272 233 
 
 ANSIT^LEVEL 
 
 LAW 
 
IN MEMORIAM 
 FLORIAN CAJORI 
 
ADJUSTMENTS 
 
 COMPASS, TRANSIT, AND LEVEL. 
 
 BY 
 
 A. V. LANE, C.E., PH.D., 
 
 ASSOCIATE PROFESSOR OF MATHEMATICS, 
 UNIVERSITY OF TEXAS. 
 
 BOSTON: 
 
 PUBLISHED BY GINN & COMPANY. 
 
 1886. 
 
Entered, according to Act of Congress, in the year 1886, by 
 
 A. V. LANE, 
 in the Office of the Librarian of Congress, at Washington. 
 
 J. S. CUSHING & Co., PRINTERS, BOSTON. 
 
PREFACE. 
 
 A N examination of those text-books in which this 
 -*--*- subject is or should be treated, reveals the fact 
 that it is for the most part very meagrely and arbitrarily 
 presented, sometimes dismissed with the statement that 
 the adjustments should be made by the maker of the 
 instrument. The method, when given, is generally with- 
 out explanation or proof that it will accomplish the 
 desired object, so that the student must either take the 
 author's word for it, get some one to explain it to him, 
 or work it out for himself ; and being usually unprepared 
 for such original work, he is in danger of adopting the 
 first course mentioned, or of leaving the matter in doubt 
 and mystery. 
 
 A great source of trouble lies in the fact that such 
 authors are expected to express themselves in accurate 
 terms and do not ; the word " half," for example, being so 
 often used for that which is at best but approximately 
 so, that the student marvels at the talisman for such 
 diverse operations being so uniformly that particular 
 fraction. 
 
 Perhaps the absence of explanation and proofs of the 
 
iv PREFACE. 
 
 correctness of some of the methods is largely due to the 
 difficulty of making the subject clear to those who have 
 not studied Descriptive Geometry. It is believed, how- 
 ever, that one whose attainments in the line of mathe- 
 matics go no further than through Elementary Trigo- 
 nometry will experience no difficulty with the following 
 discussion of the adjustments of the three principal in- 
 struments used by surveyors and engineers. 
 
 This little volume has been called forth by the neea 
 of such an exposition of the subject, felt by the author 
 for some time past in presenting the matter to his classes 
 in Engineering, and any suggestions in the line of 
 improvement will be acceptable. 
 
 A. V. LANE. 
 UNIVERSITY OF TEXAS, 
 AUSTIN, May, 1886. 
 
 
 i 
 
ABBREVIATIONS. 
 
 A, horizontal axis of telescope (Transit). 
 
 B, axis of the level bar (Level). 
 
 C, line of collimation (Transit, Level). 
 
 H and F, horizontal and vertical plane of reference. 
 
 /, intersection of A and C (Transit). 
 
 R, plane of revolution of adjusted C (Transit). 
 
 S, axis of spindle (Compass, Transit, Level). 
 
 W, intersection of cross-whcs (Transit, Level). 
 
 Y and y, longer and shorter distances from B to the centers 
 
 of the telescope's wye bearings (Level). 
 U., level tube (Compass, Transit, Level). 
 l.t.c., level tube-case (Compass, Transit, Level). 
 Lt.cM., level tube-case axis (Compass, Transit, Level). 
 
I. THE COMPASS. 
 
 1. The Plate-Levels, to so adjust them that when their 
 'bubbles are centered, the plate shall be horizontal. 
 
 FIG. 1. 
 
 Let the l.t.c. GE be turned in a vertical plane and 
 about its center G through an angle. The bubble, which 
 was at Z>, the center of the l.t., will move to a point F' 
 found by raising a vertical through the new position of 
 the center of curvature of the l.t. The arc D'F' through 
 which the bubble has moved subtends the angle 
 
 2 = DGD' = the angle between DE and D'E' ; 
 
2 THE COMPASS. 
 
 i.e., the motion of the bubble is proportional to the angle 
 through which the l.t.c. is turned about its center. 
 
 Let us suppose that the l.t.c.a. D i E 1 is not parallel to 
 the plate JKi, but makes with it an angle a. Then, when 
 the bubble is brought to the center, A-^i being horizontal, 
 will differ a from horizontally. If now the plate 
 
 D 
 
 FIG. 2. 
 
 be turned 180 on $, the l.t.c.a. will have changed ends, 
 and will lie in a vertical plane parallel to the one which 
 before contained it, and distant therefrom by twice the 
 distance of the l.t.c.a. from S. Since the two halves 
 of the l.t.c. are precisely alike, the effect of this operation 
 on the motion of the bubble is clearly the same as that 
 of a rotation in a vertical plane and about the center of 
 the l.t.c. through an angle 2 a. If then the bubble is 
 
THE COMPASS. 3 
 
 brought half-way back to the center by the screws at 
 the ends of the l.t.c., and the rest of the way by changing 
 the position of $, the l.t.c.a. will take the adjusted posi- 
 tion FG (parallel to the plate), and S will be vertical, 
 the plate also horizontal. See Methods of Adjustment, 
 p. 33. 
 
 2. The Sights, to make them vertical when the plate 
 is horizontal. 
 
 The plate being horizontal, if a vertical line is sighted 
 to and each sight made to range with it from any point 
 of the other, they will be vertical. See Methods. 
 
 3. The Needle, to so adjust it that any one vertical 
 plane may contain its end points, its center of rotation, 
 and the center of the graduated circle. 
 
 If the ends of the needle do not in every position cut 
 opposite degrees, (point to divisions which differ 180), 
 this adjustment is needed. Failure to do so is due to 
 the point of the pivot not being in a perpendicular to 
 the plane of the graduated circle at its center, or to the 
 needle being bent, or to both causes. 
 
 Suppose that, one end of the needle pointing to a divis- 
 ion G^ the other fails to point to its opposite, Ji- Let 
 the pivot be bent until the failure is corrected, when the 
 conditions will be such as are represented in the diagram. 
 Reversing the sights exactly, the points F^ D^ E^ E 
 take the positions F 2 , D 2 , E 2 , EJ. Evidently the divis- 
 
4 THE COMPASS. 
 
 ion N 2 is approximately half-way between 3/ 2 and G^ 
 the angle F 2 D.,0 differing slightly from J^DjO, owing to 
 the displacement of F and F. 2 from 0. Hence, if the 
 needle be so bent that its end point E. 2 cuts half-way back 
 to G 2J its ends will be approximately in line with its 
 center. If the pivot be now bent until the needle cuts 
 the readings J z and G. z , F 2 will be at or very near some 
 
 FIG. 3. 
 
 point as F 3 in the line GyJ*. So by repeating this process 
 the ends of the needle will be brought into line with its 
 center, which center will be somewhere in the line G^J^ as 
 at F 3 . 
 
 "When this has been done, if one end of the needle 
 be held at K 2 (90 from J 2 ) and the pivot bent towards 
 
THE COMPASS. 
 
 K 2 L 2 until the- other end cuts L$, F 3 will be brought into 
 the line K 2 L 2 and nearer to its proper position. 
 
 By repeating this last operation on the pivot with refer- 
 ence to the lines G 2 J 2 and K 2 L^ alternately, its point 
 will be adjusted. 
 
II. THE TRANSIT. 
 
 1. The Levels, Same as I. 1. 
 
 2. The Needle. Same as I. 3. 
 
 3. The Line of Collimation, to make it perpendicular 
 to the horizontal axis of the telescope, so that, by revolu- 
 tion about that axis, it will generate a plane and not a 
 conical surface. 
 
 It should be remembered that C is the line of sight 
 and is determined by the optical center of the object- 
 glass and the intersection of the cross- wires. The wires 
 should be respectively vertical and horizontal ; they may 
 be made so by causing one to range with a plumb-line 
 or known vertical line, since they are perpendicular to 
 each other. 
 
 Assume a horizontal circle with center /, and let Ci 
 differ a from IE, a perpendicular to A. Revolving Ci 
 about A^ till it again cuts the circle, it takes the position 
 C 2 , differing a from ID, and therefore 180 2 a from Ci. 
 Turning the instrument about S 180 2 a, C 2 becomes 
 C 3 in coincidence with Ci, while A 2 takes the position A 3 . 
 Revolving C 3 about A 3 until it again cuts the circle, it 
 takes the position (7 4 , differing a from C 5 perpendicular 
 
THE TRANSIT. 7 
 
 to A 5 . C 5 and ID, respectively perpendicular to A 5 and 
 AS, differ by the same angle as those positions of the 
 axis; viz., 2a. Since the arc 4 5 subtends a, 5D sub- 
 tends 2 a, and D2 subtends a, the ratio of the arc 4 5 to 
 the arc 4 2 is clearly that of a to 4 a, or one-fourth. If then 
 C 4 were at (7 5 , one-fourth of the way toward (7 2 , it would 
 
 A.AI 
 A\ ' 
 
 2^ 
 
 A A, 
 
 /? 
 
 F\" - A - 
 
 \ /^2? 
 
