 I83 938908 425 o06192 I 60 816943 242 877780 183 939163 425 060837 o / Cosine. D. Sine. D. Cotang. D. Tang. 18[0~0 490 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 59 410 1380 / Sine. D. Cosine. D. Tang. D. Cotang. I o 9.816943 242 9.877780 i83 9.939I63 425 I oo6d837 60 I 817088 242 877670 183 939418 425 060582 59 2 817233 242 877560 i83 939673 425 060327 58 3 817379 242 877450 I83 939928 425 060072 57 4 817524 241 877340 i83 940183 425 059817 56 5 817668 241 877230 I84 940439 425 05956I 55 6 817813 241 877120 184 940694 425 o59306 54 7 8i7958 241 8770o0 184 940949 425 o5g5I 53 8 8I8103 241 876899 I84 941204 425 o58796 52 9 818247 241 876789 I84 94I459 425 058541 5I 10 818392 241 876678 84 941713 425 058287 50 II 9.818536 240 9.876568 I84 9-941968 425 o.o058032 49 12 8I868i 240 876457 I84 942223 425 05777 48 13 8I8825 240 876347 I84 942478 425 057522 47 14 818969 240 876236 I85 942733 425 057267 46 15 819113 240 876125 I85 942988 425 057012 45 16 819257 240 8760I4 185 943243 425 056757 44 17 81940I 240 875qo4 i85 943498 425 056502 43 18 8i9545 239 875793 185 943752 425 056248 42 19 819689 239 875682 185 944007 425 055993 41 20 819832 239 875571 185 944262 425 055738 40 21 9.819976 239 9 875459 185 9 9445I7 425 o 0055483 39 22 820120 239 875348 185 944771 424 055229 38 23 820263 239 875237 i85 945026 424 o54974 37 24 820406 239 875126 186 945281 424 054719 36 25 820550 238 875014 I86 945535 424 054463 35 26 820693 238 874903 186 945790 424 054210 34 27 820836 238 874791 i86 946045 424 053955 33 28 820979 238 874680 i86 946299 424 053701 32 29 821122 238 874568 i86 946554 424 053446 3i 30 821265 238 874456 186 946808 424 053192 30 31 9.821407 238 9.874344 186 9-947063 424 I10052937 29 32 821550 238 874232 187 9473I8 424 052682 28 33 821693 237 874121 187 947572 424 052428 27 34 821835 237 874009 187 947827 424 052173 26 35 821977 237 873896 187 948081 424 05919g 25 36 822120 237 873784 87 948335 424 o5i665 24 37 822262 237 873672 I87 948590 424 o5141o 23 38 822404 237 873560 187 948844 424 051156 22 39 822546 237 873448 187 949099 424 050901 21 40 822688 236 873335 187 949333 4i24 050647 20 4I 9-822830 236 9-873223 187 9-949608 424 10o.50392 19 42 822972 236 87310o 188 949862 424 050o38 18 43 823114 236 872998 88 9501o6 424 049884 17 44 823255 236 872.85 i88 950371 424 049629 i6 45 823397 236 872772 188 950625 424 049375 I5 46 823539 236 873659 i88 950879 424 049121 14 47'823680 235 872547 188 95r133 424 o48867 13 48 823821 235 872434 I88 95I388 424 0486I2 12 49 823963 235 872321 I88 951642 424 o48358 ii 50 824104 235 872208 188 95I896 424 048104 10 5i 9.824245 235 9-872095 189 9952I50 424 Io'o47850 9 52 824386 235 871981 189 952405 424 047595 53 824527 235 871868 189 952659 424 047341 7 54 824668 234 871755 189 952913 424 047087 6 55 824808 234 871641 189 953i67 423 046833 5 56 824949 234 871528 189 953421 423 046579 4 57 825090 234 871414 189 953675 423 046325 3 58 825230 234 871301 189 953929 423 046071 2 59 825371 234 871187 189 954i83 423 045817 i 60 8255 1 234 871073 190 954437 423 045563 o' Cosine. D. Sine. 1. Cotang. D. Tang. 131~ 480 60 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 420 __1370 Sine. D. Cosine. D. Tang. D. Cotang. o 9.825511 234 9 871073. 90 9'954437 423 Io-o45563 60 I 825651 233 870960 190 954691 423 045309 59 2 825791 233 870846 I90 954946 423 045054 58 3 825931 233 870732 190 9552o0 423 04480o 57 4 826071 233 870618 190 955454 423 044546 56 5 826211 233 870504 190 955708 423 044292 55 6 82635i 233 870390 190 95596I 423 044039 54 7 826491 233 870076 190 956215 423 043785 53 8 826631 233 87016I 190 956469 423 04353i 52 9 826770 232 870047 191 956723 423 043277 51 10 826910 232 869933 191 956977 423 043023 5 II 9-827049 232 9.869818 I9I 9.957231 423 10-042769 49 I2 827189 232 869704 191 957485 423 042515 48 13 827328 232 869589 191 957739 423 042261 47 14 827467 232 869474 19I 957993 423 042007 46 15 827606 232 869360 191 958247 423 04I753 45 16 827745 232 869245 191 958500 423 04150o 44 17 827884 231 869130 191 958754 423 041246 43 18 828023 231 869015 I92 959008 423 o40992 42 19 828162 23I 868900 192 959262 423 040738 4I 20 828301 231 868785 192 959516 423 040484 40 2I 9.828439 231 9.868670 192 9-959769 423 10-040231 3 22 828578 231 868555 192 960023 423 039977 38 23 828716 231 868440 192 960277 423 639723 37 24 828855 230 868324 192 960530 423 039470 36 25 828993 230 868209 I92 960784 423 039216 35 26 829131 230 868093 192 961038 423 038962 34 27 829269 230 867978 193 96I292 423 038708 33 2 829407 230 867862 193 961545 423 038455 32 29 829545 230 867747 193 96I799 423 03820I 31 30 829683 230 867631 193 962052 423 037948 30 31 9829821 229 9.8675I5 193 9-962306 423 Io037694 29 32 829959 229 867399 193 962560 423 037440 2 33 830097 229 86728 93 3 962813 423 37187 27 34 830234 229 867I67 I93 963067 423 036933 26 35 830372 229 86705I I93 963320 423 o36680 25 36 830509 229 866935 194 963574 423 036426 24 37 830646 229 866'89 I94 963828 423 036I72 23 38 830784 229 866703 194 964081 423 035919 22 39 830921 228 866586 194 964335 423 035665 21 40 831058 228 866470 194 964588 422 0354I2 20 41 9.831195 228 9.866353 194 9-964842 422 1IO035i58 19 42 83I332 228 866237 194 965095 422 034905 18 43 831469 228 866120 194 965349 422 o3465i 17 44 83i606 228 866004 195 965602 422 034398 i6 45 831742 228 865887 195 965855 422 o34I45 15 46 831879 228 865770 195 966109 422 03389I 14 -1 47 832015 227 865653 195 966362 422 033638 13 48 832I52 227 865536 195 9666I6 422 o33384 12 149 832288 227 865419 195 966869 422 03313 II 5o 832425 227 865302 195 967123 422 032877 10 51 9.,3256I 227 9.865i85 I95 9-967376 422 100.32624 52 8.32697 227 865068 195 967629 422 032371 53 832833 227 864950 I95 967883 422 032117 7 54 832969 226 864833 196 968136 422 03i864 6 55 833,10 226 8647I6 I96 968389 422 o31611 5 56 833241 226 864598 196 968643 422 031357 4 57 833377 226 864481 196 968896 422 o03II4 3 58 833512 226 864363 196 969149 422 03085I 2 59 833648 226 864245 96 969403 422 030597 I 60 833783 226 864I27 0q6 969656 422 030344 0' Cosine,, Sine. D. Cotang. D. Tang. 1320 470 TABLE II. LOGARITHMIC SINES, TANGENTS, ETC. 61 430 136~ / Sine. D. Cosine. D. Tang. D. Cotang. 0 9-833783 226 9.864127 196 9.969656 422 io0o30344 60 8330I9 225 86400o I96 969909o 422 03009 5 2 834054 225 863892 197 970162 422 029838 58 3 834189 225 863774 I97 970416 422 029584 57 4 834325 225 863656 197 970669 422 029331 56 5 834460 225 863538 197 970922 422 029078 55 6 834595 225 863419 197 971175 422 028825 54 7 834730 225 863301 197 971429 422 02857I 53 8 834865 225 863i83 197 971682 422 028318 52 9 834999 224 863064 197 971935 422 028065 51 Io 835134 224 862946 198 972188 422 027812 50 II 9.835269 224 9.862827 198 9-972441 422 10I027559 49 12 835403 224 862709 198 972695 422 027305 4 13 835538 224 862590 198 972948 422 027052 47 I4 835672 224 862471 198 973201 422 026799 46 15 835807 224 862353 198 973454 422 026546 45 I6 835941 224 862234 198 973707 422 026293 44 17 836075 223 862115 198 973960 422 026040 43 I8 836209 223 861996 198 974213 422 025787 42 19 836343 223 861877 198 974466 422 025534 41 20 836477 223 861758 199 974720 422 025280 40 21 9.8366 1 223 9.861638 199 9.974973 422 10025027 3 22 836745 223 861519 199 975226 422 024774 38 23 836878 223 861400 199 975479 422 024521 37 24 837012 222 861280 199 975732 422 024268 36 25 837I46 222 86161I I99 975985 422 024015 35 26 837279 222 861041 199 976238 422 023762 34 27 837412 222 860922 199 976491 422 023509 33 28 837546 222 860802 199 976744 422 023256 32 29 837679 222 860682 200 976997 422 023003 31 30 837812 222 860562 200 97j250 422 022750 30 3I 9-837945 222 9.860442 200 9-977503 422 10-022497 2 32 838078 221 860322 200 977756 422 022244 2 33 838211 221 860202 200 978009 422 021991 27 34 838344 221 860082 200 978262 422 02I738 26 35 838477 221 859962 200 978515 422 021485 25 36 8386io 221 859842 200 978768 422 021232 24 37 838742 221 859721 201 97902I 422 020979 23 38 838875 221 859601 201 979274 422 020726 22 39 839007 221 859480 201 979527 422 020473 21 40 839140 220 859360 201 979780 422 020220 20 41 9.839272 220 9.859239 201 9.980033 422 10-019967 19 42 f 839404 220 859119 201 980286 422 019714 18 43 839536 220 858998 201 980538 422 019462 17 44 839668 220 858877 201 980791 42I 019209 16 45 839800 220 858756 202 981044 421 018956 15 46 839932 220 858635 202 981297 421 018703 I4 47 840064 219 858514 202 981550 421 018450 13 48 840196 219 858393 202 981803 421 018197 12 49 840328 219 858272 202 982056 42I 017944 II 50 840459 2I9 858i51 202 982309 421 017691 10 51 9.840591 219 9-858029 202 9-982562 42I 10-017438 52 840722 219 857908 202 982814 421 I07186 53 840854 219 857786 202 983067 42I 016933 7 54 840985 219 857665 203 983320 421 o06680 6 55 841116 218 857543 203 983573 421 016427 5 56 841247 218 857422 203 983826 42I 016174 4 57 841378 218 857300 203 984079 421 015921 3 58 841509 218 857178 203 984332 421 o05668 2 59 84i640 218 857056 203 984584 421 015416 I 60 841771 218 856934 203 984837 421 015163 o Cosine. D. Sine. D. Cotang D.. Tang. / 133~ 46~ 62 LOGARITHMIC SINES, TANGENTS, ETC. TABLE II. 440 135~ / Sine. D. CnTang. D Cosine. D. Tang. D. 0 9.841771 218 9.856934 203 9.984837 421 iooi5i63 60 I 841902 218 856812 203 985090 421 014910 5o 2 842033 218 856690 204 985343 421 I 04657 58 3 842163 217 856568 204 985596 421 014404 57 4 842294 217 856446 204 985848 421 014152 56 5 842424 217 856323 204 986101 421 013899 55 6 842555 217 856201 204 986354 421 oi013646 54 842685 217 856078 204 986607 421 013393 53 8 8428I5 217 855956 204 986860 421 o03140 52 9 842946 217 855833 204 987112 421 012888 51 o1 843076 217 855711 205 987365 421 012635 50 II 9843206 216 9.855588 205 9.9876i8 421 10012382 4 2 843336 216 85546 205 987871 421 012129 4 13 843466 216 855342 205 988123 421 011877 47 14 843595 216 855219 205 988376 421 011624 46 15 843725 216 855096 205 988629 421 11371 45 i6 843855 216 854973 205 988882 42 I oi0 18 44 17 843984 216 854850 205 089134 421 I o866 43 18 8441,4 215 854727 206 989387 421 o01063 42 19 844243 215 854603 206 989640 421 oio360 41 20 844372 215 854480 206 989893 421 010107 40 21 9.844502 215 9.854356 206 9-990145 421 io-oo9855 3 22 844631 215 854233 206 990398 421 00602 3 23 844760 215 854o09 206 990651 421 009349 37 24 844889 215 853986 206 990903 421 009097 36 25 845o18 215 853862 206 991156 421 008844 35 26 845147 2I5 853738 206 991409 421 oo859I 34 27 845276 214 853614 207 991662 421 00oo8338 33 28 8454o5 214 853490 207 990914 421 oo8086 32 29 845533 214 853366 207 992167 421 007833 31 30 845662 214 853242 207 992420 421 007580 30 31 9 845790 214 9.853ii8 207 9.992672 421 10-007328 29 32 845919 214 852994 207 992925 421 007075 28 33 846047 214 852869 207 993178 421 006822 27 34 846175 214 85274 207 993431 42 00o6569 26 35 846304 214 852620 207 993683 421 006317 25 36 846432 213 82496 208 993936 421 oo6064 24 37 846560 213 852371 208 994189 421 0o5811 23 38 846688 213 852247 208 99444I 421 oo5559 22 39 8468I6 213 852122 208 994694 421 oo5306 21 40 46944 213 851997 208 994947 421 oo5053 20 4I 9-847071 213 9.851872 208 9.995199 421 10-00480I 19 42 847199 213 851747 208 995452 421 004548 18 43 847327 213 85622 208 995705 421 004295 17 44 847454 212 851497 209 995957 421 004043 16 45 847582 212 851372 209 996210 421 003790 15 46 847709 212 851246 209 996463 421 003537 I4 47 847836 212 851121 209 996715 421 oo3285 13 48 847964 212 850996 209 996968 421 003032 12 49 848091 212 850870 209 997221 421 002779 II 50 848218 212 850745 209 997473 421 002527 10 5I 9.848345 212 9-85061 209 9'997726 421 I10002274 52 848472 211 850493 210 997979 421 002021 8 53 848599 211 85o368 210 998231 421 001769 7 54 848726 211 850242 210 998484 421 0oo156 6 55 848852 211 850i6 210 998737 421 001263 5 56 848979 211 849990 210 998989 421 ooo000I 4 57 849106 211 849864 210 999242 421 000758 3 58 849232 211 849738 210 999495 421 ooo505 2 59 849359 211 8496 1 210 999747 421 000253 1 34 82 454 6o 849485 211 84948S 20 0 4 19O4* 00 O00000 3..5 6~ 99.8 0o. TABLE III,1 OF NATURAL SINES AND TANGENTS; TO EVERY DEGREE AND MINUTE OF THE QUADRANT. IF the given angle is less than 45~, look for the degrees and the title of the column, at the top of the page; and for the minutes on the left. But if the anlgle is between 45~ and go0, look for the degrees and the title of the column, at the bottozm; and for the minutes on the rig7t. The Secants and Cosecants, which are not inserted in this table, may be easily supplied. If I be divided by the cosine of an arc, the quotient will be the secant of that arc. And if I be divided by the sine, the quotient will be the cosecant. The values of the Sines and Cosines are less than a unit, and are given in decimals, although the decimal point is not printed. So also, the tangents of arcs less than 45~, and cotangents of arcs greater than 45~, are less than a unit and are expressed in decimals with the decimal point omitted. 64 NATURAL SINES AND COSINES. TABLE III. 00 10 20 30 40 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. ooooo000 Uit. 01745 99985 o3490 99939 05234 99863 06976 99756 60 I 0002 UnTit. oI774 99984 03519 99938 05263 9986I 07005 99754 5 2 oo58 Unit. i0803 99984 o354 99937 05292 99860 07034 99752 3 00087 Unit. oI832 99983 03577 99936 05321 99858 07063 99750 57 4 oo1i6 Unit. 01862 99983 o3606 99935 o5350 99857 07092 99748 56 5 ooI45 Unit. 01891 99982 03635 99934 05379 99855 07121 99746 55 6 00175 Unit. 01920 99982 o3664 99933 o5408 99854 07150 99744 54 7 oo00204 Unit. 949 99981 03693 99932 05437 99852 07179 99742 153 8 00233 Unit. 01978 99980 03723 9993I 05466 9985I 07208 99740 52 9 oo262 Unit. 02007 99980 o3752 99930 05495 99849 07237 99738 5I I 00291 Unit. 02036 99979 03781 99929 05524 99847 07266 99736 50 ii 00320 99999 02065 99979 o381o 99927 o5553 99846 07295 99734 49 I2 oo349 99999 02094 99978 o3839 99926 05582 99844 07324 9973I 48 13 oo378 99999 02123 99977 o3868 99925 o56i I 99842 o7353 99729 47 I4 00407 99999 02152 99977 03897 99924 o564o 99841 07382 99727 46 5 00436 99999 02181 99976 03926 99923 05669 99839 07411 99725 45 i6 00465 99999 02211 99976 03955 99922 05698 99838 07440 99723 44 7 00495 99999 02240 99975 03984 99921 05727 99836 07469 9972I 43 18 00524 9999 02269 99974 o40o3 999I9 o5756 99834 07498 99719 42 9 oo553 99998 02298 99974 04042 9998 05785 99833 07527 997I6 41 20 00582 99998 02327 99973 04071 999I7 o5814 99831 07556 997I4 40 21 0oo6i 99998 02356 99972 04o00 99916 o5844 99829 07585 997I2 39 22 oo640 99998 02385 99972 04129 999I5 05873 99827 07614 99710 38 23 00669 99998 02414 99971 0459 999I3 05902 99826 07643 99708 37 24 oo698 99998 02443 99970 o4i88 99912 o593I 99824 07672 99705 36 25 00727 99997 02472 99969 04217 999II o5960 99822 07701 99703 35 26 00756 99997 02501 99969 04246 99910 o5989 99821 07730 9970I 34 2 00785 99997 0 2530 9996 04275 99909 o0608 99819 07759 99699 33 28 00814 99997 02560 99967 04304 99907 06047 99817 07788 99696 32 29 oo844 99996| o2589 99966 o4333 99906 06076 99815 07817 99694 3r 30 00873 99996 026I8 99966 04362 99905 o06o5 998I3 07846 99692 30 31 00902 99996 02647 99965 04391 99904 o0634 99812 07875 99689 29 32 oo0093 99996 2676 99964 04420 99902 06163 998o1 07904 99687 28 33 00960o 9995 02705 99963 0 4449 99901 06192 99808 07933 99685 27 34 00989 99995 02734 99963 04478 999oo 06221 99806 07962 99683 26 35 OIS 99995 02 o763 99962 o0457 99898 o6250 99804 0799I 99680 25 36 01047 9999l 02792 99961 04536 99897 o6279 998o3 08020 99678 24 37 01076 99994 02821 99960 04565 99896 o6308 99801 08049 99676 23 38 oo105 99994 o2850 9999 o4594 99894 06337 99799 08078 99673 22 39 34 99994 02879 99959 46 999 99893 6366 99797 08107 99671 21 40 oi 64 99993 02908 99958 04653 99892 o6395 99795 o8I36 99668 20 41 oI193 99993 02938 99957 04682 99890 06424 99793 o8I65 99666 I9 42 01222 99993 02967 99956 04711 99889 06453 99792 08194 99664 18 43 01251 99992 02996 99955 04740 99888 06482 99790 08223 99661 17 44 01280 99992 03025 99954 04769 99886 06511 99788 08252 99659 i6 45 01309 99991 03054 99953 04798 99885 o6540 99786 0828I 99657 5 46 O338 99999 o3083 99952 04827 99883 06569 99784 I0831o 99654 14 47 o0367 99991 03112 99952 o4856 99882 0659 99782 08339 99652 13 48 o0396 99990 o3i41 9995I o4885 99881 06627 99780 o8368 99649 I2 49 01425 99990 03170 99950 04914 99879 o6656 99778 08397 99647 I 50 oI454 99989 03199 99949' 04943 99878 o6685 99776 o08426 99644 IO I 01 o483 99989 03228 99948 0 4972 99876 o6714 99774 o8455 99642 9 52 oI53 99989 o3257 99947 0500oo 99875 06743 99772 08 8484 99639 53 oi542 99988 03286 99946 o5030 99873 06773 99770 o85i3 99637 7 54 0571 99988 o33I6 99945 o5059 99872 06802 99768 o8542 99635 6 55 01600 99987 o3345 99944 o5088 99870 0683I 99766 0857 99632 5 56 01629 99987 03374 99943 o5I17 99869 o686o 99764 o8600 99630 4 57 oi658 99986 03403 99942 0o546 99867 06889 99762 o8629 99627 3 58 o0687 99986 03432 99941 05Io75 99866 06918 99760 o8658 99625 2 59 0716 99985 03461 99940 05205 99864 o6947 99758 08687 99622 I 60o 0745 99985 i3490 99939 05234 99863 06976 99756 876 996I9 0 Cosine. Si ne. Cosine. Sine. Cosine. Sine. Cosine Sine. Cosine. Sine. 890 88~ 87~ 86~ 85~ TABLE III. NATURAL SINES AND COSINES. 65 5 60 70 8~ 90 Sine Cosine. Sine. Cosine. Sine. sin ine. Sine Cosine. Sine. Cosine. 0 108716 996I9 io453 99452 12187 99255 13917 99027 15643 98769 6o i08745 99617 10482 99449 12216 99251 13946 99023 15672 98764 59 2 08774 99614 105ii 99446 12245 99248 13975 99~0 9 15701 98760 5 3 o88o3 99612 o1054o 99443 12274 99244 14004 99015 15730 98755 57 4 o08831 99609 10o569 99440 12302 99240 14o33 99o011 15758 98751 56 5 o886o 99607 10597 99437 12331 99237 14o61 99006 15787 98746 55 6 o8889 99604 10626 99434 12360 99233 14090 99002 15816 98741 54 7 08918 99602 io655 99431I 2389 99230 14119 98998 15845 98737 53 8 08947 99599 10684 99428 I2418 99226 14148 98994 15873 98732 52 9 08976 99596 10713 99424 12447 99222 14I77 98990 15902 98728 51 10 09005 99594 10742 99421 12476 99219 14205 98986 15931 98723 50 II 09034 99591 077 I 99418 12504 99215 14234 98982 15959 98718 49 12 09063 99588 io8oo00 99415 12533 99211 I 14263 98978 15988 98714 4 I3 09092 99586 10829 99412 12562 99208 14292 98973 I6017 98709 47 I4 09121 99583 io858 99409 12591 99204 I4320 98969 I6046 98704 46 15 09150 99580 10887 g99406 2620 99200 14349 98965 16074 98700 45 16 09179 99578 10916 99402 2649 99197 14378 98961 i6io3 98695 44 17 09208 99575 10945 99399 12678 99193 14407 98957 i6132 98690 43 18 09237 99572 10973 99396 12706 99189 14436 98953 i6i6o 98686 42 19 09266 99570 I1002 99393 12735 99186 14464 98948 61689 98681 41 20 09295 99567 i1o3i 99390 12764 99182 14493 98944 16218 98676 40 21 o09324/ 99564 1io6o 99386 12793 99178 14522 98940 16246 98671 39 22 09353 99562 110o89 99383 12822 99175 i455i 98936 16275 98667 38 23 09382 99559 1111.99380 12851 99171 4580 98931.16304 98662 37 24 09411 99356 11147 99377 12880 99167 146o8 98927 16333 98657 36 25 09440 99553 11176 99374 12908 99163 14637 98923 1636r 98652 35 26 0946 995 11205 99370 12937 99160 14666 98919 16390 98648 34 27 09498 99548 11234 99367 12966 99156 14695 98914 16419 98643 33 28 09527 99545 11263 99364 12995 99152 14723 98910 16447 98638 32 29 09556 99542 11291 99360 13024 99148 14752 98906 16476 98633 31 30 09585 99540 11320 99357 13053 99144 14781 98902. 16505 98629 3o 31 09614 99537 11349 99354 13o8 99141 48o10 98897 I6533 98624 2 32 09642 99534 11378 99351 i3io 9 9137 14838 98893 16562 98619 28 33 09671 9953I 11407 99347 13139 99133 14867 98889 I6591 98614 27 34 09700 99528 1436 99344 i3i68 99129/ 14896 98884 16620 98609 26 35 09729 99526 1i465 99341 13197 99125 14925 98880 16648 98604 25 36 09758 99523 11.494 99337 13226 99122 14954 98876 16677 98600 24 37 09787 99520 11523 99334 13254 99118 14982 98871 16706 98595 23 38 09816 99517 1I552 99 933 1283 99114 15011 98867 16734 98590 22 39 09845 99514 11580 99327 13312 99110 15040 98863 16763 98585 21 40 09874 99511 11609I 69324 1334i 99106 15069 98858 16792 98580 20 4' 09903 99508 11638 99320 13370 99102 15097 98854 16820 98575 19 42 09932 99506 11667 99317 I3399 99098 15126 98849 16849 98570 i8 43 o0996 99503 11696 99314 13427 99094 15155 98845 16878 98565 17 44 09990 99500 11725 9931o 13456 99091 15184 9884I 16906 98561 16 45 10019 99497 11754 99307 13485 99087 15212 98836 16935 98556 15 46 10048 99494 11783 99303 135,4 99083 15241 98832 16964 98551 14 47 10077 99491 11812 99300 13543 99079 15270 98827 16992 98546 i3 48 oo10106 99488 11840 99297'3572 99075 15299 98823 17021 98541 12 49 io135 99485 11869 99293 i36oo 99071 15327 98818 17050 98536 ii 50o 10164 99482 11898 99290 13629 99067 i5356 98814 17078 98531 io 5i'10192 99479 11927 99286 i3658 99063 I5385 98809 17107 98526 52 10221 99476 11956 99283 13687 99059 15414 98802 17136 98521 53 1025O 99473 11985 99279 13716 9905/ 15442 988oo00 17164 98516 7 54 10279 99470 12014 99276 13744 99051 15471 98796 17193 98511 6 55 io3o8 99467 12043 99272 13773 99047 15500 98791 17222 98506 5 56 10337 99464 1207I 99269 13802 99043 15529 98787 17250 98501 4 57 io366 99461 12100 99265 1383i 99039 15557 98782 17279 98496 3 58 1o395 99458 12129 99262 i3860 99035 i5586 98778 17308 98491 2 9 10424 99455 12158 99258 13889 99031 5615 98773 17336 98486 i 60o o453 99432 I2187 992 1i39,17 99027 i5643 98769 17365 98481 0 Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 840 830 82 81~ 80 e 66 NATURAL SINES AND COSINES. TABLE III. 100 11~ 120 ~ ~130 140 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 0 17365 98481 19081 98163 20791 9785 22495 97437 24192 97030 6o 17393 98476 19109 98157 20820 07809 22523 997430 24220 97023 59 2 17422 98471 19138 98152 20848 97803 22552 97424 24249 97015 58 3 17451 98466 19167 98146 20877 97797 22580 97417 24277 97008 57 4 I7479 98461I 19195 98I40 2090 977 22608 97411 24305 97001 56 5 17508 98455 19224 98135 20933 97784 22637 97404 24333 96994 55 6 17537 98450 19252 98129 20962 97778 22665 97398 24362 96987 54 7 17565 98445 19281 98124 20990 97772 22693 97391 24390 96980 53 8 17594 9844o0 19309 98118 21019 97766 22722 97384 24418 96973 52 9 17623 98435 19338 98112 21047 97760 22750 97378 24446 96966 51 1o 1765I 9843o0 9366 98107 21076 97754 22778 97371 24474 96959 50 I 17680 98/425 19395 9810I 21104 97748 22807 97365 24503 96952 49 12 17708 98420 19423 98096 2132 197742 22835 97358 24531 96945 48 13 17737 98414 19452 98090 21161 97735 22863 97351 24559 96937 47 14 17766 98409 19481 98084 21189 97729 22892 97345 24587 96930 46 15 17794 98404 19509 98079 21218 97723 22920 97338 24615 96923 45 i6 17823 98399 19538 98073 21246 97717 22948 97331 24644 96916 44 17 17852 98394 19566 98067 21275 97711 22977 97325 24672 96909 43 i8 17880 98389 19595 98061 21303 97705 23005 97318 24700 96902 42 19 17909 98383 19623 98056 21331 97698 23033 97311 24728 96894 41 20 17937 98378 I9652 98050 21360 97692 23062 97304 24756 96887 40 21 17966 98373 19680 98044 21388 97686 23090 97298 24784 96880 39 22 17995 98368 19709 98039 21417 97680 23118 97291 24813 96873 38 23 18023 98362 19737 98033 21445 97673 23146 97284 24841 96866 37 24 180o52 98357 19766 98027 21474 97667 23175 97278 24869 96858 36 25 18081 98352 19794 98021 21502 97661 23203 97271 24897 96851 35 26 18109 98347 19823 98016 21530 97655 23231 97264 24C25 96844 34 27 18138 98341 i985i 98010 21559 97648 23260 97257 24954 96837 33 28 I'866 98336 19880 98004 21587 97642 23288 97251 24982 96829 32 29 i8i95 98331 19908 97998 21616 97636 23316 97244 25010 96822 31 30 18224 98325 I9937 97992 21644 97630 2334-5 97237 25038 96815 3o 3I I8252 98320 19965 97987 21672 97623 23373 97230 25066 96807 29 32 18281 98315 19994 97981 21701 97617 23401 97223 25094 96800 29 33 18309 983o10 20022 97975 21729 97611 23429 97217 25122 96793 27 34 18338 98304 200oo51 97969 2J758 97604 23450 97210 25151 96786 26 35 18367 98299 20079 97963 21786 97598 23486 97203 25179 96778 25 36 i8395 98294 20108 97958 21814 97592 23514 97196 25207 96771 24 37 18424 98288 20136 97952 21843 97585 23542 97189 25235 96764 23 38 18452 98283 20165 97946 21871 97579 23571 97182 25263 96756 22 39 i848i 98277 20193 97940 21899 97573 23599 97176 25291 96749 21 40 i8509 98272 20222 97934 21928 97566 23627 97169 25320 96742 20 41 i8538 98267 20250 97928 21956 97560 23656 97162 25348 96734 19 42 i8567 98261 20279. 