 \ 4 / 
 
 \ /l/4 / 
 
 \ 
 
 \ / A 
 
 4 " x - / 
 
 \Ci 
 
 \ ,,' f 
 
 ^--*. 
 
 \ ,,'''' 
 
 ^ i 
 
 > 2 
 
 FIG 
 
 . 4. 
 
 be perpendicular to the horizontal axis of the telescope, 
 and hence in adjustment. In practice the above condi- 
 tions cannot usually be perfectly realized. Thus the 
 points 1, 2, 4, 5 are not in one plane, which introduces 
 a slight error, from the fact that, until adjusted, C revolves 
 about A in a conical surface and so moves to the right 
 or left as it leaves a horizontal position ; and again, the 
 
8 THE TRANSIT. 
 
 points 2, 4, 5 are not taken exactly on a circle, but 
 in a straight line on top of a stake. The errors in each 
 case are small, as the displacements of the points from a 
 horizontal plane and the distances apart of the last three 
 are taken quite small in comparison with their distance 
 from J. 
 
 After this has been done, it may be found that the 
 intersection of the cross-wires is not in the center of the 
 field of view of the eye-piece. Since the position of O 
 depends only on the object-glass and TF, we may, with- 
 out affecting it, move the e} T e-piece by the proper screws 
 until the center of the field of view is brought into coinci- 
 dence with the intersection of the cross- wires. For the 
 eye-piece is simply a microscope with which we magnify 
 and invert the inverted image formed at the cross-wires 
 where its focus is. 
 
 Again, it is evident that, even when C is not perpen- 
 dicular to A, we may locate three points on the same side 
 of the instrument, which will either be in one vertical 
 plane (in line), if they are all at the same height as /; 
 or very nearly in line, if they do not vary much from 
 that height. Let these points be 1, / and F in the pre- 
 ceding diagram. Setting up the instrument at the middle 
 one 7, sighting to 1 , and turning Ci about A, it comes to 
 (7 2 . If C 2 were perpendicular to J. 2 , the point D would 
 be sighted to. So the point 2 is twice as far from F as 
 the point D upon which to effect adjustment. 
 
THE TRANSIT. 
 
 9 
 
 4. The Standards, to make the bearings of the hori- 
 zontal axis of the telescope equally distant from the plate, 
 so that when the instrument is levelled, the line of collima- 
 tion will, by revolution about the horizontal axis of the 
 telescope, generate a vertical plane. 
 
 FIG. 5. 
 
 Suppose the instrument set up on H, level, and with the 
 preceding adjustments made. Suppose A horizontal and 
 occupying a position ID parallel to H, but not necessarily 
 so to V. Draw the vertical IJ and, through J, the line 
 
10 THE TEANSIT. 
 
 -ffilTa in H parallel to ID. Through J draw JG perpen- 
 dicular to KK 2 ; and, through G, GP 8 perpendicular to 
 the intersection of H and V. Now .R, being perpendic- 
 ular to A or ID, is perpendicular to H, and must cut V 
 in a line perpendicular to the intersection of H and F. 
 Hence R must contain IJ, JG, and GP a . Take L in 
 6rP 3 so that GL = JCr. (7 may be directed to L, when 
 it will be parallel to H and perpendicular to the plane 
 DIJ. Since the arm DI of the right angle DIL may be 
 rotated about IL into the position A, A may take the 
 position AI in the plane DIJ, and C be still fixed upon L. 
 Since, if turned into that plane, C will take the position 
 Zffi, perpendicular to ^4 X , 1?, containing K and IL, must 
 now cut Hin a line If^ parallel to IL and therefore to 
 JG, and so must cut V in I^L. 
 
 If now the instrument be turned on S or IJ through 
 180, A takes a position A 2 , making the same angle with 
 Z7as before. A 2 being in the plane DIJ, C may still be 
 directed to L or, in that plane, to IT 2 , JK 2 being equal to 
 JK^ and the triangle Jl/fj having turned into the position 
 JIK 2 . R must therefore now cut IT in a line K 2 F, par- 
 allel to JG, and F in a line FL. 
 
 Draw PiP 2 in V and parallel to the intersection of II 
 and F. With the axis in the position AI, C could be 
 directed to P t ; with the axis in the position A 2 , to P 2 ; 
 and in the adjusted position DI, to P 3 . The wires should 
 therefore be moved toward P l by an amount P 2 P 3 . 
 
 Since JG bisects K^K^ it bisects EF, and therefore 
 
THE TRANSIT. 
 
 11 
 
 GL bisects P 2 Pi- So the wires should be moved half- 
 way from their second position, P 2 , toward their first, P 1? 
 by changing the height of one of the standards. 
 
 Practically, the ground and a vertical wall take the 
 place of H and F. (Given last, under Methods, II. 4.J 
 
 FIG. 6. 
 
 Suppose the instrument at 7J, the right-hand end of A 
 the higher, and R occupying the position P^P^K^. Imag- 
 ine the instrument turned 180 on $, so that the triangle 
 occupies the position IJK^ and R the position 
 
12 THE TEANSIT. 
 
 , just as in the preceding case. Directing C to L 19 
 imagine it turned to the left, remaining horizontal, until 
 a point L 2 is reached, such that, on revolving C about A, 
 it will again cut P x . In order for L 2 to be such a point, 
 it is evident that the triangle IJK 2 must have revolved 
 into a position /J/ig, such that perpendiculars JP 5 and 
 /jP 4 to JK 3 shall meet the intersection of H and V in 
 points P 5 and P 4 , respectively on a vertical through L 2 
 and on the prolongation of P 3 L 2 . 
 
 C was first turned down from P l to P 2 , then from 
 P 3 to P 4 , A lying in the plane IJK 3 . If A became 
 horizontal in that plane, H would occupy the position 
 L 2 P 5 J. So the wires should be moved from P 4 to P 5 , 
 by changing the height of one of the standards. We 
 desire therefore to determine the ratio of P 4 P 5 to P 4 P 2 . 
 
 Refer the point J to P 2 by JG = d, angle JGP 2 = 0, 
 GP 2 = b. Let L 2 P 5 = L l G = IJ=i. Let P,0=p, 
 and the angle I 3 P 4 P 5 = angle JP 5 G <. Then the an- 
 gle G JP 5 = 6 < ; also 
 
 P 5 N= JK 3 = JK 2 = JKi = MP 2 = b sin 0. 
 By similar triangles 
 
 /IN 
 
 P 4 p P 2 
 
 By alternation 
 
 P 2 G P 2 0' 
 
 and this, by composition, gives 
 
THE TRANSIT. 13 
 
 / 
 
 or 
 
 and in the triangle P 5 GJ 
 
 d sin < tan 
 
 whence, by (3), 
 
 6 
 = P 4 P 2 - (P 4 P 5 + &) = 
 
 Whence 
 
 PJP, = Pf t -L-b. (2) 
 
 In the triangle NPP 5 we have 
 
 and 
 
 (2). 
 
 (4) 
 
 Substituting these in (4) and clearing of fractions, we 
 have 
 
 b PP 2 (p - i) 4- bdp cos 6 
 
 i 2 - 2 Up P 4 P 2 + &y - 6V sin 2 0. 
 
14 THE TRANSIT. 
 
 Squaring, cancelling, and taking out the common factor 
 P 4 P 2 , we find 
 
 P p = 2 bdp [di + b (p ^- J) cos 0] 
 " "~&?-V(p-i)* 
 
 Substituting this in (2), we obtain 
 
 P 4 P 2 p 
 the required ratio. 
 
 If be 90 and p i = d, this ratio becomes 
 
 i i 2 -b 2 
 p 2ip 
 
 b 2 i 
 
 or, neglecting the small fraction , it becomes -- 
 
 2 ip 2p 
 
 These conditions are usually very nearly satisfied, so 
 we may consider - - quite an approximate value of the 
 true ratio. Assuming i = 5 ft. and p = 50 ft. as about 
 average values, this ratio equals ; while, even if b 
 
 b 2 
 
 could equal 6 inches, the neglected fraction - - would 
 
 2ip 
 
 only equal - (Given second, under Methods, II. 4.) 
 
 Suppose that, having as before located the point P 2 
 from the point PI, we turn the instrument about half- 
 way around on S and, revolving the telescope, fix the 
 wires again on the lower point P 2 . Now, raising them 
 
THE TEANSIT. 
 
 15 
 
 to P 4 at height of P 19 we wish to determine the ratio of 
 P 4 P 6 to P 4 P X . 
 
 For, as before, R having first the position P^/ii, on 
 turning exactly half-wa}* around, it would take the posi- 
 tion L^FK^ ; and, directing C to 1? we must turn the 
 
 FIG. 7. 
 
 instrument to the right until C strikes a point Z/ 2 , such 
 that, on revolving about A, the point P 2 is again cut. 
 
 The triangle IJjfa has now revolved into the position 
 IJKz, and R into the position P 4 P 3 K B ; while, if A be- 
 came horizontal, R would have the position P 5 G 2 J. 
 
16 THE TRANSIT. 
 
 The references of J to P 2 are now JG l = d, angle 
 J6? 1 P 2 = 6, G 1 P 2 = b. 
 