97922 21985 97553 23684 97155 25376 96727 I8 43 i8595 98256 20307 97916 22013 97547 23712 97148 25404 96719 17 44 18624 98250 2036 97910 22041 97541 23740 97141 25432 96712 16 45 18652 98245 20364 97905 22070 97534 23769 97134 25460 96705 i5 46 i8681 98240 20393 97899 22098 97528 23797 97127 25488 96697 14 47 I8710 98234 20421 97893 22126 97521 23825 97120 25516 96690 13 48 18738 98229 2045o 97887 22155 9751i 23853 97113 25545 96682 12 49 18767 98223 20478 97881 22183 975o8 23882 97106 25573 96675 Ii 50 I8795 98218 2o0507 97875 22212 97502 23910 97100 25601 96667 10 5i 18824 9822 20535 97869 2224097496 23938 97093 25629 96660 52 18852 98207 20563 97863 22268 97489 23966 97086 25657 96653 53 18881 98201 20592 97857 22297 97483' 23995 97079 25685 96645 7 54 18910 98196 20620 97851 22325 97476 24023 97072 25713 96638 6 55 i8938 98190 20649 97845 22353 97470 24051 97065 25741 96630 5 56 18967 98i85 20677 97839 22382 97463 24079 97058 25769 96623 4 57 i8995 98179 20706 97833 22410 97457 24108 97051 25798 96615 3 58 19024 98174 20734 97827 22438 97450 24136 97044 25826 96608 2 59 19052 98168 20763 97821 22467 97444 24164 97037 25854 96600 1 60 19081 98163 20791 97815 22495 97437 24192 97030 25882 06593 o Csine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 790 780 10 10 750 TABLE III. NATURAL SINES AND COSINES. 67 15o. 160g 170 180 190 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 0 25882 96593 27564 96126 29237 195630 30902 95106 32557 94552 60 1 25910 96585 27592 96118 29265 95622 30929 95097 32584 94542 5 2 25938 96578 27620 96110 29293 95613 30957 95088 32612 94533 58 3 25966 96570 27648 96102 29321 95605 30985 95079 32639 94523 57 4 25994 96562 27676 96094 29348 95596 31012 95070 32667 94514 56 5 26022 96555 27704 96086 29376 95588 3Io4o 95061 32694 94504 55 6 26050 96547 27731 96078 29404 95579 3io68 95052 32722 94495 54 7 26079 96540 27759 96070 29432 9557 3o1095 95043 32749 94485 53 8 26ro 7 96532 27787 96o062 2946o 95562 31123 95033 32777 94476 52 9 26135 96524 27815 96054 29487 95554. 3ii5i 95024 32804 94466 51 Io 26163 96517 27843 96046 29515 95545 31178 95015 32832 94457 50 Ir 26191 96509 27871 96037 29543 95536 31206 95006 32859 94447 49 12 26219 96502 27899 96029 2957I 95528 31233 94997 32887 94438 48 13 26247 96494 27927 96021 29599 95519 31261 94988 32914 94428 47 I4 26275 96486 27955 96013 29626 955ii 31289 94979 32942 94418 46 15 26303 96479 27983 96005 29654 95502 31310 94970 32969 94409 45 i6 26331 9647I 28011 95997 29682' 95493 31344 94961 32997 94399 44 17 26359 96463 28039 95989 29710 95485 31372 94952 33024 94390 43 I8 26387 96456 28067 95981 29737 95476 31399 94943 33o5i 94380 42 19 26415 96448 28o95 95972 29765 95467 31427 94933 33079 94370 41 20 26443 96440 28123 95964 29793 95459 31454 94924 33o106 94361 40 21 26471 96433 28150 95956 29821 95450 31482 94915 33134 9435i 39 22 26500 96425 28178 95948 29849 9 544 3510o 94906 33161 94342 38 23 26528 9647 28206 95940 2976 3153 94897 33189 94332 37 24 26556 96410 28234 9593I 29904 95424 3i565 94888 33216 94322 36 25 26584 96402 28262 95923 29932 95415 31593 94878 33244 94313 35 26 26612 96394 28290 95915 29960 95407 31620 94869 33271 94303 34 27 26640 96386 28318 95907 29987 95398 31648 94860 33298 94293 33 28 26668 963'79 28346 95898 300o5 95389 31675 9485i 33326 94284 32 29 26696 96371 28374 95890 30043 95380 31703 94842 33353 94274 31 3o 26724 96363 28402 95882 30071 95372 31730 94832 3338i 94264 3o 31 26752 96355 28429 95874 30098 95363 31758 94823 33408 94254 29 32 26780 96347 28457 95865 30126 95354 31786 94814 33436 04245 28 33 2680o8 96340 28485 95857 3o0154 95345 31813 94805 33463 94235 27 34 26836 96332 28513 95849 30182 95337 31841 94795 33490 94225 26 35 26864 96324 28541 95841 30209 95328 3i868 94786 33518 94215 25 36 26892 96316 28569 95832 30237 953I9 31896 94777 33545 94206 24 37 26920 96308 28997 95824 30265 953o10 31923 94768 33573 94i96 23 38' 26948 96301 28625 95816 30292 95301 31951 94758 336oo 94186 22 39 26976 96293 28652 95807 30320 95293 31979 94749 33627 94176 21 ~40 27004 96285 28680 95799 30348 95284 32006 94740 33655 94167 20 41 27032 96277 28708 9579I 30376 95275 32034 94730 33682 94157 1 42 27o60 96269 28736 95782 304o3 95266 32061 9472I 33710 94147 1 43 27088 96261 28764 95774 3o43i 95257 32089 94712 33737 94137 17 44 27116 96253 28792 95766 30459 95248 32116 94702 33764 94127 I6 45 27144 96246 28820 95757 3o486 95240 32144 94693 33792 94118 15 46 27172 96238 28847 95749 30514 95231 32171 94684 33819 94108 14 47 27200 96230 28875 95740 30542 95222 32199 94674 33846 94098 13 48 27228 96222 28903 95732 30570 95213 32227 94665 33874 94088 12 49 27256 96214 28931 95724 30597 95204 32254 94656 33901 94078 II 50 27284 96206 28959 95715 3o625 95195 32282 94646 33929 94068 10o 51 27312 96198 28987 95707 3o653 95186 32309 94637 33956 94o58 9 52 27340 96190 29015 95698 3068o 95177 32337 94627 33983 94049 8 53 27368 96182 29042 95690 30708 95168 32364 94618 34.oi 94039 7 54 27396 96174 29070 95681 30736 95159 32392 94609 34038 94029 6 55.27424 96166 29098 95673 30763 95150 32419 94599 34065 94019 5 56 27452 96158 29126 95664 30791 95142 32447 94590 34093 94009 4 57 27480 96150 29154 95656 30819 95i33 32474 94580 34120 93999 3 58 27508 96142 29182 95647 30846 95124 32502 94571 34147 93989 2 59 27536 96134 29209 95639 30874 95II5 32529 94561 34175 93979 I 60 27564 96126 29237 9563o 30902 95106 32557 94552 34202 93969 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 1740 0720 730, 1' 68 NATURAL SINES AND COSINES. TABLE III. 200 210 220 230 240 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. o 34202 93969 35837 93358 37461 92718 39073 92050 40674 91355 6o 1 34229 93959 35864 93348 37488 92707 39100 02o 39 40700 91343 59 2 34257 93949 35891 93337 37515 92697 39127 92028 40727 91331 58 3 34284 93939 35918 93327 37542 92686 39153 92016 40753 91319 57 4 34311 93929 35945 93316 37569 92675 39180 9200o 40780 91307 56 5 34339c 93919 35973 93306 37595 192664 39207 91994 408o6 91290 55 6 34366 93909 36ooo 93295 37622 92653 39234 91982 40833 91283 54 7 34393 93899 36027 93285 37649 92642 3926o0.9971 40860 91272 53 8 34421 93889 36054 93274 37676 92631 39287 91959 4886 91260 52 o 34448 93879 36081 93264 37703 92620 39314 91948 40013 91248 5i io 34475 93869 36o108 93253 37730 92609 39341 91936 40939 9I236 50 i 345o3 93859 36i35 93243 37757 92598 39367 91925 40966 91224 49 12 3453o 93849 36162 93232 37784 92587 39394 91914 40992 91212 48 13 34557 93839 3619o 93222 37811 92576 39421 91902 41019 91200 47 14 34584 93829 36217 93211 37838 92565 39448 9189I1 4o1045 1188 46 15 34612 93819 36244 93201 37865 92554 39474 91879 41072 91176 45 i6 34639 93809 36271 93190 37892 92543 39501 91868 41098 91164t 44 17 34666 93799 36298 93180 37919 92532 39528 91856 41125 91152 43 i8 34694 93789 36325 93169 37946 92521 39555 91845 41i5I 91140 42 19 34721 93779 36352 9359 37973 925o10 39581 91833 41178 91128 41 20 34748 93769 36379 93148 37999 92499 39608 91822 41204 91116 40 21 34775 93759 36406 93137 38026 92488 39635 91810 41231 91104 39 22 34803 93748 36434 93127 38053 92477 39661 9I799 41257 91092 38 23 34830 93738 36461 93116 38o8o 92466 39688 19I787 41284 91080 37 24 34857 93728 36488 93106 38107 92455 39715 91775 4i3o1 91068 36 25 34884 93718 365i5 93095 38134 92444 39741 91764 41337 91056 35 26 34912 93708 36542 93084 38i6i 92432 39768 91752 4i363 r1044 34 27 34939 93698 365609 93074 38188 92421 39795 91741 4i39o 91032 33 28 34966 93688 36596 93063 38215 92410 39822 91729 41416 91020 32 30 35021 93667 36650 93C42 38268 923 39875 91706 41469 90996 30 31 35048 93657 36677 93031 38295 92377 39902 91694 41496 90984 29 32 35075 93647 36704 93020 38322 92366 39928 91683 41522 90972 28 33 35102 93637 36731 93oo010 38349 92355 39955 91671 41549 90960 27 34 35i30 93626 36758 92999 38376 92343 39982 9I660 41570 90948 26 35 35157 93616 36785 92988 38403 92332 4ooo0008 91648 41602 90936 25 36 35184 93606 36812 92978 38430 92321 40035 91636 41628 90924 24 37 35211 93596 36839 92967 38456 92310 40062 91625 41655 90o11 23 38 35239 93585 36867 92956 38483 92299 4oo88 9'i6i3 4681 90o89 22 39 35266 93575 36894 92945 38510 92287 4oii5 91601 41707 908o7 21 40 35293 93565 36921 92935 38537 92276 40141 01590 41734 90875 20 41 35320 93555 36948 92924 38564 92265 40168 91578 41760 go863 19 42 35347 93544 36975 92913 38591 92254 40195 91566 41787 90851 1 43 35375 93534 37002 92902 38617 92243 40221 91555 41813 90839 17 44 35402 93524 37029 92892 38644 92231 40248 91543 41840 90826 i6 45 3542993514 37056 92881 38671 92220 40275 91531 41866 90814 15'46 35456 93503 37083 92870 38698 92209 4030o 91519 41892 90802 14 47 35484 93493 37110 92859 38725 92198 4o328 91508 41919 9079o0 3 * 48 35511 93483 37137 92849 38752 92186 4o355 91496 41945 90778 12 49 35538 93472 37164 92838 38778 92175 40381 91484 41972 90766 ii 50 35565 93462 37191 92827 388o5 92164 40408 91472 41998 90753 o10 5i 35.5092 93452 37218 92816 38832 92152 4o434 91461 42024 90741 I 52 3561.9 93441 37245 92805 38859 92141 40461 91449 42051 90729 3 53 35647 93431 37272 92794 38886 92130 40488 91437 42077 90717 7 54 35674 93420 37299 92784 38912 92119 4o054 91425 42104 90704 6 55 35701 93410 37326 92773 38939 92I07 4054i 91414 42130 0o692 5 56 35728.93400 37353 92762 38966 996 40567 91402 42156 90o680 4 57 35755.03389 37380 92751 38993 92085 4o0594 91390 42183 90668 3 58 35782 -93379 37407 92740 39020 92073 40621 91378 42202 90655 2 59 358io 93368 37434 92729 39046 92o62 40647 91366 42233 90643 I 6 35837 93358 3746 927 9073 9200 40674 91355 42262 90631 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 690 680 67~ 660 650 TAE III.. NATURAL SINES AND COSINES. 69 250 26~ 27 28~ 29~0 Sine. Cosine. Sine. Cosine. Sine. Cosine, Sine. Cosine. Sine. Cosine. 0 42262 90631 43837 89879 45399 89101 46947 88295 48481 8-7462 6o I 42288 9o618] 43863 89867 45425 89087 46973 88281 48506 87448 59 2 42315.90606 43889 89854 45451 89074 46999 88267 48532 87434 58 3 4234i 90594 43016 89841 45477 89061 47024 88254 48557 87420 57 4 42367 90582 43942 89828 45503 89048 47050 88240 48583 87406 56 5 42394 90569 43968 89816 45529 89035 47076 88226 48608 87391 55 6 42420 90557 43994 898o3 45554 89021.471o I 88213 48634 8737?7, 54 7 42446 90545 44020 89790 45580 89008 47127 88199 48659 87363 53 8 42473 90532 44o46 89777 45606 88995 47153 88185 48684 87349 52 9 42499 90520 44072 89764 45632 88981 47178 88172 48710 87335 51 io 4252 90507 44098 89752 45658 88968 47204 88i58 48735 87321 5o ii 42552 90495 44124 80739 45684 88955 47229 88144 4876I 87306 49 12 42578 90483 4415., 89726 4571o 88942 47255 88130 48786 87292 48 13 42604 90470 44177 89713 45736 88928 47281 88117 48811 87278 47 14 42631 90458 44203 89700 45762 88915 47306 8810o3 48837 87264 46 i5 42657 90446 44229 89687 45787 88902 47332 88089 48862 87250 45 i6 42683 90433 44255 89674 45813 88888 47358 88075 48888 87235 44 17 42709 90421 44281 89662 45839 88875 47383 88062 48913 87221 43 18 42736 90408 44307 89649 45863 88862 47409 88048 48938 87207 42 19 42762 90396 44333 89636 45891 88848 47434 88034 48964 87193 41 20 42788 90383 44359 89623 45917 88835 47460 88020 48989 87178 40 21 42815 90371 44385 89610 45942 88822 47486 88006 49014 87164 39 22 42840 90358 44411 89597 45968 88808 47511 87993 /49o040 8715o 38 23 42867 90346 44437 89584 45994 88795 47537 87979 490o65 87136 37 24 42894 9o334 44464 8957I 46020 88782 47562 87965 49090 87121 36 25 42920 90321 44490 89558 46046 88768 47588 87951 49106 87i07 35 26 42946 90309 44516 89545 46072 88755 47614 87937 4914' 87093 34 27 42972 90296 44542 89532 46097 88741 47639 87923 49166 87079 33 28 42999 90284 44568 89519 46123 88728 47665 87909 4.9092 87064 32 29 43025 90271 44594 89506 46149 88715 47690 87896 492I7 87050 3i 30 43051 90259 44620 89493 46173 88701 4776 87882 49242 87036 3o 31 43077 90246 44646 89480 46201 88688 47741 87868 49268 87021 29 32 43io4 90233 44672 89467 46226 88674 47767 87854 49293 87007 28 33 4313o 9022. 44698 89454 46252 88661 47793 87840 4.9318 86993 27 34 43156 90208 44724 89441 46278 88647 47818 87826 49344 86978 26 35 43i82 901go96 44750 89428 46304 88634 47844 87812 49369 86964 25 36 4.3209 901oi83 44776 8941/5 46330 88620 47869 87798 49394 86949 24 37 43235 90171 44802 89402 4.6355 88607 47895 87784 4.9409 86935 23 38 43261 90158 44828 89389 46381 88593 47920 87770 49445 86921 22 39 43287 90146 44854 89376 46407 88580 47946 87756 49470 86906 2I 40 43313 90133 44880 89363 46433 88566 47971 87743 494905 86892 20 41 4334o 90120 44906 89350 46458 88553 47997 87729 4.9520 86878 I 42 43366 go90io08 44932 89337 46484 88539 48022 87715 49546 86863 1 43 43392 9009 44958 89324 465o10 88526 48048 87701 49571 86849 17 44 43418 goo90082 44984 89311 46536 88512 48073 87687 49596 86834 16 45 43445 90070 45010 89298 46561 88499 48099 87673 49622 86820 15 46 43471 900goo57 45036 89285 46587 88485 48124 87650 49647 868o5 14 47 43497 goo90045 45062 89272 46613 88472 48150o 87645 49672 8679I 13 48 43523 90032 45088 89259 46639 88458 48175 87631 49697 86777 12 49 43549 90019 45i14 89243 46664 88445 48201 87617 49723 86762 II 50 43575 90007 45140 89232 46690 88431 48226 87603 49748 86748 io 51 43602 89994 45166 89219 46716 88417 48252 87589 49773 86733 q 52 43628 89981 45192 89206 46742 88404 48277 87575 49798 86719 8 53 43654- 89968 45218 89193 46767 88390 48303 8756i 49824 86704 7.54 43680 89956 45243 89180 46793 88377 48328 87546 49849 86690 6 55 43706 89943 45269 89167 46819 88363 48354 87532 49874 86675 5 56 43733 89930 45295 89153 46844 88349 48379 875i8 49899 8666i 4 57 43759 89918 45321 89140 46870 88036 48405 87548 49924 86646 3 58 43785 89905 45347 89127 46896 88322 4843o 87490 49950 86632 3 59 43811 89892 45373 89114 46921 88308 48456 87476 4997S 86617 I 60 43837 89879 45399 89101 46947 88295 48480 87462 50000oooo 86603 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 640 63 620 610 600 70 NATURAL SINES AND COSINES. TABLE III. 300 31~ 320 33 34~0 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. o 50000oooo 866o3 51504 85717 52992 84805 54464 83867 55919 82904 60 i 50025 86588 51529 85702 53017 84789 54488 83851 55943 82887 59 2 50050 86573 51554 85687 53o41 84774 54513 83835 55968 82871 58 3 50076 86559 51579 85672 53o66 84759 54537 83819 55992 82855 57 4 5oIoi 86544 51604 85657 53091 84743 54561 83804 56o016 82839 56 5 50o126 8653o 51628 85642 53115 84728 54586 83788 56040 82822 55 6 o5015 86515 51 653 85627 53140.84712 546o10 837.72 56o64 82806 54 7 50176 86501 51678 85612 53164 84697 54635 83756 56088 82790 53 8 50201 86486 51703 85597 53189 8468I 54659 83740 56121 82773 52 9 50227 86471 51728 85582 53214 84666 54683 83724 56136 82757 5i 6 50252 86457 51753 85567 53238 84650 54708 83708 5616o 82741 50o ii 50277 86442 51778 85551 53263 84635 54732 83692 561684 82724 49 12 50302 86427 518o3 85536 53288 846I9 54756: 83676 56208 82708 48 13 50327 86413 51828 85521 53312 84604 54781 8366o 56232 82692 4.7 i4 650352 86398 51852 8550o6 53337 84588 548o5 83645 56256 82675 46 i5 50377 86384 51877 85491 53361 84573 54829 83629 56280 82659 45 i6 5o4o3 86369 51902 85476 53386 84557 54854 83613 563o5 82643 I 44 17 50428 86354 51927 85461 534II 84542 54878 83597 56329 82626 43 i8 50453 86340 51952 85446 53435 845265492 8358 546390 826o10 42 j 19 50478 86325 51977 85431 53460 84511 54927 83565 56377 82593 41 20 5050o3 863io 52002 85416 53484 84495 54951 83549 564oi 82577 40 2 5o0528 86295 52026 8540o 53509 84480 54975 83533 56425 82561 39 22 5o553 86281 52051 85385 53534 84464 54999 83517 56449 82544 38 23 50578 86266 52076 85370 53558 84448 55024 83501o 56473 82528 37 24 5o6o3 -86251 52101 85355 53583 84433 55048 83485 56497 82511 36 25 50628 86237 52126 8534o 536o7 84417 55072 83469 56521 82495 35 26 5o654 86222 52151 85325 53632 84402 55097 83453 56545 82478 34 27 50679 86207 52175 85310 53656 84386 5512i 83437 56569 82462 33 28 507o4 86192 52200 85294 53681 84370 55145 83421 56593 82446 32 29 50729 86178 52225 85279 537o5 84355 55169 834o5 56617 82429 3r 3o 50754 86163 52250 85264 53730 84339 55194 83389 56641 82413 3o 31 50779 86148 52275 85249 53754 84324 55218 83373 56665 82396 29 32 50804 86133 52299 85234 53779 84308 55242 83356 56689 82380 28 33 50o829 86119 52324 85218 53804 84292 55266 83340 56713 82363 27 34 5o854 86104 52349 852o3 53828 84277 55291 83324 56736 82347 26 35 50879 86089 52374 85188 53853 84261 55315 833o8 5676o 82330 25 36 50904 86074 52399 85173 53877 84245 55339 83292 56784 82314 24 37 50929 86059 52423 85157 53902 84230 55363 83276 568o8 82297 23 38 50954 86045 52448 85142 53926 84214 55388 83260 56832 82281 23 39 50979 8603o0 52473 85127 53951 84198 55412 83244 56856 82264 21 40 51oo4 86o015 52498 85112 53975 84182 55436 83228 56880 82248 20 41 529 86ooo000 522 850o96 54ooo 84167 55460 83212 56904 82231 19 42 5o054 85985 52547 850o8 54024 84151 55484 83195 56928 82214 i8 43 51079 85970 52572 85066 54049 84135 55509 83179 56952 82198 17 44 51104 85956 52597 85o5i 54073 84120 55533 83i63 56976 82181 16 45 51129 85941 52621 85o35 54097 84104 55557 83147 57000 82165 15 46 51154 85926 52646 85020 54122 84088 55581 83131 5o024 82148 14 47 51179 85911 52671 85oo5 54146 84072 556o5 83I15 57047 82132 13 48 51204 85896 52696 84989 54171 84057 55630 83098 57071 82115 12 49 51229 85881 52720 84974 54195 84041 55654 83082 57095 82098 ii 50 51254 85866 52745 84959 54220 84025 55678 83066 57119 82082 10 5I 51279 85851 52770 84943 54244 84009 55702 83050 57143 82065 9 52 51304 85836 52794 84928 54269 83994 55726 83034 57167 82048 8 53 51329 85821 52819 84913 54293 83978 55750 83017 57191 82032 7 54 51354 858o6 52844 84897 54317 83962 55775 83o01 57215 82015'6 55 51379 85792 52869 84882 54342 83946 55799 82985 57238 81999 5 56 51404 85777 52893 84866 54366 83930 55823 82969 57262 81982 4 57 51429 85762 52918 84851 5439i 83915 55847 82953 57286 81965 3 58 51454 85747 52943 84836 54415 83899 55871 82936 57310 81949 2 59 51479 85732 52967 84820 54440 83883 55895 82920 57334 81932 I 60 51504 85717 52992 84805 54464 83867 55919 82904 57358 81915 o Cosine. Sine. 1Cosine. Sine. Cosine. Sine. Cosine. Sine. iCosine. Sine. 690 580 5~0 560 650 TABLE III. NATURAL SINES AND COSINES. __ 1 1 350 8 360 370 38~0 390 ~ Sine. Cosine. Sine. Cosine. Sine.'Cosine. Sine. Cosine. Sine. Cosine. o 535738 81915 58779 80902 60182 79864 6i566 78801 62932 77715 50 i 57381 81899 58802 8o885 60205 79846 61589 78783 62955 77696 59 2 57405 81882 58826 80867 60228 79829 61612 78765 62977 77678 58 3 57429 8i865 58849 8o85o 60251 798ii 6i635 78747 63000ooo 77660 57 4 5745 81848 58873 80833 60274 79793 61658 78729 63022 77641 56 5 57477 81832 58896 8o816 60298 79776 6i68I 78711 63045 77623 55 6 57501 81815 58920 80799 60321 79758 61704 78694 63068 77605 54 7 57524/ 81798 58943 80782 60344 79741 61726 78676 63090 77586 53 8 57548. 81782 58967 80765 60367 79723 61749 78658 63113 77568 52 9 57572 81765 58990 80748 60390 79706 61772 78640 63135 77550 51 io 57596 81748 59oi4 80730 60441 79688 61795 78622 63i58 7753 50o i 576i9 81731 5'9037 80713 60437 79671 6i8i8 78604 63180o 77513 49 12 57643 81714 59o6i 80696 60460 79653 61841 78586 63203 77494 48 13 57667 81698 59084 80679 60483 79635 61864 78568 63225 77476 47 14 5769i 8i68i 59108 80662 60506 79618 61887 78550 63248 77458 46 15 57715 8i664 59131 80644 60529 79600 61909 78532 63271 77439 45 i6 57738. 81647 59i54 8o627 60553 79583 61932 78514 63293 77421 44 17 57762 863i 59178 8o6o10 6o576 79565 6i955 78496 633i6 77402 43 18 57786 81614 5920i 80593 60599 79547 61978 78478 63338 77384 42 19 578io1 8i597 59225 80576 160622 79530 62001 78460 6336i 77366 41 20 57833 8i580 59248 8o558 o60645 79512 62024 78442 63383 77347 40 21 57857 81563 59272 8o54I 60668 79494 62o46 78424 63406.77329 39 22 5788i 81546 59295 80524 60691 79477 62069 78405 63428 77310 38 23 57904 8153o 593i8 80507 60714 79459 62092 78387 6345I 77292 37 24 57928 8i513 59342 80489 60738 79441 62115 78369 63473 77273 36 25 57952 81496 593651 80472 60761 79424 62138 78351 63496 77255 35 26 57976 81479 59389 80455 60784 79406 62160. 78333 635i8 77236 34 27 57999 81462 59412 80438 60807 79388 62183.78315 63540 77218 33 28 58023 8i445 59436 80420 6o83o 79371 62206 78297 63563 77199 32 29 58047 81428 59459 80403 60853 79353 62229 78279 63585 77181 31 30 58070 81412 59482 8o386 60876 79335 62251 78261 63608 77162 30 31 58094 81395 59506 80368 60899 79318 62274 78243 63630 77144 29 32 581i8 813 78 59529 8o35i 6o922 79300 62297 78225 63653 77125 28 33 5814i 8136i 59552 80334 6o945 79282 62320 78206 63675 77107 27 34 58i65 81344 59576 80o36 60968 79264 62342 78188 63698 77088 26 35 58i89 81327 59599 80299 60991 79247 62365 78170 63720 77070 25 36 58212 81310io 59622 80282 6ioi5 79229 62388 78152 63742 77051 24 37 58236 81293 59646 80264 610o38 79211 62411II 78134 63765 77033 23 38 5826o 81276 59669 80247 6io6i 79193 62433 78116 63787 7701o4 22 39 58283 81259 59693 80230 6io84 79176 62456 78098 638o.- 76996 21 40 58307 81242 59716 80212 61107 79i58 62479 78079 63832 76977 20 41 5833o 81225 59739 8oi95 6ii30 79140 62502 78061 63854 76959 i9 42 58354 81208 59763 80178 6II53 79122 62524 78043 63877 76940 18 43 58378 81191 59786 8oi6o 61176 79o05 62547 78025 63899 76921 17 44 584oi 81174 S98og 8oi43 61199 79087 62570 78007 63922 76903 16 45 58425 81157 59832 8o125 61222 79069 62592 77988 63944 76884 15 46 584491 81140o 59856 8o010o8 61245 79051 62615 77970 63966 76866 14 47 58472 81123 59879 80091 61268 79033 62638 77952 63989 76847 13 48 58496 81io6'59902 80073 61291 79016 62660 77934 64011 76828 12 49 5i9 5 851089 59926 8oo56 613i4 789908 62683 77916 64033 76810 II 50 58543 81 072 59949 8oo38 61337 78980 62706 77897 64o56 76791 I0 5i 58567 8io55 59972 80021 6136o 78962 62728 77879 64078 76772 9 52 5859o 8io38 59995 8ooo3 6i383 78944 62751 77861 64ioo 76754 8 53 58614 810o21'6oo019 79986 614o6 78926 62774 77843 64123 76735 7 54 58637 8oo1004 60042 79968 61429 78908 62796 77824 64145 76717 6 55 5866i 80987 6oo65 79951 61451 78891 62819 77806 64167 76698 5 56 58684 80970 60089 79934 61474 78873 62842 77788 64190 76679 4 57 587o8 8o953 60112 79916 61497 78855 62864 77769 64212 7666i 3 58 5873 i 8o936 6o35 79899 61520 78837 62887 7775I 64234 76642 2 59 58755 80919 6o0158 79881 61543 78819 62909 77733 64256 76623 I 60 58779 80902 60182 79864 61566 78801 62932 77715 64279 76604 0 Cosine Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine.. Cosine. Sine. 4~0 530 520 510 600 _72_____I__ NATURAL SINES AND COSINES. TABLE III. 400 i 410 420 430 440 Sine. Cosine. Sine. Cosine. Sin. ose. C. Sine. Cosine. Sine. Cosine. o 64279 76604 656o6 75471 66913 74314 68200 73135 69466 71934 6o I 64301 76586 65628 75452 66935 74295 68221 73116 69487 71914 59 2 64323 76567 65650.75433 66956 74276 68242 73096 69508 71894 58 3 64346 76548 65672 75414 66978 74256 68264 73076 69529 71873 57 4 64368 76530 65694 75395 66999 74237 68285 73056 69549 71853 56 5'64390go 76511 65716 75375 67021 74217 68306 73036 69570o 71833 55 6 644,2 76492 65738 75356 67043 74198 68327 73016 69591 71813 54 7 64435 76473 65759 75337 67064 74178 68349 72996 69612 71792 53 8 64457 764.55 65781 75318 67086 74159 68370 72976 69633 71772 52 9 64479 76436 65803 75299 6710o7 74139 68391 72957 696541 71752 51 io 64501 76417 65825 75280o 67129 74120 68412 72937 69675 71732 50 ii 64524 76398 65847 75261 67151 74100 68434 72917 69696 71711 49 12 64546 76380 65869 75241 67172 74080 68455 72897 69717 71691 48 13 64568 76361 65891 75222 67194 74061 68476 72877 69737 7167I 47 14 64590 76342 65913 75203 67215 74041 68497 72857 69758 71650 46 i5 64612 76323 65935 75184 67237 74022 6851 72837 69779 71630 45 I6 64635 76304 65956 75165 67258 74002 68539 72817 69800 71610 44 17 64657 76286 65978 75146 67280 73983 68561 72797 69821 71590 43 I8 64679 76267 66oo00075126 67301 73963 68582 72777 69842 71569 42 I9 64701 76248 66022 75107 67323 73944 68603 72757 69862 71549 41 20 64723 76229 66044 75088 67344 73924 68624 72737 69883 71529 40 21 64746 762o10 66o66 75069 67366 73904 68645 72717 69904 71508 39 22 64768 76192 66o88 75050 67387 73885 68666 72697 69925 71488 38 23 64790 76173 66o109 7530 67409 73865 68688 72677 69946 71468 37 24 04812 76154 6613i 750II1 67430 73846 68709 72657 69966 71447 36 25 64834 76135 66i53 74992 67452 73826 68730 72637 69987 71427 35 26 64856 76116 66175 74973 67473 73806 68751 72617 70008 71407 34 27 64878 76~ 97 66197 74953 67495 73787 68772 72597 70029 71386 33 28 64901 76078 662i8 74934 67516 73767 68793 72577 70049 71366 32 29 64923 76059 66240 74915 67538 73747 68814 72557 70070 71345 31 30 64945 76041 66262 74896 67559 73728 68835 72537 70091 71325 30 31 6Z4967 76022 66284. 