 The angle JG 2 is <, and the angle G^G 2 = <j> Q. 
 Since, by similar triangles, 
 
 P G 
 
 and we will first determine ~ 
 
 OP 3 
 
 By alternation, 
 
 and this, by composition, gives 
 
 or, 
 
 __ 
 
 4 p x oq i OQ ' 
 
 (1) 
 
 Whence 
 
 P 3 G 2 =OQ. l --b. (2) 
 
 P 
 
 In the triangle NG 2 P 3 we have 
 _ 
 
THE TRANSIT. 17 
 
 and, in the triangle 
 
 A = sin (< - Q) = cQ sin 
 
 d sin < tan </> 
 
 whence, by (3), 
 
 Ct 
 
 (4) 
 
 and 
 
 P A _^Z* by(2) . 
 
 Substituting these in (4) and clearing of fractions, we 
 have 
 bdpcosO bi- OQ 
 
 i 2 -2 Up - OQ + b 2 p 2 - by sin 2 0. 
 
 Squaring, cancelling, and taking out the common fac- 
 tor OQ, we find 
 
 d b cos 
 
 Substituting this in (2) , we obtain 
 
 OQ p 2dp d bcosOp 2d(d 
 Therefore 
 
 i OQ 2dp d-bcosO 
 
 the required ratio. 
 
18 THE TRANSIT. 
 
 If be 90, neglecting the small fraction 
 
 (P ~ & 
 
 2d 2 p 
 it becomes 
 
 p i _ 1 _ i_ 
 2p ~2 p' 
 
 or approximately -, since is usually about ; while 
 the neglected fraction, under the preceding conditions, is 
 
 . (Given first, under Methods, II. 4.) 
 
 18000 
 
 This last is the best and most generally used method. 
 The first is excellent, if divisions on the horizontal 
 circle such as the zeros may be depended on as exactly 
 opposite. In the second, the amount through which the 
 wires are to be moved for correction is such a small part 
 of a small distance that it is difficult of exact application. 
 
 5, a. The Vertical Circle's Vernier Zero, to so ad- 
 just it that when the vertical circle's zero is in line with it, 
 the line of collimation shall be horizontal. 
 
 If, instead of a full circle firmly attached to A, there 
 is only an arc of a circle, which may be turned upon and 
 clamped to it, as is the case in some instruments, this 
 adjustment depends upon and is made after that of the 
 level attached to the telescope. The method of making 
 it in that case is explained further on (after 6) . 
 
THE TRANSIT. 
 
 19 
 
 Let us suppose that the vernier zero Z^ is not in its 
 proper position (vertically below the center of the verti- 
 cal circle) ; so that, when the circle zero Zi is made to 
 coincide with it, C has the position Ci, making the same 
 angle with the horizontal O i D 1 that Ofa makes with the 
 vertical 0^. 
 
 Let P l be some point in Ci. Turning the instrument 
 180 on S, Ci, PU z x , Zi move to <7 2 , P 2 , z 2 , Z 2 . The 
 
 angle P 2 2 D 2 being equal to y, if O 2 be turned on ^4 2 
 through 180 2 y, it takes the position (7 3 , and z 2 the 
 position 2; 3 , the angles P & 3 D 3 and E 2 s z s being each 
 equal to y. The division z s ' opposite ^ 3 is now y from F 2 , 
 and 2y from ^3. Hence if Z 2 were moved half-way to z 3 ' 
 and then 3 ' brought into coincidence with it, (7 3 would 
 
20 THE TRANSIT. 
 
 take the horizontal position 3 7> 3 , and, the zeros being 
 also in coincidence, the adjustment would be effected. 
 If A 3 were turned into the position A 4 < and C 3 directed to 
 P 4 , the angle P0D being slightly different from y, the 
 ratio corresponding to the above would not be exactly 
 one-half, yet by taking the distance of P 1 from the instru- 
 ment sufficiently great in comparison with A, the point P x 
 may be used for P 3 with but little error ; and it is so used, 
 the process being repeated until the whole error becomes 
 inappreciable. 
 
 6. a. The Level attached to the Telescope, to so ad- 
 just it that when its bubble is at the center, the tine of 
 collimation shall be horizontal. 
 
 A full vertical circle being present, and its zero in 
 coincidence with the vernier zero after the preceding 
 adjustment, if the bubble is brought to the center by 
 means of the nut at either end of the level, the adjust- 
 ment is effected. 
 
 5. b. The Level attached to the Telescope, If, how- 
 ever, the instrument has the movable arc, the level is 
 first adjusted as follows : 
 
 Let J9, J], -E7, and J s be four points in line, and in the 
 order named ; also equally distant horizon tall}'. Suppose 
 the instrument set up at /i, and C l directed to a graduated 
 rod held on the stake Z>, giving a reading T^Z). Turning 
 the instrument 180 about , without otherwise disturbing 
 
THE Til AX SIT. 
 
 21 
 
 Ci, C 2 will cut the rod held on the stake E at a reading 
 F. 2 E, the points F, F 2 being at the same height, since 
 Cj^ and C' 2 make equal angles with the vertical I^J^ and 
 
 the positions of the rod are equidistant from 7j. Let 
 the instrument be now set up at / 3 , and, with C in a posi- 
 
22 THE TRANSIT. 
 
 tion <7 3 , the readings F 3 E and F 4 D noted. Draw 
 parallel to F 2 F^ and therefore horizontal. 
 
 F^ = F 4 D- FJ) and F & F 2 = F S E - F 2 E being known, 
 let x = LF l = KF 2 , and we have, by similar triangles, 
 
 - x LI 3 
 
 whence 
 
 _ 
 
 JU 
 
 is known. 
 
 So, by directing C to a reading Z^ 4- a?, it takes the 
 horizontal position (7 4 , and the level on the telescope may 
 be adjusted by bringing its bubble to the center, by means 
 of the nut at either end. 
 
 6. b. The Vertical Arc's Vernier Zero. The vertical 
 arc may now be turned on A and clamped with its zero in 
 coincidence with the index zero, effecting its adjustment. 
 
 For convenience of application as to sign, etc., it is well 
 to take the first reading just above the top of the higher 
 of the stakes D, E, and to set up at J 3 so that J 3 shall be 
 higher than F^ F 2 ; taking the readings F&E, F 4 D greater 
 than F 2 E, FJ) respectively. 
 
 By driving the stakes at D and E so that their tops are 
 at F l and F^ the first two readings reduce to zero, and 
 F 3 F 2 , F^ are the only readings taken ; they are used as 
 stated above, x being itself the reading of the target at 
 FI for horizon tali t} T . 
 
III. THE LEVEL. 
 
 1. The Line of Collimation, to so adjust it that it 
 will pass through the centers of the circles of the wye bear- 
 ings of the telescope. 
 
 This adjustment is a first step toward making C per- 
 pendicular to S. 
 
 FIG. 10. 
 
 Let D and E be the centers of the circles of the wye 
 bearings and C l the position of C sighting to some point 
 as P!, distant PiP 3 from the line ED produced. If now 
 C l be turned 180 about ED, it will take the position <7 2 , 
 making with ED an angle equal to that made by Ci, and 
 sighting to some point as P 2 at the same distance as P 1 
 from and P 3 . Clearly now for C, to coincide with ED, 
 W. 2 must be brought to TFg, half-way back to TFi, its for- 
 mer position, by moving it until C 2 takes the position (7 3 
 sighting to a point P 3 , half-way from P 2 towards P lf 
 This may be best accomplished by adjusting each wire 
 in succession, if they be much in error, by means of some 
 
24 THE LEVEL. 
 
 line sighted to, until each is nearly right, then completing 
 the adjustment of each. This amounts to substituting 
 for P! a line through P x and perpendicular to the plane 
 of the paper, for P 2 a like but imaginary line, and for 
 WH Wz, and TF^one of the cross-wires, also perpendicular 
 to the plane of the paper and through those points. 
 
 As already explained in II. 3, the center of the field of 
 view should now be brought into coincidence with W by 
 moving the proper screws, and the correctness of the 
 centering may be tested by turning the telescope in the 
 wyes, when the object should not appear to shift its posi- 
 tion. 
 
 2. The Level, to make the level tube-case axis par- 
 allel to the line of collimation. 
 
 In the explanation of I. 1 it was shown that in the 
 operation of reversing the l.t.c.a. with reference to any 
 plane or line to which its ends were referred, the bubble 
 moves from the center in the first position to some 
 point in the second position, such that the arc between 
 the center and that point is double the arc through which 
 the bubble should be moved back in order to make the 
 l.t.c.a. parallel to the line or plane of reference. And 
 this, too, whether that line or plane be horizontal or not. 
 
 Since now C contains the centers of the circles of the 
 wye bearings, by reversing the telescope end for end as 
 to the wyes, we may effect this adjustment ; the ends of 
 the l.t.c.a. being referred to C. But there is a disturbing 
 element to be considered in this connection. 
 
THE LEVEL. 
 