74876 67580 73708 68857 72517 701.12 7I305 29 32 64989 76003 663o5 74857 67602 73688 68878 72497 70132. 71284 2 33 65o1.750984 66327 74838 67623 73669 68899 72477 70153 71264 27 34 65o33 75965 66349' 74818 67645 73649 68920 72457 70174 71243 26 3 5055 75946 66371- 74799 67666 73629 68941 72437 70195 7I223 25 36 65077 72927 66393 74780 67688 73610 68962 72417 70215 71203 24 37 65ioc 70908 66414 74760 67709 73590 68983 72397 70236 71182 23 38 65122 75889 66436 74741 67730 73570 69004 72377. 70257 71162 22 39 6514.4 75870 66458 74722 67752 73551 69025 72357 70277 71141 21 40 65i66 75851 66480 74703 67773 73531 69046 72337 70298 71121 20 41 65188 75832 665oI 74683 67795 73511 69067 72317 70319 71100 19 42 65210 75813 66523 74664 67816 73491 69o88 72297 70339 71080 I8 43 65232 75z794 66545 74644 67837 73472 69109 72277 70360 71059 17 44 65254 75775 66566 74625 67859 73452 69130 72257 70381 71039 i6 45 65276 70756 66588 74606 67880 73432 69151 72236 70401 71019 i5 46 65298 75738 66610o 74586 67901 73413 69172 72216 79422 70998. 14 47 65320 75719 66632 74567 67923 73393 69193 72196 70443 70978 13 48 65342 75700 66653 74548 67944 73373 69214 72176 70463 70957 12 49 65364 75080 66675 74528 67965 73353 69235 72156 70484 70937 II 50o 65386 75661 66697 74509 67987 73333 69256 72136 70505 70916 io 51 65408 75642 66718 74489 68oo008 73314 69277 72116 70525 70896 9 52 65430 75623 66740 74470 68029 73294 69298 72095 70546 70875 53 65452 75604 66762 74451 6805i 73274 69319 72075 70567 70855 7 54 65474 75585 66783 74431 68072 73254 69340 72055 70587 70834 6 55 65496 75566 66805 74412 68093 73234 69361 72035 70608 70813 5 56 655i8 75547 66827 74392 68ii5 73215 69382 72015 70628 70793 4 57 65540 75528 66848 74373 68i36 73195 69403 71995 70649 70772 3 58 65562 75509 66870 74353 68157 73i75 69424 71974 70670 70752 2 59 65584, 75490 66891 74334 68179 73155 6944 7i95 4 706954 70731 I 60 65606 75471 66913 74314 68200 73135 69466 71934 70711 70711 0 Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 49~ 48~ 410 460 45~ TABLE III. NATURAL TANGENTS AND COTANGENTS. 73, Q _00 _ 10 20 30 / Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o0 ooo0 Infinite. 01746 57'2900 03492 28-6363 05241 19.0811 60 1 00029 3437 75 oI775 56.3506 03521 28.3994 05270 18.9755 59 2 00058 I718.87 oi8o4 55.44I5 o3550 28.1664 05299 I8'8711 58 3 00087 1145.92 oi833 54.56I3 03579 27.9372 05328 I8.7678 57 4 ooIi6 859-436 oI862 53.7086 03609 27.7117 05357 I8.6656 56 5 00145 687.549 o8g91 52..8821 03638 27.4899 05387 18.5645 55 6 00175 572.957 01920 52.0807 o3667 27 27151 054i6 18.4645 54 7 00204 491 10o6 0I 49 5 13032 03696 27 0566 o5445 I8.3655 53 8 00233 42971I8 01978 50.5485 03725 26.8450 05474 18.2677 52 9 00262 381.971 02007 49 8157 03754 26'6367 05503 18.708 5I IO 00291 343.774 02036 49 Io39 03783 26.431 6 o5533 I8.o750 50 I 0oo320 312.521 02066 48.4121 03812 26.2296 05562 17 9802 49 I2 00349 286.478 02095 47-7395 03842 26.0307 0559 17.8863 48 13 00378 264.441 021 24 47o0853 o3871 25.8348 05620 17 -7934 47 I4 00407 245.552 02153 46.4489 o39oo 25.6418 05649 17.7015 46 15 00436 229I82 02182 45.8294 03929 25.4517 o567 17.6106 45 6 00465 214-858 02211 45.2261 03958 25.2644 o5708 17-5205 44 17 oo495 202.219 02240 44-6386 o3987 25.0798 o5737 I7-4314 43 I8 00524 190.984 02269 44-066I o40o6 24.8978 05766 17-3432 42 19 oo553 80o. 932 02298 43.50o8 o4046 24.7185 05795 I7.2558 41 20 o0582 171 885 02328 42.9641 04075 24.5418 05824 17-1693 40 21 00611 I63.700 02357 42.4335 04104 24.3675 o5854 17-0837 39 22 oo640 156.259 02386 41i.9158 0433 24-1957 o5883 169990 38 23 oo669 I49.465 02415 41 *4106 04162 24-0263 05912 16.9150 37 24 00698 I43-237 02444 40-9174 04191 23.8593 0594I i6 8319 36 25 00727 I37.507 02473 4o.4358 04220 23.6945 05970 I6 7496 35 26 00756 132.2I9 02502 39-9655 04250 23.5321 05999 16.6681 34 27 00785 127-321 0253I 39.5059 04279 23.37I8 o6029 I6.5874 33 28 o0084 I22.774 02560 39-o568 04308 23.2137 o6058 i6.o505 32 29 00844 118.540 02589 38.6I77 04337 23.0577 06087 16.4283 31 30 oo00873 114589 02619 38.i885 04366 22.9038 06116 16.3499 30 31 00902 IIo1 892 02648 37.7686 04395 22.75I9 06145 16.2722 29 32 00931 107-426 02677 37-3579 04424 22.6020 06175 16.1952 28 33 00960 104 171 02706 36.9560 o4454 22.454I 06204 16.1190 27 34 o0098og o9 11 7 02735 36.5627 o4483 22.3C81 06233 i6.o435 26 35 oiOI8 98.2179 02764 36. 776 04512 22.1640 06262 15.9687 25 36 O1047 95-4895 02793 35.8006 0454I 22.0217 06291 15.8945 24 37 01076 92.9085 02822 35.43 3 04570 21.8813 0632I 15.821I 23 38 o1105 90.4633 02851 35.o695 04599 21.7426 06350 15.7483 22 39 01135 88.1436 02881 34.715I 04628 2 16056 06379 15.6762 21 40 o1164 85.0398 02910 34.3678 o4658 21.4704 o6408 I5.6048 20 41'oii93 83.8435 02939 34 0273 04687 21.3369 06437 I5.5340 I9 42 01222 81.8470 02968 33.6935 04716 21.2049 06467 15.4638 I8 43 OI251 79 9434 02997 33.3662.04745 21 o747 06496 15.3943 17 44 01280 78.1263 03026 33.0452 04774 20.9460 06525 15.3254 16 45 01309 76-3900 o3055 32.7303 o4803 20.8188 o6554 15.2571 15 46 oi338 74-7292 o3084 32.42I3 04832 20o6932 o6584 I5.1893 14 47 01367 73.1390 0o314 32.1181 04862 20.5691 066i3 15.I222 13 48 oI396 7'6I51I o3i43 3I.8205 04891 20.4465 06642 I5.o557 12 49 OI425 70.I533 03172 31.5284 04920 20.3253 06671 14.9898 ii 50 O1455 68.7501 03201 31.2416 04949 20.2056 06700 14.9244 10 5I 01484 67-4019 03230 30.9599 04978 20.0872 06730 14.8506 9 52 oi513 66.o055 03259 30.6833 o50o7 I9-9702 06759 14-7904 8 53 0o542 64-858o 03288 3o.4II6 o5037 Ig98546 06788 14-7317 7 54 01571 63.6567 03317 30o.446 0o5066 19.7403 068I7 i4.6685 6 55 oi600 62-4992 o3346 29-8823 05095. g196273 o6847 I4-6059 5 56 01629 6I.3829 03376 29.6245 05124 Ig.5I56 06876 I4-5438 4 57 1o658 6o.3058 03405 29-3711I 0553 19-4051 o69o5 I4.4823 3 58 o1687 59.2659 o3434 29-I220 o0582 19-2959 06934 1 4-4212 2 59 o I7I6 58.26I2 03463 28.8771 05212 19-1879 06963 143607 I 60 01746 57-2900 03492 28.6363 05241 19g081 06993 14-3007 0 Cotang. Tangent. Cotang. Tanen CoCotang. Tangent. Cotang. Tangent. 890 88~ 87o 860 74 NATURAL TANGENTS AND COTANGENTS. TABLE III. 40 50 6 _ _ 1 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. 0 06993 14.300o 08749 II 43oi io5Io 9.51436 12278 81I4435 60 I 07022 14-2411 08778 II.39I9 I0540 9-4878I 12308 8.12481 5 2 0705I I4-1821 08807 II.3540 10569 9-4614I 12338 8.io536 58 3 07080 14-1235 08837 II.3I63 10599 9.43515 12367 8.08600 57 4 07110 I4.0655 08866 II.2789 10628 9 40904 I2397 8-06674 56 5 7I39 14-o0079 08895 11-2417 10657 9.38307 I2426 8-04756 55 6 0768 I3.9507 08925 112048 Io687 9.35724 12456 8.02848 54 07197 7 3.-8940 08954 II.I68I 10716 9-33154 12485 8o00948 53 8 07227 13-8378 08983 ii.i3i6 10746 9.30599 12515 7.99058 52 9 07256 13.782I 09013 1.o0954 I0775 9-28058 12544 7.97176 5I 10 07285 13.7267 09042 11-0594 10805 9-25530 12574 7.95302 50 II 07314 13.67I9 09071 11.0237 10834 9-23016 12603 7.93438 49 12 07344 13.6174 09101 10-9882 o0863 9-20516 12633 7-91582 48 13 07373 I3.5634 09g30 I0-9529 10893 9- 8028 12662 7-89734 47 I4 07402 I3.5098 09159 Io19178 I0922 9.15554 12692 7.87895 46 I5 07431 I3 4566 09189 0o.8829 10952 9. 3093 12722 7.86064 45 i6 07461 13-4039 09218 Io.8483 o198I 9.10646 12751 7.84242 44 17 07490 i3 35I5 09247 10-8I39 Iioi 9 08211 12781 7.82428 43 I8 o75i9 I3.2996 09277 10-7797 1Io40 9.o5789 128IO 7.80622 42 19 07548 13-248o o0306 10-7457 11070 9-03379 12840 7-78825 41 20 07578 I3.1969 09335 o0.7119 11099 9.oo983 12869 7.77035 40 21 07607 13-I46I 09365 10.6783 11128 8.98598 12899 7.75254 3 22 07636 13.o958 o9394 io.6450 11158 8.96227 12929 7-73480 38 23 07665 3.o0458 09423 Io-6II8 11187 8.93867 12958 7.717I5 37 24 07695 12-9962 09453 10-5789 11217 8.91520 12988 7.69957 36 25 07724 12.9469 09482 o105462 11246 8-89185 I30I7 7.68208 35 26 07753 I2.8981 og95I 10 o536 11276 8.86862 13047 7.66466 34 27 07782 12-8496 09541 I1o48I3 11305 8-84551 13076 7.64732 33 28 07812 12-8014 09570 10.4491 II335 8 82252 i3Io6 7.63005 32 29 07841 12-7536 09600 IO4172 11364 8.79964 i3136 7.61287 31 30 07870 12.7062 09629 o103854 11394 8.77689 I3165 7.59575 30 31 07899 12-6591 09658 Io.3538 11423 8.75425 I3195 7-57872 2 32 07929 12-6124 09688 Io 3224 II452 8-73172 I3224 7 56I76 2 33 07958 12-5660 097I7 10o2913 11482 8-7093I I3254 7.54487 27 34 07987 12.5199 09746 10-2602 11511 8-6870I I3284 7-52806 26 35 08017 12 4742 09776 I0o2294 1154I 8.66482 13313 7.51132 25 36 08046 12.4288 09805 10o1988 11570 8.64275 13343 7-49465 24 37 08075 12-3838 09834 1o.I683 11600 8-62078 13372 7-47806 23 38 08104 12-3390 o9864 0-.138I II629 8-59893 13402 7-46154 22 39 08I34 12-2946 09893 io.0o8o 11659 8.57718 13432 7.44509 21 40 08i63 12-2505 09923 10-0780 11688 8-55555 13461 7-42871 20 41 08192 12-2067 09952 Oo-0483 11718 8-53402 13491 7-41240 19 42 08221 I2-I632 09981 10-OI87 11747 8-51259 13521 7.396i6 18 43 08251 12-120I IOOII 9.98930 11777 8.49128 i3550 7.37999 17 44 08280 12-0772 10040 9-96007 ii806 8.47007 I3580 7.36389 I 45 08309 12-0346 I0069 9.93IOI 11836 8-44896 13609 7 3478 15 46 o833g 119923 100o9 9-90211 I1865 8-42795 13639 7.33i90 14 47 08368 II9504 IOI28 9.87338 11895 8 40705 I3669 7.31I600 13 48 08397 1.90o87 10158 9-84482 11924 8.38625 13698 7.30018 12 49 08427 11.8673 o1187 9-8I64I 11954 8.36555 13728 7-28442 11 50 08456 11-8262 102I6 9-788I7 Ig983 8.34496 13758 7-26873 io 5I 08485 11.7853 10246 9.76009 I20o3 8.32446 I3787 7-2531 o 52 08514 117448 10275 9-73217 12042 8.30406 13817 7 23754 53 08544 II-7045 io305 9-70441 I2072 8-28376 13846 7-22204 7 54 08573 II-6645 10334 9-67680 12101 8-26355 13876 7-20661 6 55 08602 II-6248 I0363 9.64935 12131 8-24345 13906 7-19125 5 56 08632 II.5853 10393 9.62205 12160 8-22344 13935 7.7594 4 57 0866 I I.546i 10422 9.-o 549 120 29 820352 I3965 71I6071 3 58 8690 I-5072 I0452 95679I 12219 8-18370 13995 7-14553 2 59 08720 11 4685 1048I 9.54106 12249 8-16398 14024 7*13042 I 6o 08749 II 43o01 10o5 9-51436 12278 8.14435 i4054 7-11537 0 Cotang. Tangent Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. ______ _______ _______ ________ ______ i 850 840 830 820 TABLE III. NATURAL TANGENTS AND COTANGENTS. 76 8~ 90 ~ 1 11~ Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o I4054 7-11537 15838 6.31375 I7633 5-67128 19438 5. 4455 60 I 14084 7.o0038 15868 6.30oi89 7663 5.66I65 19468 5.13658 5 2 14113 7.08546 15898 6.29007 17693 5.65205 19498 5.12862 58 3 I4i43 7-07059 15928 6.27829 17723 5.64248 19529 5.12069 57 4 14173 7.05579 15958 6.2665 17753 5.63295 19559 5.11279 56 5 14202 7.0410I 15988 6.25486 17783 5.62344 19589 5.10490 55 6 14232 7.02637 I6017 6.24321 17813 5.6i397 I9619 5.09704 54 7 14262 7.01I74 I6047 6.23160 17843 5.60452 19649 5.08921 53 8 I429I 6.99718 16077 6.22003 I7873 5.5951I I9680 5.08I39 52 9 14321 6.98268 16107 6.2085I 17903 5.58573 I9710 5.07360 51 1o I435I 6.96823 I6137 6.19703 I7933 5.57638 I9740 5.06584 50 Ii I438I 6.95385 16167 6.18559 17963 5.56706'I9770 50o5809 49 12 14410 6.93952 16196 6'174I9 17993 5.55777 Ig801 5.05037 48 i3 14440 6.92525 I6226 6.16283 18023 5.5485I I983I 5.04267 47 14 I4470 6.91104 I6256 6.i5i5i i8053 5.53927 19861 5.03499 46 15 14499 6.89688 16286 6.14023 i8083 5.53007 19891 5o02734 45 16 14529 6.88278 i63i6 6.12899 18113 5.52090 19921 5.0197I 44 17 14559 6.86874 16346 6.11779 18143 5.5II76 19952 5.012IO 43 i8 14588 6.85475 16376 6.10664 18173 5.50264 19982 5.oo451 42 19 i46I8 6.84082 I6405 6.o9552 18203 5.49356 20012 4.99695 41 20 14648 6.82694 I6435 6.o8444 18233 5.4845I 20042 4.98940 40 21 14678 6.8I312 16465 6.07340 18263 5.47548 20073 4.98I88 39 22 14707 6.79936 16495 6.06240 18293 5.46648 20103 4.97438 38 23 14737 6.78564 16525 6.05I43 I8323 5.45751 20I33 4.96690 37 24 14767 6.77I99 i6555 6.o4o05. 18353 5.44857 o0164 4.95945 36 25 I4796 6.75838 i6585 6.02962 I8383 5.43966 20I94 4.9520I 35 26 14826 6.74483 i6615 6.o 878 184I4 5.43077 20224 4-94460 34 27 1I4856 6.73133 16645 6.00797 I8444 5.42192 20254 4.93721 33 28 i4886 6.7I789 16674 5.99720 18474 5.4I309 20285 4.92984 32 29 14915 6.70450 16704 5.98646 i8504 5.40429 20315 4.92249 31 30 14945 6.6916 16734 5.97576 18534 5.39552 20345 4.9g516 30 31 14975 6.67787 I6764 5.965o1 18564 5.38677 20376 4.90785 29 32 I5005 6.66463 I6794 5.95448 I8594 5.37805 20406 4.9oo56 28 33 15034 6.65144 16824 5.94390 I8624 5.36936 20436 4.89330 27 34 i5064 6.6383I 16854 5.93335 I8654 5.36070 20466 4.88605 26 35 15094 6.62523 I6884 5.92283 18684 5.35206 20497 4.87882 25 36 I5124 6.612I9 16914 5.91235 18714 5.34345 20527 4.87I62 24 37 I5I53 6.5992I I6944 5.9oi09 18745 5.33487 20557 4.86444 23 38 I5183 6.58627 I6974 5.89151 18775 5.32631 20588 4.85727 22 39 I5213 6.57339 17004 5.8814 I88o5 5.3I778 20618 4.850I3 2I 40 I5243 6.56055 17033 5.87080 i8835 5.30928 20648 4.84300 20 41 i5272 6.54777 17063 5.8605I i8865 5.30080 20679 4.83590 19 42 15302 6.53503 17093 5.85024 18895 5.29235 20709 4.82882 18 43 15332 6.52234 17123 5.8400o 18925 5-28393 20739 4-82175 I7 44 15362 6.50970 17153 5.82982 18955 5.27553 20770 4.8147I i6 45 15391 6.49710 17183 5.81966 I8986 5.26715 20800 4.80769 15 46 15421 6.48456 17213 5.80953 g19o6 5.25880 20830 4.80068 14 47 i545I 6.47206 17243 5.79944 19046 5.25048 20861 4.79370 13 48 I548I 6.45961 17273 5.78938 19076 5.24218 20891 4.78673 12 49 15511 6.44720 17303 5.77936 19106 5.23391 20921 4-77978 II 50 15540 6.43484 17333 5.76937 19136 5.22566 20952 4.77286 io 51 i5570 6.42253 17363 5.75941 g1966 5.21744 20982 4-76595 9 52 15600 6.41026 17393 5.74949 I9197 5.20925 21013 4.75906 8 53 i5630 6.39804 17423 5.73960 19227 5.20Io7 2o043 4.75219 7 54 I5660 6.38587 17453 5.72974 19257 5.19293 2o073 4.74534 6 55 i5689 6.37374 17483 5.71992 19287 5.18480 2II04 4-7385I 5 56 15719 6.36I65 17513 5.71013 19317 5I.7671 21134 4.73I70 4 57 15749 6.34961 17543 5.70037 19347 5'i6863 21164 4.72490 3 58 15779 6.33761 17573 5.69064 19378 5.16058 21195 4.71813 2 59 I58o9 6.32566 17603 5.68094 19408 5.15256 21225 4.71137 I 60 15838 6.31375 17633 5.67128 19438 5.I4455 21256 4.70463 o Cotang. Tangent. Cotang. Tanent tang. Tangent. Cotang. Tangent. 810 800 70~ 780 76 NATT RAL TANGENTS AND COTANGENTS. TABLE III. 12~ 13~ 14~ 15~ Tang ent. Cotang.. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 21256 4-70463 23087 4.33I48 24933 4oio078 26795 3-73205 60 I 21286 4-69791 23117 4.32573 24964 4-00582 26826 3-72771 59 2 21316 469I121 23148 4-32001 24995 4-ooo86 26857 3-72338 58 3 21347 4-68452 23179 4.31430 25026 3.99592 26888 3-71907 57 4 21377 4.67786 23209 4.30860 25056 3.99099 26920 3-71476 56 5 21408 4-67121 23240 4.30291 25087 3.98607 26951 3-71046 55 6 21438 4.66458 23271 4.29724 25118 3-98117 26982 3-70616 54 7 2469 4.65797 23301 4.29I59 25149 3-97627 27013 3-70188 53 8 21499 4-65I38 23332 4-28595 25180 3-97139 27044 3-69761 52 9 21529 4.64480 23363 4i 28032 25211 3.96651 27076 3.69335 5I o0 2I560 4.63825 23393 4-27471 25242 3-96165 27I07 3-68909 50 II 2I590 4-6317I 23424 4-269 1 25273 3-95680 27I38 3-68485 49 12 21621 4-62518 23455 4-26352 25304 3-95196 27169 3-6806 48 13 21651 4-6I868 23485 4-25795 25335 3.94713 27201 3-67638 47 I4 21682 4.61219 235I6 4.25239 25366 3.94232 27232 3-67217 46 i5 2I712 4.60572 23547 4-24685 25397 3-93751 27263 3.66796 45 i6 21743 4.59927 23578 4.24132 25428 3.9327I 27294 3-66376 44 17 21773 4-59283 23608 4-23580 25459 3-92793 27326 3.65957 43 18 21804 4-5864I 23639 4-23030 25490 3-923I6 27357 3-65538 42 19 21834 4-5800I 23670 4-2248I 25521 3-91839 27388 3.65121 41 20 21864 4.57363 23700 4-21933 25552 3-91364 27419 3.64705 40 21 2189.5 4-56726 23731 4-21387 25583 3-90890 27451 3-64289 39 22 21925 4-56091 23762 4-20842 2564 390417 27482 3.63874 38 23 2I956 |4-55458 23793 4-20298 25645 3-89945 275I3 3-63461 37 24 21986 4-54826 23823 4I9756 25676 3-89474 27545 3-63048 36 25 22017 4.54196 23854 4-I9215 25707 3-89004 27576 3.62636 35 26 22047 4-53568 23885 4-18675 25738 3.88536 27607 3-62224 34 27 22078 4.52941 23916 4.I8137 25769 3.88068 27638 361814 33 28 22I08 4.523i6 23946 4-I7600 25800 38j60oI 27670 3-61405 32 29 22139 4.5I693 23977 4.I7064 2583I 3-87I36 2770I 3-60996 31 30 22I69 4-5I07I 24008 4.i6530 25862 3 86671 27732 3-60o88 30 31 22200 4-5045I 24039 4.I5997 25893 3-86208 27764 3.60oI8 29 32 2223I 4-49832 24069 4. 5465 25924 3.85745 27795 3'59775 28 33 22261 4.492I5 24zO1 4-14934 25955 3-85284 27826 3-59370 27 34 22292 4.48600 24I3I 41I4405 25986 3-84824 27858 3-58966 26 35 22322 4.47986 24162 4-13877 26017 3-84364 27889 3.58562 25 36 22353 4.47374 24193 4-3350 26048 3-83906 27920 3-58I60 24 37 22383 4-46764 24223 4.12825 26079 3.83449 27952 3-57758 23 38 22414 4.46I55 24254 4-12301 26110 3.82992 27983 3-57357 22 39 22444 4-45548 24285 4-11778 26141 3-82537 28015 3.56957 21 40 22475 4.44942 24316 4-11256 26172 3-82083 28046 3-56557 20 41 22505 4.44338 24347 4-10736 26203 3.8i630 28077 3-56I59 19 42 22536 4-43735 24377 4-102I6 26235 3.81177 28109 3-55761 i8 43 22567 4.43134 24408 4-9699 26266 3.80726 28140 3.55364 17 44 22597 4.42534 24439 4-09182 26297 3-80276 28I72 3-54968 i6 45 22628 4.41936 24470- 4o8666 26328 3-79827 28203 354573 15 46 22658 4.4I340 2450I 4-08152 26359 3-79378 28234 3-54I79 14 47 22689 440745 24532 4-07639 26390 3-78931 28266 3-53785 13 48 22719 4.40152 24562 4.07I27 2642I 3-78485 28297 3.53393 12 49 22750 4.39560 24593 4-066i6 26452 3-78040 28329 3-53001 o ii 50 22781 4.38969 24624 4o06107 26483 3-77595 28360 3-52609 IO 51 22811 4.38381 24655 4.05599 26515 3-77152 28391 3-52219 9 52 22842 4-37793 24686 4-o5o92 26546 3.76709 28423 3'5I829 8 53 22872 4.37207 247I7 40o4586 26577 3.76268 28454 3-5I44I 7 54 22903 4.36623 24747 4-0408I 26608 3.75828 28486 3-5I053 6 55 22934 436040 24778 4-o3578 26639 3.75388 28517 3-50666 5 56 22964 4-35459 24809 14-03075 26670 3-74950 28549 3.50279 4 57 22995 4.34879 24840 4-02574 267o0 3-74512 28580 3-49894 3 58 23026 434300 24871 402074 26733 374075 28612 3-49509 2 59 23056 4.33723 24902'4-0I576 26764 3.73640 28643 3-49125 I 60 23087 4.-33148 24933 401078 26795 3.73205 28675 3 48741 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. I 160 P450 1 40 TABLE III. NATURAL TANGENTS AND COTANGENTS. 7 16~ 1o7 180 19~ Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. 0 28675 3 4874i 30573 3 27085 32492 3 o7768 34433 2-90421 6o I 28706 3 48359 30605 3 326745 32524'3-07464 34465 290147 59 2 28738 3.47977 30637 3.26406 32556 3-0o760 34498 2-89873 58 3 28769 3 47596 30669 3-26067 32588 3-o6857 34530 2-89600 57 4 28800 3.472i6 30700 3.25729 32621 30o6554 34563 2.89327 56 5 28832 3.46837 30732 3.25392 32653 30o6252 34596 2.89055 55 6 28864 3.46458 30764 3-25055 32685 3.o595o 34628 2.88783 54 7 28895 3-46080 30796 3.24719 32717 30o5649 34661 2-8851 53 8 28927 3 45703 30828 3.24383 32749 30o5349 34693 2.88240 52 9 28958 3345327 3o86o 3.24049 32782 3o05049 34726 2 87970 5 Io 28990 3-44951 30891 3 23714 32814 3-o4749 34758 2-87700 50 II 29021 3.44576 30923 3.23381 32846 3.04450 34791 2.87430 49 12 20053 3.44202 30955 3.23048 32878 3-04I52 34824 2. 87161 48 13 29o84 3 43829 30987 322715 32911 30 3854 34856 2.86892 47 14 29116 3.43456 31019 3.22384 32943 3-o3556 34889 2.86624 46 15 29147 3.43084 31051 3-22053 32975 3-03260 34922 2.86356 45 i6 29179 3-42713 3o183 3.2I722 33007 3 -02963 34954 2 86089 44 17 29210 3.42343 3i115 3.21392 33040 3.02667 34987 2.85822 43 i8 29242 3.41973 3II47 3.2Io63 33072 3-02372 35019 2.85555 42 19 29274 3-41604 31178 3 -20734 33o14 3.-02077 35052 2 285289 41 20 29305 3.41236 31210 3 20406 33136 3.o0783 35o85 2.85023 40 21 29337 3.40869 31242 3.20079 33169 3.01489 35117 2.84758 39 22 29368 3.40502 31274 3 19752 3320I 3 II196 35150 2.84494 38 23 29400 3-40i36 3i306 3.19426 33233 3oo00903 3583 2-84229 37 24 29432 3.3977I 3i338 3. 9100 33266 3-oo006 35216 2.83965 36 25 29463 3.39406 31370 3.18775 33298 3.oo319 35248 2.83702 35 26 29495 3'39042 3I402 3.I845I 33330 3-00028 35281 2.83439 34 27 29526 3.38679 3I434 3.8127 33363 2.99738 353I4 2.83176 33 2 29558 3.38317 31466 3.17804 33395 2.99447 35346 2.82914 32 29 29590 3-37953 3I498 3.I748i 33427 2-9958 35379 2-82653 31 30 29621 3.37694 31530 3.17159 33460 2.98868 35412 2-8239g 30 31 29653 3.37234 31562 3;.6838 33492 2.9858o 35445 2.82130 29 32 29685 3.36875 31594 3.I6517 33524 2'98292 35477 2.81870 28 33 29716 3'36516 31626 3'16197 "33557 2'98004 355o0 2.8I610 27 34 29748 3.36.58 3i658 3.I5877 33589 2-97717 35543 2.81350 26 35 29780 3-35800 31690 3.i5558 33621 2-97430 35576 2-81091 25 36 29811 3.35443 31722 3-I5240 33654 2-97144 35608 2.80833 24 37 29843 335087 31754 3.I4922 33686 2.96858 35641 2-80574 23 38 29875 3-34732 31786 3.14605 33718 2-96573 35674 2.80316 22 39 29906 3.34377 31818 3.14288 3375I 2'96288 35707 2.80059 21 40 29938 3.34023 31850 3.-3972 33783 2-96004 35740 2-79802 20 4I 29970 3-33670 31882 3.13656 33816 2-95721 35772 279545 19 42 3000oo 3.333317 3 194 3.1i334I 33848 2.95437 35805 2.79289 I8 43 30033 3.32965 31946 3.13027 3388i 2.95I55 35838 2.79033 17 44 30065 3.32614 31978 3. 12713 33913 2.94872 35871 2.7 778 16 45 30097 3.32264 32010 3.12400 33945 294590 35904' 2.78523 I5 46 30I28 3.3I914 32042 3.I2087 33978 2-94309 35937 2-78269 14 47 30160 3.3I 65 32074 3.11775 34010 2-94028 35969 2.780I4 13 48 30192 3-312I6 32106 3. 1464 34043 2.93748 36002 2.77761 12 49 30224 3.30868 32139 3.zIt53 34075 2-93468 36035 2.77507.11 50 30255 3.3052I 32171 3.10842 34o18 2-93189 36068 2.77254 1o 5I 30287 3.30I74 32203 3.1o532 34I40 2.929I0 36IOI 2.77002 9 52 30319 3'29829 32235 3.10223 34I73 2'92632 36I34 2.76750 8 53 3035i 3.29483 32267 30o9914 34205 2.92354 36I67 2.76498 7 54 30382 3.29139 32299 3.09606/ 34238 2.92076 36I99 2.76247 [ 55 304i4 3-28795 32331 30o9298 34270 2'9I799 36232 2.75996 5 56 30446 3.28452 32363 30o8991 34303 2-91523 36265 2.75746 4 57 30478 3.28o09 32396 3.o8685 34335 2.91246 36298 2.75496 3 30509 3.27767 32428 3.08379 34368 2.9097I 36331 2.75246 2 59 3054i 3.27426 32460 3.08073 34400 2.90696 36364 2.74997 I 6o 30573 3.27085 32492 3.07768 34433 2.90421 36397 2.74748 0 Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. I ___ ____ _____ 2 __ __ 73~ 720 710 70 78 NATURAL TANGENTS AND COTANGENTS. TABLE III. 20~_0 210~ 1 22~ 23~ - Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 36397 2.-74748 38386 2.60509 40403 2-47509 42447 2.35585 60 I 36430 2-74499 38420 2.60283 40436 2-47302 42482 2.35395 59 2 36463 2.74251 38453 2.60057 40470 2-47095 42516 2.35205 58 3 36496 2.74004 38487 2-5983I 40504 2.46888 42551 2-35015 57 4 36529 2-73756 38520 2.59606 40538 2.46682 42585 2i34825 56 5 36562 2-73509 38553 2.59381 40572 2.46476 42619 2.34636 55 6 36595 2.73263 38587 2.59156 40606 2.46270 42654 2.34447 54 7 36628 2.730I7 38620 2.58932 40640 2.46065 42688 2.34258 53 8 36661 2.7277I 38654 2.58708 40674 2-45860 42722 2-34069 52 9 36694 2-72526 38687 2.58484 40707 2.45655 42757 2.33881 5I io 36727 2-72281 38721 2.58261 40741 2-45451 42791 2.33693 50 ii 36760 2-72036 38754 2.58038 40775 2.45246 42826 2.33505 49 12 36793 2-71792 38787 2-57815 40809 245043 42860 2.33317 48 13 36826 2.71548 38821 2.57593 40843 2-44839 42894 2-33130 47 I4 36859 2.71305 38854 2.57371 40877 2.44636 42929 2.32943 46 15 36892 2.71062 38888 2-57I50 40911 2-44433 42963 2.32756 45 I6 36925 2.70819 38921 2.56928 40945 2-44230 42998 232570 44 I7 36958 2.70577 38955 2.56707 40979 2.44027 43032 2.32383 43 I8 3699I 2-70335 38988 2.56487 410o3 2.43825 43067 2.32197 42 19 37024 2.70094 39022 2.56266 41047 2-43623 43IoI 2-32012 41 20 37057 2.6985 39055 2-56046 4108I 2.43422 43 I36 2.31826 40 21 37090 2.69612 39089 2.55827 41115 2.43220 43170 2.31641 39 22 37124 2-69371 39122 2.55608 41149 2'43019 3205 2.31456 38 23 37157 2-69I31 39156 2.55389 41183 2.42819 43239 2.3127I 37 24 37190 2.68892 39I90 2-55I70 41217 2.426I8 43274 2.3I086 36 25 37223 2.68653 39223 2.54952 4125I 2-42418 43308 2.30902 35 26 37256 2-684I4 39257 2.54734 41285 2.42218 43343 2.30718 34 27 37289 2.68175 39290 2.54516 41319 2-420I9 43378 2.30534 33 28 37322 2.67937 39324 2.54299 41353 2-41819 43412 2.30351 32 29 37355 2-67700 39357 2.54082 41387 2.41620 43447 2.30167 3I 30 37388 2-67462 39391 2.53865 41421 2 41421 43481 2-29984 30 3I 37422 2.67225 39425 2.53648 41455 2-41223 435i6 2.2980I 29 32 37455 2-66989 39458 2.53432 41490 2-41025 43550 2-29619 28 33 37488 2 66752 39492 2 -53217I 41524 2 40827 43585 2-29437 27 34 37521 2.665I6 39526 2-53001 41558 2.40629 43620 2-29254 26 35 37554 2-66281 39559 2.52786 41592 2.40432 43654 2-29073 25 36 37588 2-66046 39593 2.5257I 41626 2.40235 43689 2.28891 24 37 37621 2-65811 39626 2.52357 41660 2.40038 43724 2.28710 23 38 37654 2.65576 39660 2.52142 41694 2.39841 43758 2-28528 22 39 37687 2.65342 39694 2.51929 41728 2.39645 43793 2.28348 21 40 37720 2-65109 39727 2.