 25 
 
 The l.t.c. has a screw at one end for lateral adjustment, 
 to bring the Lt.c.a. into one plane with (7, and another, 
 at the other end, for vertical adjustment to make these 
 lines equidistant throughout. If we remove the clips from 
 the wyes, and reverse the telescope end for end, it may 
 be that we do not put it down with the vertical screw 
 immediately below (7, as before, but slightly rotated to 
 one side. We must therefore examine the effect on the 
 bubble of a slight rotation of the Lt.c.a. about 0, with a 
 view to ascertaining how much if any of the bubble's mo- 
 tion on reversal is due to this cause. 
 
 J) 
 
 FIG. 11. 
 
 Imagine the plane of the paper vertical, and containing 
 C and the Lt.c.a. Draw DE tangent to the l.t. at E. 
 The highest point of the l.t. being that of contact with a 
 horizontal tangent plane, it is evident that, being hori- 
 zontal, this highest point will lie on a cross-section of the 
 l.t. at E for any revolution less than 90 about (7, 
 whether the Lt.c.a. is equidistant from Cor not. For its 
 position does not affect the symmetry of the parts of the 
 l.t. on each side of E, but simply their amounts. 
 
26 THE LEVEL. 
 
 Suppose C is not horizontal. After a revolution of 
 90, #iGy and KJQ turning about G 2 ' and Ay respec- 
 tively, into a horizontal position, G 1 will be at the same 
 height as 6r 2 ', and therefore higher than K^ will be (same 
 height as K. 2 ') . So the bubble will, as the rotation begins, 
 start towards the end G- 
 
 FIG. 12. 
 
 Again, in Fig. 11, suppose L to be slightly in front of 
 the plane of the paper, and F a like amount behind. If 
 we rotate towards us, L rises and F falls, the bubble 
 therefore running toward /f, instead of remaining at E. 
 as before ; while, if we rotate the other way, the reverse 
 occurs. Also, in Fig. 12, if we make a like supposition, 
 since the bubble before moved toward G^ whichever 
 way rotation took place, while the effect of this new sup- 
 position tends to make it move towards KI or GI (accord- 
 ing as we rotate towards us or away) , it is clear that these 
 causes may combine to move the, bubble one way, or may 
 oppose and even neutralize each other. 
 
 Thus it is evident that, knowing nothing of the actual 
 state of the positions of C and the l.t.c.a., we cannot 
 
THE LEVEL. 
 
 27 
 
 predicate anything as to how much of the motion of the 
 bubble on reversal may be due to the cause just exam- 
 ined. Nevertheless the lateral screw may be, by trial-, so 
 adjusted as to eliminate the effect of this slight rotation, 
 and finally to bring the Lt.c.a. into the same plane with 
 C, as in the method given farther on. 
 
 3. The Wyes, to so adjust them, as to the distances 
 of the centers oj their circular bearings from the axis of 
 the level-bar, that the axis of the spindle shall be perpen- 
 dicular to the line of collimation. 
 
 B is supposed to be made perpendicular to S, so that 
 if Y be made equal to y, C will be parallel to J5, and 
 therefore perpendicular to S. But if B differs by some 
 angle /3 from perpendicularity to S, by reason of bad 
 construction or some strain received, Y and y must be 
 given such values as to counteract this and make C per- 
 pendicular to S. 
 
 FIG. 13. 
 
28 
 
 THE LEVEL. 
 
 From the diagram it is easily seen that the necessary 
 condition is a = /5, with opposing effects on the bubble 
 (a special case of which is a = 0, /2 = 0, when G is 
 parallel to B as well as perpendicular to S) , or, in other 
 words, C making the same angle with B that B does with 
 a perpendicular to $, and S making an acute angle with 
 that half of B which carries Y. 
 
 2/3 
 
 FIG. 14. 
 
 Let us suppose S to have a position Si, making an 
 acute angle 90 p with the y half of B. We see that, 
 by reversing about S^ B and Ci take the positions B 3 
 and Ci; while if S t had the position (S 2 ) perpendicular 
 to BI, B and Ci would take the positions (B. 2 ) and (O 2 ), 
 on reversal. Rotating (B a ) through 2/3 into the position 
 BS, (C 2 ) rotates through an equal angle into the position 
 
THE LEVEL. 29 
 
 v / 
 
 C 3 . Since, then, C 3 makes an angle 2/5 with (O 2 ), and 
 (<7 2 ) makes 2 a with d, C 3 makes 2 a + 2/5 with d. 
 
 This may also be shown thus : Drawing DE parallel to O 3 
 and DF parallel to J? 3 , we have EDF= a. But FDG = 20, since 
 J^D is parallel to B 3 and D<7 to B x . Therefore the (acute) angle 
 between D F and ( (7 2 ) equals 2 # o. Hence the (acute) angle 
 between DE and (<7 2 ) or C 3 and (O,) equals a + 2)8 a = 20; 
 whence the (acute) angle between C 3 and <7 X equals 2 a+ 2/3. 
 
 Thus we see that, on reversing about S^ the bubble 
 moves from the center toward Y over an arc corresponding 
 to 2 a + 2/5, the effects of a and /5 being in this case 
 cumulative. Remembering that only 2 a of this is due 
 to (7s not being parallel to -B, and that, if the bubble 
 were moved back over a, C 3 would be parallel to B 3 , we 
 see" that if we shorten Y until the bubble comes half-way 
 back to the center, we move it over i(2a-}-2/5) = a-r-/5; 
 i.e., /5 more than the proper amount (a) to make O 3 parallel 
 to B 3 . Thus we have replaced the error a by another, /5, 
 and made that wye which was the longer, now the 
 shorter ; having decreased Y too much for parallelism 
 of C 3 and J5 3 . Since Y and y have interchanged ends, 
 8 now makes an acute angle with the Fhalf of jB, and 
 the angle between O 3 and _B 3 is equal to that between B 
 and a perpendicular to $, precisely the conditions of 
 Fig. 13 (except that S is not vertical). So that the 
 adjustment is effected with B not perpendicular to S. 
 
 Suppose the effects of a and /5 on the bubble are oppos- 
 
30 
 
 THE LEVEL. 
 
 ing, that a is greater then (3 (S making an acute angle 
 with the ]Thal^ of J5). Reversal about Si brings Z>\ and 
 Ci into the positions B 3 and C 3 ; while revel-sal about (S 2 ) 
 perpendicular to BI would have brought them into the 
 positions (B 2 ) and (C 2 ). Introducing the effect of /?, 
 
 FIG. 15. 
 
 (<7 2 ), which had separated from Ci, returns toward but 
 not to it, since a > /?. The combined effect of a and /? 
 is thus to move the bubble over an arc corresponding to 
 2a 2/3 toward T^. Since, in bringing it half-way back 
 to the center, it is moved over a (3 (ft less than the 
 proper amount to make (7 3 parallel to J3 3 ) , there remains 
 an error ft between C and B. But since Y 3 has not been 
 made shorter than y 8 , S is still making an acute angle 
 with the Fhalf of B, and so the adjustment is effected, 
 as in Fig. 13. 
 
THE LEVEL. 31 
 
 Finally, suppose a and /5 opposing, and /2>a. The 
 effect of a being to make the bubble run toward the ( Y 2 ) 
 end of (C4), and fi causing (C 2 ) to turn back through 
 Oi to C 3 (making the bubble run away from the Y 3 end 
 of (7 3 ), their combined effect will move it 2/3 2 a toward 
 the Y s end of O 3 . To move it half-way back we must 
 now lengthen Y^ (or shorten ?/ 3 ), and so increase a by 
 -|-(2/3 2a) ; i.e., by /? a, making the angle between 
 
 -B 3 and 3 a + (/8 a) = /3, >S still making an acute 
 angle with the Y half of J5, again the conditions of 
 Fig. 13. 
 
 Thus we see that, when a and ft are cumulative in 
 effect, the process results in making them opposing, and 
 if they are at first opposing, they result opposing. For, 
 in the first case, we make the longer wye the shorter, thus 
 introducing a change ; while in the latter, although the 
 length of one of the wyes is changed, the longer one 
 
32 THE LEVEL. 
 
 remains the longer. In each case a is changed to the 
 amount /3 (there being no facilities for changing /3), aii'd 
 so they are left equal and opposing. 
 
 It should be noted that, while C is made perpendicular 
 to , by thus bringing the bubble half-way back to the 
 center, it must be brought the rest of the way back by 
 the leveling-screws ; then C will be horizontal, and if it is 
 made so in two intersecting positions, S will be vertical. 
 
METHODS OF ADJUSTMENT. 
 
 I. THE COMPASS. 
 
 1. The Levels. Set up the instrument, and bring the 
 bubbles 'to the center by pressure of the hands on the 
 plate. Reverse the sights, and if the bubbles remain at 
 the center, the levels are in adjustment. If the}" do not, 
 bring each half-way back to the center by means of the 
 screws at the ends of the level tube-case, the rest of 
 the way by means of the plate, and repeat. Or perform 
 this operation with one level at a time until it is nearly 
 adjusted, then with the other, finally completing the adjust- 
 ment of each and seeing that both will reverse correctly 
 and remain in the center during an entire revolution of 
 the plate. 
 
 2. The Sights. Observe through the slits a good 
 plumb-line and if either sight fails to range with it, 
 make it do so by whatever means the instrument calls 
 for, usually, filing a little off of one side of the surface 
 of contact of the sight with the plate. 
 
 3. The Needle. If the needle will not in various 
 positions cut opposite degrees, this adjustment is needed. 
 
34 METHODS OF ADJUSTMENT. 
 
 Having removed the glass top of the compass-box, with 
 a splinter of wood bring one end of the needle to any 
 prominent graduation, as the zero, having the eye nearly 
 in the plane of the graduated circle, and see if the other 
 end corresponds to the opposite division. If not, bend 
 the center-pin (pivot) with a small wrench, about one- 
 eight of an inch below its point, until that other end of 
 the needle will cut the other zero. Reverse the zeros 
 but not the ends of the needle, and, holding with the 
 splinter the same end of the needle at the new zero, 
 note what division the other end cuts. Bend the needle 
 until that end cuts a division half-way back to the adja- 
 cent zero. This puts the needle's ends very approxi- 
 mately in line with its center. Bend the center-pin again, 
 until the ends of the needle will cut the zeros. Repeat 
 until perfect reversion is obtained. 
 
 Bring one end of the needle to the 90 division, and 
 if the other end does not cut the opposite division, bend 
 only the pivot until it does, and repeat, using alternately 
 the line of zeros and that of the 90 divisions, until it 
 will cut opposite degrees in any position. 
 
 II. THE TRANSIT. 
 
 1, The Levels. Adjust by reversal as in I. 1, accom- 
 plishing the reversal by means of the readings of the 
 horizontal circle and moving the plate by means of the 
 leveling-screws. 
 
METHODS OF ADJUSTMENT. 35 
 
 2. The Needle. Same as I. 3. 
 
 3. The Line of Collimation. Having the instrument 
 set tip and leveled on tolerably level ground, make the 
 wires respectively horizontal and vertical by loosening 
 the proper screws and turning the ring around until the 
 vertical wire ma} 7 be made to coincide with some known 
 vertical line as a plumb-line, or the vertical edge of a 
 building from two to five hundred feet distant. Make 
 the screws tight again. 
 
 Select or locate a point from two to five hundred feet 
 distant, clamp the plates, revolve the telescope on its 
 axis, and locate a point on a stake on the other side of 
 and at the same distance from the instrument as the 
 first. Uiiclamp the plates and turn about the spindle 
 until the wires can be again fixed on the first point. 
 Clamp the plates and again revolve the telescope on its 
 axis. If the intersection of the wires strikes the second 
 point, the line of collimation is in. adjustment. If not, 
 the. intersection of the cross-wires should be brought 
 one-fourth of the way back to this second point by 
 means of the pair of cross-wire screws, on the sides of 
 the telescope, which move the vertical wire. 
 
 The operator, in loosening one of these screws and 
 tightening the other, should remember that they have 
 their bearings in the cross-wire ring, that a non-inverting 
 telescope inverts the relations of the cross-wires to the 
 object, and vice versa. He must therefore, in a non- 
 inverting telescope, proceed as if apparently to move 
 
36 METHODS OF ADJUSTMENT. 
 
 the vertical wire in the opposite direction from that 
 desired. 
 
 Test by repetition. 
 
 If, when this has been done, the intersection of the 
 cross-wires is not in the center of the field of view, move 
 the latter until they are, by means of the screws which 
 control the eye-piece, loosening and tightening them in 
 pairs ; the movement being now direct or as it appears 
 it should be. 
 
 Another method is to locate three points in line and 
 all on the same side of the instrument. Then setting up 
 over the middle point, sight to one of the end ones, and 
 clamping the plates, revolve the telescope to sight to 
 the other end one. If the intersection of the wires fails 
 to strike it, move that intersection half-way to the point 
 by means of the vertical wire, as just explained, and 
 repeat. Then center the eye-piece as in the preceding 
 method. 
 
 4. The Standards. Select a tolerably level piece of 
 ground in front of a tall spire, tower, or like object, that 
 shall afford from top to base a long range in a vertical 
 direction. Set up and level the instrument, so that it 
 will be about as far in front of the structure as its 
 telescope is below a good sight-point near the top. 
 Clamp to the spindle, and, fixing the wires on the point 
 selected, clamp the plates and lower the wires to some 
 point found or marked at the base of the structure. (If 
 
METHODS OF ADJUSTMENT. 37 
 
 the ground is not very level, take the point in the face of 
 the building, and at about the height of the ground on 
 which the instrument stands.) Unclamp the plates, and, 
 turning the instrument about half-way around, revolve 
 the telescope, and again fix the wires on the lower 
 point. Clamp the plates and raise the wires to the 
 height of the upper point. If they cut it, the standards 
 are in adjustment. If they do not, bring them half- 
 way to it, by raising the right-hand end (or lowering 
 the left) of the horizontal axis of the telescope, if the 
 wires are to the right of the point ; by raising the left 
 (or lowering the right), if they are to the left. Most 
 instruments have a means of making this adjustment 
 at one end of the horizontal axis, a movable bearing. 
 If the instrument has no such means, file equally a little 
 off of the feet of the higher standard. Repeat until the 
 adjustment is perfected. 
 
 Another method is to sight to an upper point, and 
 lowering the wires, fix a lower point at the base of the 
 structure just as before ; but, on turning the instrument 
 about half-way around and revolving the telescope, fix 
 the wires again on the upper point. Clamping, and 
 lowering the wires to the height of the lower point, if 
 they do not strike it, move them back towards it over 
 that fractional part of the distance, which the height of 
 the instrument above the ground is of double the height 
 of the upper point above the ground. This is to be 
 done by raising the right-hand end (or lowering the left) 
 
38 METHODS OF ADJUSTMENT. 
 
 of the horizontal axis, if the wires strike to the left of 
 the point ; raising the left (or lowering the right) , if they 
 come to the right. Repeat until the adjustment is 
 perfected. 
 
 Another method (which is dependent on the accuracy 
 of the graduation of the horizontal circle) is as follows : 
 Setting up and leveling the instrument, clamp the plates 
 at zero or some other convenient reading, and fix the 
 wires on an elevated (or depressed) point of sight, 
 clamping to the spindle. Unclamp the plates, and, 
 turning through exactly 180, as shown by the horizontal 
 circle reading, clamp again. If on now raising (or 
 lowering) the wires to the point, they cut it, the stand- 
 ards are in adjustment. If they do not, bring them half- 
 way to it, by changing the height of one of the standards 
 in such a way as to raise the right-hand end (or lower 
 the left) of the horizontal axis, if the wires strike to the 
 right of the point ; raising the left (or lowering the right) , 
 if they come to the left. 
 
 5. a. The Vertical Circle. (If only an arc is present, 
 see 56.) Bring its zero into coincidence with the zero 
 of the vernier attached to the standards, and with the tele- 
 scope find or place some point or- horizontal line cut by 
 the horizontal wire and about two or three hundred feet 
 distant. Turn the instrument about half-way around, 
 revolve the telescope, and fixing the wires upon the 
 point, or the horizontal wire upon the line first selected, 
 
METHODS OF ADJUSTMENT. 39 
 
 clamp the telescope and note if the zeros are again in 
 coincidence. If not, loosen the screws that attach the 
 vernier to the standards, and, moving it so as to bring 
 its zero half-way to the vertical circle's zero, make it 
 secure again. Repeat until no error can be detected. 
 Instead of moving the vernier zero, the circle zero may 
 be brought into coincidence with it, and the horizontal 
 wire moved back over half the amount by which it has 
 been thus displaced, provided the error is so slight as 
 not to appreciably throw the intersection of the cross- 
 wires out of the center of the field of view as previously 
 adjusted. Repeat as before. 
 
 6. a. Level on Telescope. Level the instrument care- 
 fully, and with the clamp-aud-tangent movement to the 
 horizontal axis, bring the zero of the vertical circle into 
 coincidence with the vernier zero, and, by means of the 
 screws at each end of the level, bring its bubble to the 
 center, taking care not to jar the instrument out of 
 level. 
 
 5. b. Level on Telescope. (When the instrument 
 has only a vertical arc and not a full circle.) On toler- 
 ably level ground stake four points in line and equi- 
 distant (about 100 feet), calling them (say) jD, 7i, E, J~ 3 , 
 consecutively. (It is well that D should not be lower 
 than E, nor J :] than Jj.) Set up the instrument at Ji, 
 and direct the line of sight to a graduated rod held, as 
 
40 METHODS OF ADJUSTMENT. 
 
 nearly as may be, vertical on the higher of stakes Z), E 
 (say D), taking a small reading d-i. Clamp the telescope, 
 and turning the instrument around, take the reading e t 
 on E. Set up and level the instrument at J~ 3 , so that 
 the height of the telescope shall be greater than that 
 corresponding to the readings d 1? e^. Unclamp the tele- 
 scope and direct the line of sight so that, without changing 
 its position, readings e 2 , d 2 , respectively greater than 
 e 1? dj, are obtained on the rod held successively on E 
 and D. From three times e 2 e\ take d 2 di and divide 
 the result by 2. Set the target at a reading greater by 
 this calculated amount than d 1? and, holding the rod 
 on 7), bisect the target by the line of sight and clamp 
 the telescope. By means of the screws at the end of 
 the level, bring its bubble to the center, and see if the 
 line of sight still bisects the target. If it has been 
 jarred out of position, put it back and again bring the 
 bubble to the center. When the line of sight bisects 
 the target and the bubble is in the center, the adjustment 
 is complete. 
 