-5715 41763 2-39449 43828 2.28167 20 4i 37754 2.64875 3976I 2-51502 41797 2.39253 43862 2-27987 19 42 37787 2.64642 39795 2.51289 4I83I 2-39058 43897 2-27806 I8 43 37820 2-644I0 39829 2.5I076 41865 2.38862 43932 2-27626 17 44 37853 2.64I77 39862 2.50864 41899 2.38668 43966 2-27447 i6 45 37887 2.63945 39896 2.50652 41933 2.38473 44001 2-27267 i5 46 37920 2.63714 39930 2.50440 41968 2.38279 44036 2-27088 14 47 37953 2.63483 39963 2.50229 42002 2.38084 44071 2-26909 i3 48 37986 2.63252 39997 2.50018 42036 2-37891 44105 2-26730 12 49 38020 2-63021 40031 2-49807 42070 2.37697 44140 2-26552 II 50 38053 2.62791 40065 2.49597 42105 2.37504 44175 2-26374 I0 51 38086 2.62561 40098 2.49386 42139 2-373II 44210 2-26196 9 52 38120 2623 432 32 2.49177 42173 2-37II8 44244 2-26018 8 53 38153 2.62I03 40166 2-48967 42207 2.36925 44279 2.25840 7 54 38i86 2.61874 40200 2.48758 42242 2.36733 44314 2.25663 6 55 38220 2.6I646 40234 2.48549 42276 2-3654I 44349 2-25486 5 56 38253 2-614I8 40267 2-48340 42310 2.36349 44384 2-25309 4 57 38286 2.61190 4030o 2-48I32 42345 2-36I58 444i8 2-25132 3 58 38320 2.60963 40335 2.47924 42379 2.35967 44453 2.24956 2 59 38353 2.60736 40369 2-47716 42413 2.35776 44488 2.24780 I 60 38386 2-.6509 43403 2-47509 42447 2.35585 44523 2-24604 0 Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. / ___90 80 66 69~ 68~ 6^7~ 660 TABLE III. NATURAL TANGENTS AND COTANGENTS. 79 240 250 260 2_ 0 0_ Tangent. Cotang. Tangent. Cotang. Tangent. Cota. Tangent. Cotang. o 44523 2 24604 46631 2 I445I 48773 2 o5030 50953 I 96261 6o 44558 2.24428 46666 2.1 4288 48809 2-04879 50989 1.96120 59 2 44593 2 24252 46702 2-I4125 48843 2 04728 51026.95979 58 3 44627 2.24077 46737 2'13963 48881 2-04577 51o63 1.95838 57 4 44662 2.23902 46772 2.I3801 48917 2-04426 51099 95698 56 5 44697 2-23727 46808 2-13639 48953 2-04276 5ii36 I.95507 55 6 44732 2 23553 46843 2.I3477 48989 2 04125 51173'95417 54 7 44767 2.23378 46879 2-I3316 49026 2-03975 51209 I95277 53 8 44802 2.23204 46914 2.-3154 49062 2-03825 51246 I 95137 52 9 44837 2.23030 46950 2.12993 49098 2-03675 51283 I-94997 51 10 44872 2.22857 46985 2.12832 49134 2-03526 51319 I.9488. 50 II 44907 2.22683 47021 2.12671 49I70 2-03376 5I356 194718 49 12 44942 2.22510 47056 2I12511 49206 2-03227 51393 I-94579 48 13 44977 2-22337 47092 2-12350 49242 2-03078 51430 I194440 47 14 450I2 2.22164 47128 2.12190 49278 2-02929 51467 I-9430I 46 15 45047 2-21992 47163 2-12030 49315 2o02780 51503 1-94162 45 I6 45082 2.21819 47199 2-II871 49351 2-02631 51540 I-94023 44 17 45117 2-21647 47234 2-11711 49387 2-02483 51577 I-93885 43 18 45152 2.21475 47270 2.1I552 49423 2-02335 51614 I 93746 42 19 45187 2-2I304 47305 2 - 392 49459 202187 5165I.93608 41 20 45222 2-2JJ32 4734I 2-11233 49495 2-02039 5i688 I-93470 40 21 45257 2-2096I 47377 2-II075 49532 2-0189g 5I724 -I93332 39 22 45292 2-20790 47412 2.Og916 49568 201743 51761 I.93I95 38 23 45327 2.20619 47448 2.10758 49604 2-I0596 51798 I'93057 37 24 45362 2.20449 47483 2-o0600 49640 2 01449 51835 192920 36 25 45397 2.20278 47519 2.0o442 49677 201302 51872 1.92782 35 26 45432 2.20108 47555 2-10284 49713 2-01155 51909 192645 34 27 45467 2-I9938 47590 2101I26 49749 2I01008 5I946 1-92508 33 28 45502 2.19769 47626 2og9969 49786 2-00862 51983 1-92371 32 29 45537 2.I9599 47662 2-09811 49822 2-00715 52020 I192235 31 30 45573 2.19430 47698 2-09654 49858 2-oo569 52057 I192098 30 31 45608 2-1926I 47733 209498 49894 2-00423 52094 Iz91962 29 32 45643 2.19092 47769 2-09341 49931 2.00277 52131 Ig91826 28 33 45678 2-18923 47805 2.09184 49967 2-00I3I 52168 I9gI690 27 34 45713 2.18755 47840 2o09028 50004 I-99986 52205 I-91504 26 35 45748 2.18587 47876 2.o8872 50040 1-99841 52242 1I91418 25 36 45784 2.18419 47912 2-087 50076 6 99695 52279 -.91282 24 37 45819 2-1825I 47948 2-o8560 50o13 1 99550 52316 gI91147 23 38 45854 2.18084 47984 2.08405 50149 I99406 52353 -I91012 22 39 45889 2.I7916 480I 2-08250 50180 19926I 52390 I-90876 21 40 45924 2 I7749 48055 2o8094 50222 1 99116 52427 I-90741 20 41 45960 2.17582 48091 2-07939 50258 198972 52464 I-90607 I9 42 45995 2.I7416 48127 2.07785 50295 -98828 5250o I 90472 I8 43 46030 2-I7249 48163 2.07630 50331 I-98684 52538 I 90337 17 44 46065 2-17083 48198 2.07476 50368 I-98540 52575 I190203 16 45 46101 2 gI6917 48234 2-07321 50404 I198396 52613.900o69 15 46 46136 2.16751 48270 2-07167 50441 I-98253 52650 189935 14 47 46171 2-16585 48306 2-07014 50477 I-98I1 52687 I-8980I 13 48 46206 2-16420 48342 2-06860 505I4 -97966 52724 I-89667 12 49 46242 2.I6255 48378 2.06706 50550 I97823 5276I -189533 II 50 46277 2.16090 484I4 2-06553 50587 -197680 52798 1-89400 o1 51 46312 2.15925 48450 2.06400 50623 I97538 52836 I-89266 9 52 46348 2-15760 48486 2-06247 50660 -97395 52873 -189133 8 53 46383 2.I5596 48521 2-06094 50696 1-9723 529IO 1-8 000 7 54 46418 2-15432 48557 2-o5942 50733 I-97II 52947 i.88867 6 55 46454 2-15268 48593 2-05790 50769 I96969 52984 1-88734 5 56 46489 2.15I04 48629 2-05637 50806 I96827 53022 i.88602 4 57 46525 2-14940 48665 2-05485 50843 1-96685 53059 -88469 3 58 46560 214777 48701 2-05333 50879 196544 53096 I-88337 2 59 46595 2.14614 48737 2-05I82 509I6 I-96402 53I34 I-88205 1 60 46631 2-14451 48773 2-05030 50953 -I9626I 53171 1-88073 0 Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 650 640 63~ 620 S-9 NATURAL TANGENTS AND COTANGENTS. TABLE IIL 280 290 300 310 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 53171 I'88073 5543i I-80405 57735 1-73205 6oo0086 1-66428 60 I 53208 I-87941 55469 1-8028I 57774 1-73089 60126 I 66318 59 2 53246 I.87809 55507 1-80158 57813 1-72973 6oi65 1-66209 58 3 53283 1-87677 55545 800oo34 5785I 1-72857 60205 1.66099 57 4 53320 I-87546 55583 6179911 57890 1-72741 60245 I-65990 56 5 53358 1.87415 55621 I-79788 57929 1-72625 60284 i 6588I 55 6 53395 1-87283 55659 1-79665 57968 172509 60324 I.65772 54 7 53432 1-87152 55697 I-79542 58007 1-72393 60364 i.65663 53 8 53470 1-87021 55736 1-79419 58046 I.72278 60403 i.65554 52 9 53507 I-86891 55774 I-79296 58085 I 72163 60443 i-65445 5I 10 53545 I-86760 558I2 1*79174 58124 1-72047 60483 I.65337 5o ii 53582 i-8663o 5585o0 I7905I 58162 I719I32 60522 I 65228 49 12 53620 I-86499 55888 I-78029 58201 I 71817 60562 I 65I20 48 13 53657 I86369 55926 1-78807 58240 1-71702 60602 I.65011 47 14 53694 I-86239 55964 I-78685 58279 I-71588 60642 -I64903 46 i5 53732 I-86109 56oo003 I-78563 58318 I-71473 6068i I-64795 45 I6 53769 I-*85979 56041 1 - 78441 58357 1-71358 60721 -64687 44 17 53807 i-85850 56079 1-78319 58396 1-71244 60761 I.64579 43 I8 53844 I-85720 56117 1-78198 58435 1I71129 6080oi I-64471 42 19 53882 I-8559I 56i56 178077 58474 I 71015 60841 1.64363 4I 20 53920 I.85462 56194 1-77955 585i3 1-70901 6088 1.64256 40 2I 53957 I-85333 56232 1-77834z 58552 1-70787 60921 1.64148 39 22 53995 I-85204 56270 1.77713 58591 1-70673 60960 I-64041 38 23 54032 I-85075 56309 i-77592 5863i 1I70560 6iooo i-63934 37 24 54070 1-84946 56347 -77471 58670 1-70446 6040 I-63826 36 25 54107 1-84818 56385 I-77351 58709 170332 6io080 I637193 5 26 54145 I-84689 56424 I-77230 58748 1-70219 61120 1.63612 34 27 54183 I-8456i 56462 1-77110 58787 1-70106 6ii6o i 635o5 33 28 54920 1-84433 565oo00 I76990 58826 1-I69992 61200 I-63398 32 29 54258 I 843o5 56539 I-76869 58865 1.69879 61240 I-63292 3i 30 54296 I-84177 56577 I'76749 58904 1-69766 61280 I.63185 30 31 54333 I-84049 56616 I-76630 58944 1-69653 61320 I 63079 29 32 54371 I-83922 56654 I.76510 58983 1.69541 6I360. I62972 28 33 54409 -183794 56693 I.76390 59022 1-69428 61400 I.62866 27 34 544446 I-83667 56731 I-7627I 59061 1.69316 6I440 I.62760 26 35 54484/ i.8354o 56769 I-76151 59101 69203 6i480 I.62654 25 36 54522 I-834I3 56808 I-76032 59140 I 69091 61520 I 62548 24 37 54560 I-83286 56846 -75913 59179 I-68979 6I56I -62442 23 38 54597 -83159 56885 1-75794 59218 -68866 6i6oi i.62336 22 39 54635 i-83o33 56923 1-75675 59258 I-68754 6164I -I62230 21 40 54673 182906 56962 1I75556 59297 I.68643 6i68i -I62125 20 41 547ii I-82780 57000 I-75437 59336 i-6853i 61721 1.62019 19 42 54748 -I82654 57039 1-75319 59376 I.68419 61761.6 I914 18 43 54786 1-82528 57078 1-75200 594I5 i.683o8 6i8oi 1 61i8o8 17 44 54824 1i-82402 57116 1-75082 59454 i-68i96 61842 1i61703 16 45 54862 1-82276 57155 174964 59494 i-68085 61882 I 61598 1I5 46 54900 I-82150 57193 I-74846 59533 I-67974 61922 I-61493 14 47 54938 i-82025 57232 1-74728 59573 i-67863 6i962 i-6I388 13 48'54975 I 81899 57271 I-74610 59612 I-67752 62003 I61283 12 49 550o3 I81774 57309 I74492 59651 I-67641 62043 i-61179 II 50o 5505i i81649 57348.74375 5969I I.6753o 62083 I.61074 10 51 55o89 -81524 57386 174257 59730 i-67419 62124 1.60970 9 52 55127 I-81399 57425 I-74140 59770 I-67309 62164 I-60865 8 53. 55i65 1-81274 57464 I-74022 59809 167198 62204 1.6076I 7 54 55203 i-8ii5o 57503 I.73905 59849 I.67088 62245 1-60657 6 55 55241 1.81025 57541 1.73788 88 5988 i66978 62285 -16o553 5 56 55279 1.8090oi 57580 i.73671 59928 1.66867 62325 I-60449 4 57 553o5 7 1-80777 57619 I-73555 59967 1I66757 62366.60345 3 58 55355 1-80653 57657 1-73438 60007 I.66647 62406 1.60241 2 59 55393 1-80529 57696 I.73321 6oo0046 i66538 62446 I60137 I 60 5543I 1.80402 57735 I 73205 6oo0086 66428 62487 I-60033 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. o t __ ___.1__ _ _ Oo.__ __i___ ~ 61~ 60~ 59~ 568~ TABLE III. NATURAL TANGENTS AND COTANGENTS. 81 8320 330 340 350 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 62487 I 60033 64941 I5386 6745I 1 48256 70021 142815 60 I 62527 1.59930 64982 5388 67493 I.48163 70064 1.42726 5 2 62568 1I59826 65023 I 5379I 67536 1'48070 70107 1-42638 58 3 62608 1.59723 65065.53693 67578 1.47977 70I51 I-42550 57 4 62649.-59620 65o06 1.53595 67620 I'47885 70194 1.42462 56 5 62689 1.595I7 65148 I'53497 67663 I147792 70238 I.42374 55 6 62730 1.59414 65I89 I.534oo 67705 I.47699 70281 1.42286 54 7 62770 1.59311 65231 I.53302 67748 1.47607 70325 1.42I98 53 8 6281I 1159208 65272 1.53205 67790 1.47514 70368 1.42110 52 9 62852 I'59I05 65314 1.53107 67832 I.47422 70412 I.42022 51 o 62892 I 59oo2 65355 I.53oio 67875 1.47330 70455 I 4I934 50 I 62933 -58900 65397 I.52913 679I7 Ir47238 70499 1' 4147 49 12 62973 I.58797 65438 I-52816 67960 1.47146 70542 I 41759 48 13 630o4 I.58695 65480 1.5279I 68002 I.47053 70586 i'4672 47 I4 63055 I 58593 65521 I.52622 68045 I.46962 70629 I.41584 46 15 63095 I.58490 65563 I.52525 68088 I.46870 70673 -41497 45 i6 63136 i.58388 65604 I-52429 68130 1-46778 70717 I14I409 44 I7 63177 1.58286 65646 1.52332 68173 1.46686 70760 1.41322 43, i8 63217 I.58I84 65688 1.52235 68215 1-46595 70804 I-4I235 42 I9 63258 1-58083 65729 I.52139 68258.146503 70848 1 41148 41 20 63299 I 5798I 65771 I-52043 6830I r.464I 70891 I.4I0o6I 40 21 63340 1 57879 658i3 I.51946 68343 I 46320 70935 I 40974 39 22 63380 I 57778 6 65854 8386 46229 70979 40887 3 23 63421 157676 65896 1'51754 68429 I-46I37 71o23.4o0800 37 24 63462 1-57575 65938 I.5I658 68471 I.46046 71066 I-40714 36 25 63503 I 57474 65980 I51I562 685I4 -145955 71110 1.40627 35 26 63544 I 57372 66021 i-51466 98557 I.45864 71154 40o540 34 27 63584 1-5727I 66063 -5I37 68600 686o 45773 7I198 I.40454 33 28 63625 I 57170 66io5 I 51275 68642.1.45682 71242 I.40367 32 29 63666 1-57069 66147 I-51179 68685 I-45592 7I285 14028I 3I 30 63707 156969 66189 i.5o084 68728 I.455o0 71329 1.40195 30 31 63748 I-56868 66230 1.50988 68771 I.454I0 71373 1.40109 29 32 63789 I.56767 66272 I.50893 688I4 I 45320 71417 1.40022 28 33 63830 I-56667 66314 1.50797 68857 I-45229 71461 I.39936 27 34 63871 i.56566 66356 1.50702 68900 I 45I39 715035 139850 26 35 63912 I.56466 66398 I.50607 68942 I-45049 71549 1 39764 25 36 63953 I-56366 66440 1.50512 68985 I 44958 71593 I 39679 24 37 6399/4 56265 66482 1-50417 69028 I 44868 7I637 I-39593 23 38 64o05 i.56I65 66524 I.50322 69071 I 44778 71681 I 39507 22 39 64076 I-56065 66566 I.50228 69114 I.44688 71725 I.39421 21 40 64117 I-55966 66608 I-50o33 69I57 I-44598 7I769 I-39336 20 41 64158.