 A simplification of the proceeding is as follows : 
 Instead of taking the first two readings, drive stakes so 
 that their tops are cut by the line of sight, the telescope 
 being clamped. Their tops are then in the same hori- 
 zontal line. Making the line of sight as nearly horizontal 
 as possible by estimation, take a reading on the nearer 
 of these two stakes, and then, holding the rod on the 
 farther one (target at same reading) , if the line of sight 
 
METHODS OF ADJUSTMENT. 41 
 
 does not bisect the target, turn the telescope by the 
 tangent-screw so that it will ; repeating this until it will 
 bisect the target held on the far stake at the same reading 
 as on the near one. Then bring the bubble to the center 
 by the screws at the ends of the level- tube as before. 
 
 6. b. The Vertical Arc. With the bubble of the level 
 on the telescope at the center, clamp the vertical arc to the 
 axis of the telescope, with its zero in coincidence with 
 that of the vernier, and it is in condition to correctly 
 measure vertical angles. 
 
 III. THE LEVEL. 
 
 1. The Line of Collimation. Set the tripod firmly, 
 remove the wye-pins from the clips, so that the telescope 
 may be turned in its bearings, and, by means of the 
 leveling and tangent screws, bring either of the wires 
 into coincidence with a clearly marked edge of some 
 object, from two to five hundred feet distant. Then 
 turn the telescope half-way around in the w}'es, so that 
 the same wire may be compared with the edge selected. 
 If it now coincides with the edge, it is in adjustment ; 
 if not, bring it half-way to it by moving the capstan- 
 head screws at right angles to the wire in question, 
 remembering the inverting property of the eye-piece. 
 Repeat until it will reverse correctly. Then adjust the 
 other wire in the same manner ; or, if their errors are 
 
42 METHODS OF ADJUSTMENT. 
 
 great, make them nearly correct before exactly adjusting 
 either. 
 
 When this has been effected, unscrew the covering of 
 the eye-piece centering-screws, and move each pair in 
 succession so as to bring the center of the field of view 
 to coincide with the intersection of the cross- wires, test- 
 ing the centering by revolving the telescope in the wyes, 
 when the object should not appear to move. The screws 
 in this case are to be moved as it appears they should 
 be. Replace the covering of the screws. 
 
 2, The Level-Bubble. Having the plate about hori- 
 zontal, place the telescope over either pair of leyeling- 
 screws, and, clamping the instrument, remove the wye- 
 pins and bring the bubble to the center by means of this 
 pair of leveling-screws. Reverse the telescope end for 
 end as to the wyes, replacing it with the level-tube 
 immediately beneath, and note whether the bubble 
 remains at the center. Now rotate the telescope slightly 
 to each side, and see if this causes the bubble to move 
 toward either end. If, after the reversal it was at the 
 center, and remained so during the slight rotation, the 
 level-bubble is in adjustment. 
 
 If the rotation causes it to move from its position after 
 reversal toward either end, adjust, by trial, the horizontal 
 screw so that this will not be the case. Repeat this 
 reversa[, etc., until no further adjustment of the horizontal 
 screw is needed. Then (the telescope being in the 
 
METHODS OF ADJUSTMENT. 43 
 
 second position) bring the bubble half-way to the center 
 by means of the vertical screw-nuts at one end of the 
 level tube-case, and repeat the whole process until the 
 bubble will remain at tlie center after reversal and slight 
 rotation. 
 
 3. The Wyes. Having the telescope in its normal 
 position as to the wyes, place it over a pair of the level- 
 ing-screws and bring the bubble to the center by means 
 of them. Turn the instrument half-way around on the 
 spindle, and, if the bubble runs toward either end, bring 
 it half-way back by the wye-nuts on either end of the 
 bar and the rest of the way by the pair of leveling-screws. 
 Then place the telescope over the other pair, and proceed 
 in the same way, changing to each pair successively 
 until the adjustment is completely effected for each, so 
 that the bubble will remain at the center during an entire 
 revolution of the telescope on the spindle. 
 
188 
 
 MATHEMATICS. 
 
 Peirce's Three and Four Place Tables of Loga- 
 
 rithinic and Trigonometric Functions. By JAMES MILLS PEIRCE, 
 University Professur of Mathematics in Harvard University. Quarto. 
 Cloth. Mailing Price, 45 cts. j Introduction, 40 cts. 
 
 Four-place tables require, in the long ran, only half as much time 
 .s five-place tables, one-third as much time as six-place tables, and 
 one-fourth as much as those of seven places. They are sufficient 
 for the ordinary calculations of Surveying, Civil, Mechanical, and 
 Mining Engineering, and Navigation ; for the work of the Physical 
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 Elements of the Differential Calculus. 
 
 With Numerous Examples and Applications. Designed for Use as a 
 College Text-Book. By W. E. BYEKLY, Professor of Mathematics, 
 Harvard University. 8vo. 273 pages. Mailing Price, $2.15 ; Intro- 
 duction, $2.00. 
 
 This book embodies the results of the author's experience in 
 teaching the Calculus at Cornell and Harvard Universities, and is 
 intended for a text-book, and not for an exhaustive treatise. Its 
 peculiarities are the rigorous use of the Doctrine of Limits, as a 
 foundation of the subject, and as preliminary to the adoption of the 
 more direct and practically convenient infinitesimal notation and 
 nomenclature ; the early introduction of a few simple formulas and 
 methods for integrating; a rather elaborate treatment of the use of 
 infinitesimals in pure geometry ; and the attempt to excite and keep 
 up the interest of the student by bringing in throughout the whole 
 book, and not merely at the end, numerous applications to practical 
 problems in geometry and mechanics. 
 
 James Mills Peirce, Prof, of 
 Math., Harvard Univ. (From the Har- 
 vard Register} /In mathematics, as in 
 other branches of study, the need is 
 now very much felt of teaching which 
 
 is general without being superficial; 
 limited to leading topics, and yet with- 
 in its limits; thorough, accurate, and 
 practical; adapted to the communica- 
 tion of some degree of power, as well 
 
MA THEM A TICS. 
 
 189 
 
 as knowledge, but free from details 
 which are important only to the spe- 
 cialist. Professor Byerly's Calculus 
 appears to be designed to meet this 
 want. . . . Such a plan leaves much 
 room for the exercise of individual 
 judgment; and differences of opinion 
 will undoubtedly exist in regard to one 
 and another point of this book. But 
 all teachers will agree that in selection, 
 arrangement, and treatment^, it is, on 
 the whole, in a very high degree, wise, 
 able, marked by a true scientific spirit, 
 and calculated to develop the same 
 spirit in the learner. . . . The book 
 contains, perhaps, all of the integral 
 calculus, as well as of the differential, 
 that is necessary to the ordinary stu- 
 dent. And with so much of this great 
 scientific method, every thorough stu- 
 dent of physics, and every general 
 scholar who feels any interest in the 
 relations of abstract thought, and is 
 capable of grasping a mathematical 
 idea, ought to be familiar. One who 
 aspires to technical learning must sup- 
 plement his mastery of the elements 
 by the study of the comprehensive 
 theoretical treatises. . . . But he who is 
 thoroughly acquainted with the book 
 before us has made a long stride into 
 a sound and practical knowledge of 
 the subject of the calculus. He has 
 begun to be a real analyst. 
 
 H. A. Newton, Prof, of Math, in 
 Yale Coll., New Haven : I have looked 
 it through with care, and find the sub- 
 ject very clearly and logically devel- 
 oped. I a/n strongly inclined to use it 
 in my class next year. 
 
 S. Hart-, recent Prof, of Math, in 
 Trinity Coll., Conn. : The student can 
 hardly fail, I think, to get from the book 
 an exact, and, at the same time, a satis- 
 factory explanation of the principles on 
 which the Calculus is based; and the 
 introduction of the simpler methods of 
 
 integration, as they are needed, enables 
 applications of those principles to be 
 introduced in such a way as to be both 
 interesting and instructive. 
 
 Charles Kraus, Techniker, Pard- 
 tibitz, Bohemia, Austria : Indem ich 
 den Empfang Ihres Buches dankend 
 bestaetige muss ich Ihnen, hoch geehr- 
 ter Herr gestehen, dass mich dasselbe 
 sehr erfreut hat, da es sich durch 
 grosse Reichhaltigkeit, besonders klare 
 Schreibvveise und vorzuegliche Behand- 
 lung des Stoffes auszeichnet, und er- 
 vveist sich dieses Werk als eine bedeut- 
 ende Bereicherung der mathematischen 
 Wissenschaft. 
 
 De Volson Wood, Prof, of 
 Math., Stevens' Inst., Hoboken, N.J.: 
 To say, as I do, that it is a first-class 
 work, is probably repeating what many 
 have already said for it. I admire the 
 rigid logical character of the work, 
 and am gratified to see that so able a 
 writer has shown explicitly the relation 
 between Derivatives, Infinitesimals, and 
 Differentials. The method of Limits 
 is the true one on which to found the 
 science of the calculus. The work is 
 not only comprehensive, but no vague- 
 ness is allowed in regard to definitions 
 or fundamental principles. 
 
 Del Kemper, Prof, of Math., 
 Hanipden Sidney Coll., Va. : My high 
 estimate of it has been amply vindi- 
 cated by its use in the class-room. 
 
 R. H. Graves, Prof, of Math., 
 Univ. of North Carolina : I have al- 
 ready decided to use it with my next 
 class ; it suits my purpose better than 
 any other book on the same subject 
 with which I am acquainted. 
 
 Edw. Brooks, Author of a Series 
 of Math. : Its statements are clear and 
 scholarly, and its methods thoroughly 
 analytic and in the spirit of the latest 
 mathematical thought. 
 
190 
 
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 Equations, may prove of service to working mathematicians. 
 
 H. A. Newton, Prof, of Math., 
 Yale Coll. : We shall use it in my 
 optional class next term. 
 
 Mathematical Visitor : The 
 subject is presented very clearly. It is 
 the first American treatise on the Cal- 
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MA Til EM A TICS. 
 
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 $2.00. 
 
 The chief aim has been to meet the wants of beginners in the 
 class-room. The Elements and Lectures of Sir W. R. Hamilton 
 are mines of wealth, and may be said to contain the suggestion 
 of all that will be done in the way of Quaternion research and 
 application : for this reason, as also on account of their diffuseness 
 of style, they are not suitable for the purposes of elementary instruc- 
 tion. The same may be said of Tait's Q^^aternions, a work of 
 great originality and comprehensiveness, in style very elegant but 
 very concise, and so beyond the time and needs of the beginner. 
 The Introduction to Quaternions by Kelland contains many exer- 
 cises and examples, of which free use has been made, admirably 
 illustrating the Quaternion spirit and method, but has been found, 
 in the class-room, practically deficient in the explanation of the 
 theory and conceptions which underlie these applications. The 
 object in view has thus been to cover the introductory ground more 
 thoroughly, especially in symbolic transformations, and at the same 
 time to obtain an arrangement better adapted to the methods of 
 instruction common in this country. 
 
 FROM COLLEGE PROFESSORS. 
 
 James Mills Peirce, Prof, of 
 Math., Harvard Coll. : 1 am much 
 pleased with it. It seems to me to 
 supply in a very satisfactory manner 
 the need which has long existed of a 
 clear, concise, well-arranged, and logi- 
 cally-developed introduction to this 
 branch of Mathematics. I think Prof. 
 Hardy has shown excellent judgment 
 in his methods of treatment, and also 
 in limiting himself to the exposition 
 
 and illustration of the fundamental 
 principles of his subject. It is, as it 
 ought to be, simply a preparation for 
 the study of the writings of Hamilton 
 and Tail. 
 
 Charles A. Young, Prof, of 
 Astronomy, Princeton Coll. : I find it 
 by far the most clear and intelligible 
 statement of the matter I have yet 
 seen. 
 
192 
 
 MA THEM A TICS. 
 
 Elements of the Differential and Integral Calculus. 
 
 With Examples and Applications. By J. M. TAYLOR, Professor c.f 
 Mathematics in Madison University. 8vo. Cloth. 249 pp. Mailing 
 price, $1.95; Introduction price, $i.So. 
 
 The aim of this treatise is to present simply and concisely the 
 fundamental problems of the Calculus, their solution, and more 
 common applications. Its axiomatic datum is that the change of a 
 variable, when not uniform, may be conceived as becoming uniform 
 at any value of the variable. 
 
 It employs the conception of rates, which affords finite differen- 
 tials, and also the simplest and most natural view of the problem of 
 the Differential Calculus. This problem of finding the relative 
 rates of change of related variables is afterwards reduced to that of 
 finding the limit of the ratio of their simultaneous increments ; and, 
 in a final chapter, the latter problem is solved by the principles of 
 infinitesimals. 
 
 Many theorems are proved both by the method of rates and that 
 of limits, and thus each is made to throw light upon the other. 
 The chapter on differentiation is followed by one on direct integra- 
 tion and its more important applications. Throughout the work 
 there are numerous practical problems in Geometry and Mechanics, 
 which serve to exhibit the power and use of the science, and to 
 excite and keep alive the interest of the student. 
 
 Judging from the author's experience in teaching the subject, it 
 is believed that this elementary treatise so sets forth and illustrates 
 the highly practical nature of the Calculus, as to awaken a lively 
 interest in many readers to whom a more abstract method of treat- 
 ment would be distasteful. 
 
 Oren Root, Jr., Prof, of Afath., 
 Hamilton Coll., N.Y.: In reading the 
 manuscript I was impressed by the 
 clearness of definition and demonstra- 
 tion, the pertinence of illustration, and 
 the happy union of exclusion and con- 
 densation. It seems to me most admir- 
 ably suited for use in college classes. 
 I prove my regard by adopting this as 
 our text-book on the calculus. 
 
 C. M. Charrappin, 8.J., St. 
 
 Louis Univ. : I have given the book a 
 thorough examination, and I am satis- 
 fied that it is the best work on the sub- 
 ject I have seen. I mean the best 
 work for what it was intended, a text- 
 book. I would like very much to in- 
 troduce it in the University. 
 (Jan. 12, 1885.) 
 
MATHEMATICS. 
 
 193 
 
 J. G. Fox, Prof, of Civil Eng., La- 
 fayette Coll., Easton, Pa.: It has some 
 very good points in its favor, such as, 
 the arrangement of the subject-matter, 
 the " numerous practical problems," 
 etc. (Feb. 21, 1885.) 
 
 J. Howard Gore, Prof, of Math., 
 
 Columbian Coll., Washington, D.C. : 
 From a careful inspection I think that 
 in very many respects it is a marked 
 improvement on the various works on 
 calculus now in use. I have always 
 thought that integral and differential 
 calculus should be studied at the same 
 time. This is not feasible except when 
 the author arranged the subject-matter 
 with that plan in view, as in this book. 
 At present I regard with favor the 
 introduction of this work in my classes 
 next session. (Jan. 9, 1885.) 
 
 C. H. Judson, Prof, of Math., 
 Fur man Univ., Greenville, S.C.: I find 
 it to be one of the most accurate, logical, 
 and carefully prepared text-books that 
 I have met with. I believe there is no 
 better text-book for teachers who adopt 
 the theory of " Rates " as the basis of 
 the calculus. (Dec. 30, 1884.) 
 
 O. C. Gray, Prof, of Math., Ark. 
 Indus. Univ. : Thus far, I am very 
 much pleased with it, particularly in 
 the fact that chapters on Differentiation 
 and Integration immediately follow 
 each other. Such an arrangement was 
 needed. (Jan. 6, 1885.) 
 
 P. L. Morse, Prof, of Math.. 
 Hanover Coll., Ind. : The matter is 
 certainly admirably chosen, and the 
 arrangement natural and unique. The 
 Integral Calculus is placed in proper 
 order, and the practical application to 
 Mechanics will do much to clear away 
 the mysteries of which the student often 
 justly complains. (Dec. 25, 1884.) 
 
 London Schoolmaster: This 
 easy but comprehensive treatise on the 
 
 calculus differs in its methods from 
 similar text-books produced on this 
 side of the Atlantic. Altogether, con- 
 sidering the extent of the ground it 
 covers, it is one of the easiest and 
 clearest text-books on the calculus 
 we know. 
 
 Boston Advertiser : It reflects a 
 high measure of credit upon the au- 
 thor. He certainly has succeeded to a 
 degree that may well insure for the 
 present volume extended use as a text- 
 book in our colleges. He has shown a 
 thorough, comprehensive grasp of his 
 subject, and has brought this to bear 
 with singular force in his pointed defi- 
 nitions, and in the clear reasoning of 
 his demonstrations. 
 
 The Nation, New York: It has 
 two marked characteristics. In the 
 first place, it is evidently a most care- 
 fully written book. There is nothing 
 vague or slipshod in it. Nearly every 
 sentence, certainly every theorem, seems 
 I to have been constructed with a stren- 
 uous effort to give it clearness and pre- 
 cision. This constant attention to the 
 form of expression has enabled the 
 author to be concise without becoming 
 obscure. We are acquainted with no 
 text-book of the calculus which com- 
 presses so much matter into so few 
 pages, and at the same time leaves the 
 impression that all that is necessary 
 has been said. In the second place, 
 the number of carefully selected ex- 
 amples, both of those worked out in 
 full in illustration of the text, and of 
 those left for the student to work out 
 for himself, is extraordinary. From 
 this point of view those teachers and 
 pupils who are accustomed to or prefer 
 a different text-book, would still do 
 well to provide themselves with this, 
 regarding it merely as a collection of 
 examples and without any reference to 
 the text. 
 