-55866 66650.500o38 69200.44508 7I813.39250 19 42 64199 I-55766 66692 I-49944 69243 I.44418 71857 I.39165 18 43 64240 I 55666 66734 I.49849 69286 1.44329 71901 1-39079 17 44 64281 I-55567 66776 1 49755 69329 I.44239 71946 I.-38994 16 45 64322 I-55467 66818 I.49661 69372 I-44149 71990 1-38909 15 46 64363 I.55368 66860 I.49566 69416 I-44060 72034 1.38824 14 47 64404 1-55269 66902 1 49472 69459 I.43970 72078 1-38738 13 48 64446 I.55170 66944 I 49378 69502 I 43881 72122 I.38653 12 49 64487 I155071 66986 1.49284 69545 I-43792 72166 I.38568 ii 50 64528 1.54972 67028 1-49190 69588 I.43703 72211 1.38484 o0 5i 64569 1-54873 67071 1-49097 6963I I.436I4 72255 I-38399 9 52 646io0 I54774 67113 z 49o003 69675 I.43525 72299 I 38314 8 53 64652 I.54675 67155 1 489 69718 I.43436 72344 I-38229 7 54 64693 I.54576 67197 1-48816 69761 I.43347 72388 I-38I45 6 55 64734 I-54478 67239 I.48722 69804 I.43258 72432 i.38060 5 56 64775 1.54379 67282 1-48629 69847 I-43I69 72477 I.37976 4 57 64817 5428I 67324 1.48536 6989 I-43o80 72521 I-3789I 3 5 64858 I54183 67366 1-48442 69934 I.42992 72565 I-37807 2 59 64899 I -54085 67409 1-48349 69977 I-42903 726Io0 I37722 I 60 64941 1-53986 67451 i148256 70021 1-42815 72654 I-37638 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 5 7 56~ 55~ 54~ 82 NATURAL TANGENTS AND COTANGENTS. TABLE III.360 ____ 0 __ -380 - 390 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent.| Cotang. 0 72654 I 37638 75355 1.32704 78129 1 27994 80978 1 23490 60o 1 72699.37554 7540 1.32624 78170 1 27917 81027 I 23416 59 2 72743 1.37470 75447 I.32544 78222 1I27841 81075 1.23343 58 3 72788 1.37386 75492 1.32464 78269 1.27764 81123 1 23270 57 4 72832 1 37302 75538 1.32384 78316 1 27688 811-7 1 23196 56 5 72877.372I8 75584.32304 78363 I.27611 81220 1 23123 55 6 72921 I 37I341 7629 I'32224 78410 1.27535 81268 I23050 54 7 72966 1I37050 75675 J-32144 78457 I-27458 8I3I6 I22977 53 8 73010.36967 75721 1 32064 78504 1.27382 8I364 I 22904 52 9 73055 i.36883 75767 I-31984 78551 I127306 81413 1i22831 5I 10 73100 i~368oo 75812 1.31904 78598 1.27230 8146I 1.22758 50 II 73144 1-367I6 75858 1.3I825 78645 I 27153 8I5Io 1.22685 49 12 73189 -36633 75904 1.31745 78692 I127077 8I558 1.22612 48 I3 73234 I.36549 75950 I131666 78739 1-2700i 8I606 1.22539 47 14 73278 1.36466 75996 I-31586 78786 1-26925 8i655.22467 46 15 73323 i-36383 76042 I 3I507 78834 I26849 81703 1223941 45 i6 73368 i-363oo 76088 I131427 78881 I.26774 81752 I-22321 44 17 73413 1.362I7 76I34 1.31348 78928 1.26698 81800 1.22249 43 a,8 73457 I.36I33 76I80 1.31269 78975 1.26622 8I84 1.22I76 42 19 73502 I.3605i 76226 1.31I90 79022 1.26546 81898 I.22104 41 20 73547 1.35968 76272 i.3III0 79070 1.26471 81946 1.22031 40 21 73592 I.35885 76318 -1.303ii 7917 1.26395 81995 -2i959 39 22 73637 1.35802 76364 1.30952 79164 126319} 82044 1.21886 38 23 7368I I.35719 764I0 1.30873 79212 I20244/ 82092 1 21814 37 24 73726 I.35637 76456 -130795 79259 1-26169 82141 1.21742 36 25 73771 i.35554 76502 1.30716 79306 1.26093 82Q10 1 21670 35 26 73816 I-35472 76548 1.30637 79354 1.26018 82238 1.2I598 34 27 7386I 1.35389 76594.30o558 79401 1.25943/ 82287 1.21526 33 28 73906 1 35307 76640 1.30480 79449 1.25867 82336 1.21454) 32 29 73951 1.35224 76686 1.30401 79496 1.25792 82385 121382 31 30 73996 I35142 76733 130323 79544.25717 82434 121310 30 3I 74041 1-35060 76779 1-30244 79591 1.25642 82483 1.21238 2 32 74086 1-34978 76825 30oi66 79639 1.25567 82531 1.21166 28 33 7413I 1-34896 76871 1.30087 79686 1-25492 82580 1.21094 27 34 74176 1.34814 76918 130009 79734 1253417 82629 1.2I023 26 35 74221 1 34732 76964 1.29931 79781 1.25343 82678 1.20951 25 36 74267 I.34650 77010 1.29853 79829 1.25268 82727 1.20879 24 37 74312 -34568 7057 1.29775 79877 125193 82776 1.20808 23 38 74357 -34487 77103 1.29696 79924 1.25118 82825 1.20736 22 39 74402 i 34405 77149 I129618 79972 1.25044 82874 1.20665 21 40 74447 134323 77196 I.29541 80020 1.24969 82923 1.20593 20 4I 74492 1.34242 77242 I.29463 80067 1.24895| 82972 1.20522 19 42 74538 I-3416o 77289 I.29385 I 8oii5 1-24820 83022 1-20451 18 43 74583 i-34079 77333 129307 80o63 1-24746 83071.20379 17 44 74628 1-33998 77382 1.29229 80211 I-24672 83120 1-20308 16 45 74674 I33916 77428 1.29152 80258 1.24597 83169 I-20237 I 46 74719 i-33835 77475 1 29074 80306 1.24523 83218 1.20166 14 47 74764 i.33754 77521 1.28997 80354 1.24449 83268 1.20095 3 48 74810 1.33673 77568 1.28919 80402 -.24375 83317 1.20024 12 49 74855 1.33592 776I5 1.28842 8o45c I12430I 83366 1I.9953 i 50 74900 1i.335II 77661 1.28764 80498 1.24227 834I5 I.9882 1O S5I 74946 1.33430 77708 1.28687 80546 1.24153 83465 i.1i981 9'52 74991 1-33349 77754 1.28610 80594 1.24079 83514.19740 8 53 75037 1.33268 77801 1 28533 80642 1.24005 83564 1.19669 7 54 75082 1.33187 77848 1.28456 80690 1.23931 83613 119599 6 55 75128 1.33107 77895 1.28379 80738 1.23858 83662 1.i9528 5 56 75173 1.33026 77941 1.28302 80786 1.23784 83712 1.19457 4 5 75219 1-32946 77988 1-28225 80834 1.23710 83761 I1.9387 3 58 75264 I.32865 78035 I-28148 80882 1.23637 838I11 I.9316 2 59 75310 i.32785 78082 1-28071 80930 1.23563 8386o II9246 I oo 75355 1.32704 78129 I127994 80978 I123490 8391o 1ig975 0 Cotang. Tangent. Cotag. Tanent. Cota Tangent. Cotang. Tangent. 53I 5___ 51_ _ _ _0 53~ 02~ 51~ 500 TABLE II. NATURAL TANGENTS AND COTANGENTS. 83 1400 I.410 420 ______3______ a Cotang. anent. Cotan. Tangent. Cotang. Tanent. Cotan.g. __tangent. _.......... o 839 10 1-19175 18629 i150o37 90040 Iiio6i 93252 1I'07237 60 I 83960 1-19105 86980 1.14969 90093 110996 93306 1-07174 59 2 84009 1 19035 87031 1-14902 90146.o10931 93360 1-07112 58 3 84059 1-18964 87082 1-14834 90199 1.10867 93415 1.0o049 57 84 108 1.18894 87133 -14767i 9025I 10802 93469 -10698.7 56 5 84158 1s18824 87184 1.14699 o90304 1-10737 93524 1.O6925 55 6 842o8 1-18754 87236 -14632 j90357 1 10672 93578 1i06862 54 7 84258 I-i8684 87287 i1.4565 90410.10o607 93633 I.o68oo 53 8 84307 1I18614 87338 114498 90463 Ii10543 93688 1-06738 52 9 84357 1-18544 87389 I1.443o 90516 IO10478 93742 I.06676 5i o19 84407 118474 87441 14363 90569 10o414 93797.o6613 50o 1I 84457 1-18404 87492 1-14296 90621 1 10349 93852.o06551 40 12 84507 1-18334 87543 1.14229 90674 110285 93906 1~o06489 48 13 84556 1-18264 87595.1I4162 90727 1-10220 93961 I.06427 47 14. 846o6 1-18194 87646 I-14095 90781 1-10156 94016 ~o6365 46 i5 84656 1-18125 87698 I140281 90834 1.oo10091 94071.o63o3 45 i6 84706 1-18055 87749 I113961 90887 1.10027 94125 1-06241 44 17 84756 1-17986 87801 II3894 90940 1-09963 94180 1-06179 43 18 4 84806 1-17916 87852 II3828 90o993.-.09899 94235 1-061171 42 19 84856 1.17846 87904 I.13761 91046 1.09834 94290 i1o6o56 41 20 84906 117777 87955 1-13694 91099 1.09770 94345 I.05994 40 21 84956 1177o08 88007 1.13627 91153 -.09706 94400 1-05932 39 22 850o6 1-17638 88059 I'i356i 912o6 i.09642 94455 1.05870 38 23 85057 1.17569 88iIo 1-13494 91259 1.09578 94510 1-05809 37 24 85107 1/17500 88162 1-13428 91313 109514 94565 1-05747 36 25 85157 1-17430 88214 11.336I 91366.09450 94620 i-05685 35 26 85207 1.17361 88265 1-13295 91419 -09386 94676 1-05624 34 27 85257 1-17292 88317 1.13228 91473 I09322 94731 1-o5562 33 28 85307 1 17223 88369 9 113162 91526 1-09258 94786.o55oi 32 29 85358 I.17154 88421 1-13096 91580 -.09195 94841 1.05439 31 30 85408 I1-7085 88473 1-13029 91633 1o09131 94896 I.o5378 3o 31 85458 1-17016 88524 1 I2963 91687 1-09067 94952 1-05317 29 32 85509 1-16947 88576 1-12897 91740 I~9ooo3 95007 I-05255 28 33 85559 1-16878 88628 1.1283I1 91794 108940 95062 1-0 o5194 27 34 85609 1-16809 8868o0 112765 91847 1-08876 958 i-o5i3 26 35 8566o 1-16741 88732 1-12699 91901 o88i3 9517.3 1-05072 25 36 85710 1 -6672 88784 1-12633 91955 1-08749 95229 I -05010 24 37 8576i I-i66o3 88836 1.12567 92008 i.o8686 95284 1-04949 23 38 858i, i.i6535 88888 I-12501 92062 1.08622 95340 1-04888 22 39 85862 1-16466 88940 1.12435 92116 oS08559 95395 1-04827 21 40 85912 1-.6398 88992 1-12369 92170 108496 9543I 1-04766 20 4r 85963 1.16329 89045 -1230o 3 92223 1-08432 955o6 1-04705 19 42 86014 1-16261 89097 1-12238 92277 1.08369 95562 1-04644 18 43.86064 1/ 16192 89149.-12I72 92331 i.o83o6 95618 1-04583 17 44 86115 16124 89201 1.-12106 92385 1.08243 95673 1.04522 16 45 86i66 i-i6056 89253 1-12041 92439 1.08179 95729 1-04461 15 46 86216 1-15987 89306 1-11975 92493 i-o8ii6 95785 1-04401 14 47 86267 1-15919 89358 1-11909 92547 I.o8o53 95841 1-04340 13 48 863i8 i-i585i 89410 1-11844 92601 1-07990 95897 I-04279 I2 49 86368 1-15783 89463 -11778 92655 1-07927 95952 i-04218 i 50 86419 i-i57i5 89515 -11713i 927-0 1i-07864 96008 i-04i58 10 5i 86470 1-15647 89567 1-I1648 92762 1-07801 96064 1-04097 52 86521 -15579 89620. III582 92817 1-07738 96120 -.04036 53 86572 i-i55i 89672 I-11517 92872 1-07676 96176 -.03976 7 54 86623 i-i5443 89725 I.11452 92926 1.07613 96232.o3915 6 55 86674.15375 89777 I.11387 92980 I-07550 96288.o3855 5 56 86725.i53o8 89830 1-11321 93034 I-07487 96344 I-037941 4 57 86776 1-15240 89883 1-II256 93088 1-07425 96400 -.03734 3 58 86827 1.i5172 89935 1-11191 93143 1-07362 96467 1-03674 2 59 86878 i'51io4 89988 111126 93197 1-07299 96513 i036i3 6o 86929'150o37 90040 i-io6i1 93202 1.07237 96569 i.o3553 o Cotag. I Tangent. Cotang. Tangent. Cotang. I Tangent. Cotang. Tangent. 490 48~ I 47~ 46~ 84 NATURAL TANGENTS AND COTANGENTS. TABLE III. 440 440 Tangent. Cotang. Tangent. Cotang. 0 96569 i.o3553 60 31 98327 I-01702 29 I 96625 1.o3493 59 32 98384 I 01642 28 2. 96681 I0o3433 58 33 9844 1 ioi583 27 3 96738.o3372 57 34 98499 I-01524 26 4 96794 1 03312 56 35 98556 I.oi465 25 5 96850 1 03252 55 36 98613 I.oi 406 24 6 96907 -o3192 54 37 98671 IOI347 23 7 96963 I 03132 53 38 98728 I 01288 22 8 97020 0o3072 52 39 98786 I-I0229 21 9 97076 1i03012 51 40 98843 I-0117o 20 10 97133 1 02952 50 41 98901 I-01112 19 11 97189 1 02892 49 42 98958 1oio053 18 12 97246 I -02832 48 43 99016 I oo00994 17 13 97302 1-02772 47 44 99073 1 00oo935 16 14 97359 I-02713 46 45 99131 1-00876 15 15 97416 1-02653 45 46 99189 Ioo8i8 1 16 97472 I-02593 44 47 99247 I-00759 13 17 97529 1r02533 43 48 99304 1.00701 12 i8 97586 1 02474 42 49 99362 00oo642 II 19 97643 1 02414 41 50 99420 i.oo583 10 20 97700 1 02355 40 51 99478 1.oo00525 9 21 97756 1-02295 39 j2 99536 1oo00467 8 22 97813 1-02236 38 53 99594 I00408 7 23 97870 1-02I76 37 54 99652 i.oo350 6 24 97927 -02117 36 55 99710 1-00291 5 25 97984 1-02057 35 56 99768 1-00233 4 26 98041 I-01998 34 57 99826 1 I-00175 3 27 98098 i 01939 33 58 99884 I-oo116 2 28 98155 1-01879 32 59 99942 1-00ooo58 I 29 98213 1-01820 31 60 Urnit. Unit. o 30 98270 1.01761 30 Cotang. Tangent. Cotang. Tangent. 450 450 TABLE OF CONSTANTS. Base of Napier's system of logarithms =.........-........ O = 2.718281828459 Mod. of common syst. of logarithms =.... com. log. E = M = 0-434294481903 Ratio of circunference to diameter of a circle =............. = 3I4I592653590 log. 47r 0-497I4987269 4 " = 9.869604401089........ = 1-772453850906 Arc of same length as radius =.......... 180~0 ~ r = o080oo' 7r == 648000"' 7r 1800 * =- 570-2957795130............................. log. = 1758122632409 o10800' - 7r = 3437'7467707849.........................log. = 3.536273882793 648000" X = 206264" 8062470964,....................log. = 5-314425133176 Tropical year = 365d. 5h. 48m. 47s..588 365d. 242217456, log. 25625810 Sidereal year = 365d. 6h. 9m. Ios..742 = 365d. -256374332, log. = 2.5625978 24h. sol. t.=24h. 3m. 56s. 555335 sid. t.=24h.XI.00273791, log. I o002=-O00oI874 24h. sid. t.=24h. -(3m.55s. o90944) sol. t.=24h. X 0 9972696, log. 0o997=9.9988126 British imperial gallon = 277 274 cubic inches...............log. = 2-4429091 Length of sec. pend., in inches, at London, 39-I3929; Paris, 39.1285; New L t of se. Pn in 3 York, 39-1285. French metre = 3-2808992 Englishfeet == 39-3707904 intches. i cubic inch of water (bar. 30 inches, Fahr. therm. 620) = 252'458 Troy grains.