194 MA THEM A TICS. 
 
 Metrical Geometry : An Elementary Treatise on 
 
 Mensuration. By GEORGE BRUCE HALSTED, Ph.D., Prof. Mathema- 
 tics, University of Texas, Austin. I2mo. Cloth. 246 pages. Mailing 
 price, $1.10; Introduction, $1.00. 
 
 This work applies new principles and methods to simplify the 
 measurement of lengths, angles, areas, and volumes. It is strictly 
 demonstrative, but uses no Trigonometry, and is adapted to be taken 
 up in connection with, or following any elementary Geometry. It 
 treats of accessible and inaccessible straight lines, and of their inter- 
 dependence when in triangles, circles, etc. ; also gives a more rigid 
 rectification of the circumference, etc. It introduces the natural 
 unit of angle, and deduces the ordinary and circular measure. 
 Enlisting the auxiliary powers which modern geometers have recog- 
 nized in notation, it binds up each theorem also in a self-explanatory 
 formula, and this throughout the whole book on a system which 
 renders confusion impossible, and surprisingly facilitates acquire- 
 ment, as has been tested with very large classes in Princeton College. 
 In addition to all the common propositions about areas, a new 
 method, applicable to any polygon, is introduced, which so simplifies 
 and shortens all calculations, that it is destined to be universally 
 adopted in surveying, etc. In addition to the circle, sector, segment, 
 zone, annulus, etc., the parabola and ellipse are measured ; and be- 
 sides the common broken and curved surfaces, the theorems of 
 Pappus are demonstrated. Especial mention should be made of the 
 treatment of solid angles, which is original, introducing for the first 
 time, we think, the natural unit of solid angle, and making spherics 
 easy. For solids, a single informing idea is fixed upon of such 
 fecundity as to place within the reach of children results heretofore 
 only given by Integral Calculus. Throughout, a hundred illustrative 
 examples are worked out, and at the end are five hundred carefully 
 arranged and indexed exercises, using the metric system. 
 
 OPINIONS. 
 
 Simon Newcomb, Nautical Al- 
 manac Office, Washington, D.C.: I am 
 much interested in your Mensuration, 
 and wish I had seen it in time to have 
 sonje exercises suggested by it put into 
 my Geometry. (Sept. 8, 1881.) 
 
 Alexander MacParlane, Exam- 
 iner in Mathematics to the University 
 of Edinburgh, Scotland : The method, 
 figures, and examples appear excellent, 
 and I anticipate much benefit from its 
 minute perusal. 
 
MA THE MA TICS. 1 95 
 
 Elementary Co-ordinate Geometry. 
 
 By W. B. SMITH, Professor of Physics, Missouri State University. I2mo. 
 Cloth. 312 pp. Mailing price, $2.15; for Introduction, $2.00. 
 
 While in the study of Analytic Geometry either gain of knowledge 
 or culture of mind may be sought, the latter object alone can justify 
 placing it in a college curriculum. Yet the subject may be so pur- 
 sued as to be of no great educational value. Mere calculation, or the 
 solution of problems by algebraic processes, is a very inferior dis- 
 cipline of reason. Even geometry is not the best discipline. In all 
 thinking the real difficulty lies in forming clear notions of things. 
 In doing this all the higher faculties are brought into play. It is this 
 formation of concepts, therefore, that is the essential part of mental 
 training. He who forms them clearly and accurately may be safely 
 trusted to put them together correctly. Nearly every seeming mis- 
 take in reasoning is really a mistake in conception. 
 
 Such considerations have guided the composition of this book. 
 Concepts have been introduced in abundance, and the proofs made 
 to hinge directly upon them. Treated in this way the subject 
 seems adapted, as hardly any other, to develop the power of 
 thought. 
 
 Some of the special features of the work are : 
 
 1. Its SIZE is such it can be mastered in the time generally 
 allowed. 
 
 2. The SCOPE is far wider than in any other American work. 
 
 3. The combination of small size and large scope has been secured 
 through SUPERIOR METHODS, modern, direct, and rapid. 
 
 4. Conspicuous among such methods is that of DETERMINANTS, 
 here presented, by the union of theory and practice, in its real 
 power and beauty. 
 
 5. Confusion is shut out by a consistent, and self-explaining 
 
 NOTATION. 
 
 6. The ORDER OF DEVELOPMENT is natural, and leads without 
 break or turn from the simplest to the most complex. The method 
 is heuristic. 
 
 7. The student's grasp is strengthened by numerous EXERCISES. 
 
 8. The work has been TESTED at every point IN THE CLASS- 
 ROOM. 
 
196 MA THEM A TICS. 
 
 Determinants. 
 
 The Theory of Determinants : an Elementary Treatise. By PAUL H. 
 HANUS, B.S., Professor of Mathematics in the University of Colorado. 
 8vo. Cloth, ooo pages. Mailing price, $0.00; for Introduction, $0.00. 
 
 This is a text-book for the use of students in colleges and tech- 
 nical schools. The need of an American work on determinants 
 has long been felt by all teachers and students who have extended 
 their reading beyond the elements of mathematics. The importance 
 of the subject is no longer overlooked. The shortness and elegance 
 imparted to many otherwise tedious processes, by the introduction 
 of determinants, recommend their use even in the more elementary 
 branches, while the advanced student cannot dispense with a knowl- 
 edge of these valuable instruments of research. Moreover, deter- 
 minants are employed by all modern writers. 
 
 This book is written especially for those who have had no previous 
 knowledge of the subject, and is therefore adapted to self-instruction 
 as well as to the needs of the class-room. To this end the subject 
 is at first presented in a very simple manner. As the reader ad- 
 vances, less and less attention is given to details. Throughout the 
 entire work it is the constant aim to arouse and enliven the reader's 
 interest by first showing how the various concepts have arisen 
 naturally, and by giving such applications as can be presented with- 
 out exceeding the limits of the treatise. The work is sufficiently 
 comprehensive to enable the student that has mastered the volume 
 to use the determinant notation with ease, and to pursue his further 
 reading in the modern higher algebra with pleasure and profit. 
 
 In Chapter I. the evolution of a theory of determinants is touched 
 upon, and it is shown how determinants are produced in the process 
 of eliminating the variables from systems of simple equations with 
 some further preliminary notions and definitions. 
 
 In Chapter II. the most important properties of determinants are 
 discussed. Numerous examples serve to fix and exemplify the prin- 
 ciples deduced. 
 
 Chapter III. comprises half the entire volume. It is the design 
 of this chapter to familiarize the reader with the most important 
 special forms that occur in application, and to enable him to realize 
 the practical usefulness of determinants as instruments of research. 
 
 [Ready June i. 
 
MATHEMATICS. 197 
 
 Examples of Differential Equations. 
 
 By GEORGE A. OSBORNE, Professor of Mathematics in the Massachusetts 
 Institute of Technology, Boston. I2mo. Cloth, viii -f 50 pp. Mail- 
 ing price, 60 cts.; for Introduction, 50 cts. 
 
 Notwithstanding the importance of the study of Differential Equa- 
 tions, either as a branch of pure mathematics, or as applied to 
 Geometry or Physics, no American work on this subject has been 
 published containing a classified series of examples. This book is 
 intended to supply this want, and provides a series of nearly three 
 hundred examples with answers systematically arranged and grouped 
 under the different cases, and accompanied by concise rules for the 
 solution of each case. 
 
 It is hoped that the work will be found useful, not only in the 
 study of this important subject, but also by way of reference to 
 mathematical students generally whenever the solution of a differen- 
 tial equation is required. 
 
 Elements of the Theory of the Newtonian Poten- 
 
 tl'al Function. By B. O. PEIRCE, Assistant Professor of Mathematics 
 and Physics, Harvard University. I2mo. Cloth. 154 pages. Mailing 
 price, $l.6o; for Introduction, $1.50. 
 
 A knowledge of the properties of this function is essential for 
 electrical engineers, for students of mathematical physics, and for 
 all who desire more than an elementary knowledge of experimental 
 physics. 
 
 This book, based upon notes made for class-room use, was written 
 because no book in English gave in simple form, for the use of 
 students who know something of the calculus, so much of the theory 
 of the potential function as is needed in the study of physics. 
 Both matter and arrangement have been practically adapted to the 
 end in view. 
 
 CHAPTER I. The Attraction of Gravitation. 
 
 II. The Newtonian Potential Function in the Case of Gravitation. 
 
 III. The Newtonian Potential Function in the Case of Repulsive 
 
 Forces. 
 
 IV. Surface Distributions. Green's Theorem. 
 
 V. Application of the Results of the Preceding Chapters to 
 Electrostatics. 
 
" 
 
 HEO.CI8. JAN! 
 
 FORWNO.D06. 
 
I 1026 
 
 TA 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY