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Drpu.v M,n,..tct / . . i^ / TOPCXiRAPHICAL SURVEYS BRANCH ' I BULLETIN 41 I TESTS OF SMALL TELESCOPES AT THE LABORATORY OF THE DOMINION LANDS SURVEYS BY E. DEVILLE. LL.D., SLRVEYOR GENERAL OF DOMINION LANDS \ OTTAWA J. |.i; I,ABi:i)Qri:RIK TACHK PRINTER TO THK KING'S MOST KXCia.LKNT MA.IKSTT llil« ;<()7G4- I tMMt/tlCT iiiiinii 1 1 its'" i5Vafl03ft 5 METHOD OF TESTING. Introductory Note. In sur\(\inK Dominion liimls, a»trononii»al olist-rvations arc ni.i to half a mile with r.Kl^ divided to tenths of hnks. Moth in. ihtwls reciuire lelesmiH.'^ of exieilrMt (iiiaUty in view of the limited dimensions lomvatible willi ordinary survey instruments. The staff of the Surveys ialH)rat«iry ha\e thus In-en led lo devote considerahle alt( ntion to tests of small telestDiKs. It is pioiMised lo desirihe their miihtwl of testing and iiuidenl.illy, to jjive ihi' ex)Hrinuiilal d.Ua nathered in workinji oul the methiMl. The investigations were londucted \>\ \V. C Way, who i> in ihar^*' of the laboratory. .\11 tin- d.iia were olisir\ed li\ him and liy his asM-.tant>. The Essential Characteristics of a Surveyor's Telescope. The telescoiK"> features which are of interest to the survt\or are the l.riKlu- ness, the magnification, the field of view, the resolution or rex.lvinv; power, and, to a lesser decree, the fall in resolution from the centre to the mafnin of the field. Distortion, which would affect the readings of ,i rod near l' marnin, is of no imi)ort.uue. rods hein^c read alw.iys in the centre. The various alH-r- rations do not interest the surveyor otherwise than as they affect the resolution. The measurement of the inannifi(ation and of the field of view is a >im)>le matter. DisrcRardinK the loss of light through al)sori)ti(in and refieclion, the brightness ma\ , for the surveyor's purposes, he taken as ])roiH>rtional to the area of the exit ])upil of the tel scope, the diameter of which can lie either measured or calculated from the objective's diameter and the maKuificalion. .\ll that remains to be ascertained by the tests is the resolution. The Resoli >n. In the s\steni adopti by the Sni\eys I rcsolvi'"^ 1M)W er is the resolution ot unas^i M.itorv, the unit of resolution or arenc> 4« x 45 ctntiimtres ,)re(*ntinK h K-ries of ■ets of clear circular «iotH on an o])aque grouml. It in illuniinate«l from h.hitid by eight 4a-watt tunRsten Iaiui». Three suitably hjM.td Kroiind kIhhh plates between the lamps and the transiwrency diffuse the Lght an«l produce even illumination. The dots in each set are uniformly siwced, their diameter king one-tenth of their interval. This interval decreases in Kionutric i)roKression from one set to the adjoining one, the ratio of the progression lieing: -1 105 Each of the smaller sets contains 2,5tK) dots. To save space, the larger imes with coarse dots have not (luite so many. The upper jwrt of figure 8 reiiresents portions of four adjoining sets. Beyond the limit of resolution, a 8< t apjHjars as a surface uniformly illum- inated. The test consists in ascertaining the finest set in which structure can be detected. The ai)paratus is installed in the building of the comparator for measures of length. The transjiarency is against the wall at one end of the building ami the telescope stand near the other end. 42-80 metres awa>. A small lamp behind the observer gives just enough light for manipulating the • scope: otherwise the room is in complete darkness. Numbering of the Sets. That set in which structure can just be detected from the telescope stand by unassisted vision is numbered one. The angle subtended by the dot interval of this set was found by the laboratory observers to 1^ • 000 405 or— about ,,-,„ , . 2170' 1 23 . It 18 one inch at 206 feet, 8 inches at 25 chains, and 12-8 inches at half a mile. An> other set is numbered R when the angle subtended by the dot interval is 0000 405 R It follows that the resolution of a telescope is the serial numl)er of th in which structure can just be detected through the telescope. The constant 000 405. from which the serial numbers are calculated is a physiological factor which varies with individuals: it may even vary with the physical condition of the observer. When proceeding with resolution deter- minations, the observer must, from time to time, ascertain his personal equation or factor by observing witii a telescope of known resolution. All the results of observation are multiplied by the personal factor found. ObwrvBtion of the Resolution. Til.- .,hsfnalU.n f.,r rfM.lutic.n is fir^t nuult- l.y ixaminint; th.- imag.- of the dots .„ thr . .^urv of • uhl ami notinR the- MhaliiM .i-t in whi. I ruitur.. can (H- clfttTtid. With .. ., K. toK-sc«,K-. H.runur*. i. ,,uiti- plain .nt- Mt aiul al.«^.l in thf acljou.inu "m-. Thf crrin numUr i. iiu. riM.laft,! .H.iniation With a ixK.r ti.|est„,H-. the dit.rminati.,.. i, more un.ertain: ihi. hiuertainiy is a tiiirh K'mhI test of (juality. The fallinK off in tiehniti,)n away from the centre is mea^u^ta l.y lli< .1= rtaM in resolution rea.l ujx.n an c.eentrie inane. H> adoplinR a conMant anxle for t direction of the eccenl imaRe he rsults with differ.nl maRnifications or telescojx's are comparable. TelescoiRs of ecjual rew)lutioii d- not always disclose e«,ual detail und-r all conditions. If the ohject ^icwed is feebly Illuminated, the teles«„,x- having the largest exit pupil and the brixhtest i.naKe shows greater detail and the other tele«o,,es apiK-ar to have less resolving ,K,wer. but with the intense illumination provided l»ehind the transjKirency .lots, the changes in brightness due t.. differ.' in area of the exit pupil do not. within practical limits, affect the re»o! .!(.: .f the telescope. Aberrations. The nature of the aljerrations is disclosed by the changes which they . ause in the aspect of the tbts in and out of f.Kus. So far. little attenti.m has l,een devoted to the subject at the lalxjratory as it is n.>t of material importance for the puriK)se in view. Reading of Graduated Rods. Among the problems which present themselves to the surveyor is to find the resolution necessary for reading graduated rods. Let d be the width of one division of the rod and L the distance. The angle subtended by one division being j-, the resolution required for detecting structure is: — 0000 405 X L d This resolution is just enough to perceive that the rod is graduated For counting and reading the divisions, the laboratory experiments indicate that about one-fourth more resolution is required. ^^j5 ^ 0000 405 X L 4 d or approximately : — 30764—21 R. 2000d (1) A number of probloms can be solved by means of the alxjve relation between I lie distance of a rod, the width of the divisions, and the resolution needed to read them. Example 1 .—What resolution is needed for measuring half a mile with a stadia rod graduated to tenths of a link? In equation (1), L and d are 40 and OOOl chains respectively, hence: — R = 20. Example 2 . — How far can a rod divided to hundredths of a foot be read with a resolution of 15? Equation (1) gives: — L = 300 feet. Example 3. — What is the smallest division which can be read on a rod at 440 metres with a resolution of 22? Equation (1) gives: — d = -01 metres. EXPERIMENTAL DATA. -,- 1 Further Definitions. In working out the method of testing just (lescrii)ed, the laboratory staff liad to make a number of exjieriments which throw some light ui)oii the causes utTecting the resolution and ma>- be related as connected with the subject. The definition of " resolution " may be extended to the case of an objective: the resolution may be called R when the dots of the set number R are se])arated in the focal plane image. By definition, the resolution of the eye a full aperture is unity. The resolution becomes less than unity when the etTective ajierture of the i)i!|)il is sufficiently reduced, as when looking through a small hole or through the exit l)Ui)il of a telescope of high magnification. Notation. R Resolution or resolving power of a telescope or of an ol»jt-ciive. It lia> alreadv been defined. J '.I iV/-Magnification of a telescope. It is t\ uio of the focal lengths of ohicctivo and ocular, or of the diameters of eflfective ajwrture and exit pupil. r>— Diameter of an objective or effective ajx^rture of a telesrope, in millimetres. F— Focal length of objective in millimetres. /—Equivalent focal length of ocular in millimetres. e-Diameter of the exit pupil of a telescope or of a circular diaphragm placed in front of the eye, in millimetres. R_ ^ —Resolution factor of an objective or telescoix.-, or resolution |)er millimetre of aperture. M_l ^, .^ . — -—Magnification factor of a telescojx; or magnification ix>r millimetre of aperture. It is equal to the reciprocal of the diameter of the exit pupil. r —Resolution or resolving power of the eye when looking through a circular diaphragm or a telescope's exit jjupil. -—Resolution factor of the e>e when the diameter of the pupil is e. It varies with the pupil aperture. T, /—Angle subtended by a dot interval or by the radius of a diffraction ring. X — Wave length of light, in microns. General Principles. The angle / subtended by the dot interval of the set R is:— 0- 000 405 t = R Seen through a telescope, the angle is magnified M times and becomes :- M r = 000 405 R than To be able to resolve this angle, the resolution of the eye must be not less or:- 000 405 f R ^Mr R_ M (2) /^•' j Hence: — ' ' (1) The resolution of a telescope cannot be greater than the product of the resolution of the eye corresponding to the size of the exit pupil, by the magnification. With high magnifications, the exit pupil is small and the resolution of the eye less than unity. Hence: — (2) With high magnifications, the resolution of a telescope is always less than the magnification. The aperture D of the telescope being eciual to Me, equation 2 may be changed to: — R ^r^ /) ~ e Hence : — (3) The resolution factor of a telescope cannot be greater than the resolution factor of the eye for a pupil aperture equal to the exit pupil of the telescope, nor can it be greater than the resolution factor of the objective. It must therefore be the lower of the two factors. According to these rules, the resolution of a telescope is limited either by the resolution of the objective, or by the resolution of the eye in conjunction with the ocular. Each must be investigated separately. Diffracted Images. It is shown by the mathematical theory of diffraction that the image of a lumindiis point in the focal plane of an objective is not a point, but a bright circular patch surrounded by alternately dark and bright rings. The angular radius t of the first dark ring is: — /=1220^ in which X is the wave length of light and D the diameter of the objective. It is generally accepted that two luminous points are seen separated when the first dark ring of one image passes through the centre of the other image, that is, when the angular distance of the points is equal to the radius of the first dark diffraction ring. The illumination midway between the ]>oints is then 74 per cent of the maximum. If this rule were applicable to the dot transparency, the resolution of the objective would be: — „ 0-000 405 0-322 R = =_— £> / X Taking 0-56 microns for the wave length of the brightest part of the spectrum : — • R = 059 D The rule is not strictly applicable to the dot transparency which must give slight' greater resolution. Moreover, the value of the coefficient depends upon the quality of the definition required in the image.'" is:- The highest value derived from visual experiments by the laboratorj- staff R = 063 D The illumination of the darkest spots of the image is then 59 jxr cent of the brightest spots. Resolution of the Eye. .\s already stated the angle subtended by the dot interval of transparency set number one, which can just be resolved by unassisted vision from the telescope stand, was found to be 000 40.S. This coefficient can also be obtained by observation of the dots at closer range. No material change was noticed until the distance from the trans- parency became smaller than 7 • 50 metres, when the coefficient commenced to decrease. The observations could not be made at distances less than one metre because of the impracticability of producing by photography sets of dots sufficiently fine, the dots being enlarged by diffraction. The difficulty was overcome by observing with an ocular of long focal length the image in the focal plane of a good objective of large numerical aperture; the mean of a great number of observations gave • 000 37. It would thus appear that the vision of the observers was ten per cent keener when looking at an image through an ocular than when looking at the object itself with the naked eye. According to this, a telescope of unit magnification, consisting of an objective and an ocular of equal focal lengths, would have a resolution of 110 and would show objects ten per cent better than unassisted vision. This deduction has not been verified experimentally. The resolution of the eye at full aperture, which is unity when looking at objects from a distance exceeding 7-50 metres, becomes 1-10 at close range. Reverting to equation (2), RSMr the following additional rule may now be formulated: — (4) At low magnifications, the power of a telescope may be ten per cent greater than the magnification, but no more. This rule has been confirmed experimentally. Whether the increase in the acuity of vision at close range is an actual fact or whether the effects observed are due to other causes and may be explained otherwise, is immaterial for the purposes of this investigation. ( 1 ) The Resolving Power of Objectives, by P. G. .Vuttins— Bulletin of the Bureau of Standards. Vol. 6, No. 1. 10 Relation Between the Pupil Aperture and the Resolution of the Eye. Several series of experiments were undertaken for ascertaining the relation between the effective pupil aperture and the resolution of the eye. Circular holes were drilled through a thin brass plate and carefully measured with a micrometer microscope. The plate was fixed at various distances from the transparency and the dots examined through the holes. Tlie resolution was obtained by multii)lying the number of the observed set by: — 42-80 ^-■■^■i L being the distance in metres between the perforated plate and the trans- parency. For data at very close range, the plate was placed at the exit pupil of good telescopes of large numerical aperture and low magnification. Two of the series are given below, one at a distance of 7-50 metres from the transparency and the other with oculars of 32 mm. focal length and smaller. The figures are the mean of a large number of observations. REL.ATION BETWEEN THE PUPIL APERTURE .AND THE RESOLUTION OF THE EYE. Pupil Aperture, e, in Millimetres. 0-25 0-.?7 0-5.? 0-65 0-77 102 118 1 19 1-34 1-56 1-84 2 02 2-50 2-75 Resolution, r, at a Distance of -50 metres. 0-380 0-454 0-588 0-661 0-738 0-829 0-916 0-962 1-000 1000 032 nu'tres and less. 0157 0-230 0-336 0-404 0-485 0-627 0-726 0-805 0-890 0-956 1-041 1-102 1104 .vl 11 The observations are plotted in Fi^. 1. At 7 50 metres from the trans- parency, the resolution of t,,, eye for afK-rtures between and 0-9 mm. is pro- portional to the aperture, the equation being: — /• = ()-59 e It is the same ctiuation as for a good objective with a s .newlial smaller resolution factor. PudNs ,p«.1u'. 075 05 10 15 20 25 FiQ. 1— Relation between the pupil's ap<'rture and the resolution of the (!ye. Above 0-9 mm. the resolution increases more slowly than the aperture until it reaches unity for an aperture of 2 • 5 mm. Beyond 2 • 5 mm, there is no increase in the resolution; it remains constant at unity, which agrees with the definition of resolution. With oculars at close range, the resolution for apertures between and 0-9 mm. is also proportional to the aperture, but the equation is: — r = 0-63 e This is precisely the equation for an objective of good quality. Above 0-9 mm, the resolution increases more slowly than the aperture until it reaches a value of 1-10 for an aperture of 2-5 mm. Beyond 2-5 mm. the resolution re -ns constant a*^ 1 • 10. The resolution of the eye at full aperture, which, by d- on, is 1 • 00 for distant objects, becomes 110 at close range, thus showing keent. vision at close range than at a distance. Taking 15 mm. for the equivalent in air of the focal length of the eye, the pupil's diameters of 0-9 mm. and 2-5 mm. correspond to numerical a(x>rtures of /•717 and F/6. Up to F/17, its resolution is, like that of a lens, proportional to the diameter of the pupil, but it falls short of a good lens at greater apertures. ■*, 12 For application *^ telescope tests, It is more convenient to plot the resolution and magnification factors. The data rre as follows : RELATION BETWEEN THE RESOLUTION AND MAGNIFICATION FACTORS OF THE EYE. Magnification Factok. Resolution Factor, I > at a Distance of t J. e 7-50 metres. 0-032 metres and less. 0.364 0-364 0-402 400 0400 0-441 0-495 0-476 0516 0-544 0-498 0-520 0-641 0-532 0-570 0-746 0-551 0-601 0-84 0-610 0-85 0-560 0-98 0-577 0-615 1-30 0-590 0630 1-54 0-585 0-622 1-89 0-634 2-70 0-622 4-00 0-628 The data are plotted in Fig. 2. The observations at 7"? SO are marked with a small circle : those at short range with a cross. The smooth curves A and B represent very nearly the two sets of observations. — B Magnirication p«r milMm«tr« of pupil's apcrtur* CI 0.2 03 04 05 06 07 08 0.3 10 l.r 1,2 15 0.1 0.2 03 55 51 OS 07 SS 53 To il ti iT T4 IT" Fig. 2 — Relntion bftween the resolution and magnification (actors of the eye. 13 At 7? 50, the magnification and resolution factors are equal so long as they are below 0-4 (pupil 2 -5 mm or more). Alx)ve 0-4 the resolution factor increases more slowly than the magnification factor .mtil it reaches a maximum and constant value o* 0-59 for a magnification factor of 1 • 10. In the test . . .lose range, the magnification and resolution factors are pro- portional so long as they are below 0-4 hut they arc not equal. The relation is: — r 1 -=— X 110 e e which means that the resolution is constant and equal to 1 • 10. Above 0'4, the resolution factor increases more slowly than the magnification factor until it reaches a maximum value of 0-63 for a magnification factor of 1-10, beyond which it remains constant. The end of the curve B is not shown in the figure: it merely extends to the right as a parallel to the axis of abscissae, as far as magnification factor 4-00. This maximum value of 0-63 compared with 0-59 at 7" 50, illustrates again but in a different manner, keener vision at close range. The values of the resolution factor at close range, measured from curve B of Fig. 2, are given below: they are the values to be used in telescope tests. MAGNIFICATION AND RESOLUTION FACTORS OF THE EVE AT CLOSE RANGE, Magnification Factor. Resolution Factor. Magnification Factor. Resolution Factor. 02 220 0-7 0-587 0-3 0-330 0-8 0-607 0-4 0-440 0-9 0-619 0-5 0-511 1-0 0-627 0-6 0-556 1-1 0-630 Resolution of a Telescope. It is now possible to understand how a telescope should work. In Fig. 3, O D F is the curve of the eye's resolution factor (curve B of Fig. 2). The resolution factor of the objective Ijeing a constant, it is represented upon this diag am by a parallel to the axis of abscissae which may occupy one of three pof )ns. If the factor is less than 0-63, the line j4 E cuts the curve at D. For a magnification factor comprised between and H, the telescope's resolution factor is that of the eye: for a greater magnification, it is that of the objective. The resolution is limited by the objective: the eye could see more than the objective can resolve. B F 1 A D/-^^ E i / |H u Magnification Factor Fio. 3 — Resolution of a telescope 14 V.l ".V An olijoctive with a resolution factor of 0-6.? i>, rfprfstMited by the line B F tanKent to the curve. The resolution factor of the telescojK' is that of the eye, whatever the inagnitication. CG re|)resents an objective with a resolution factor greater than 0-63. The telescoix's resolution factor must alsK NO. 13167. f = 255.8mm K KOK OltJKCriVE ApEKTI HKS OK .WOmm 30-9mm j 23-9mm 21-3mm I9.2iiim l,S-4niiii i _i ' i 32- 00mm 20-95 7-99 12-21 8 - 53 121 .S-.53 U-8 8 • 53 8 • 39 15-75 16-24 13-8 14-5 12-7 120 9 ■ 06 28 ■ 23 21-7 IS -7 14-6 13-2 4-10 j 62-39 24-0 19-6 14-7 13-2 10-9 i 10-9 8-17 7-79 10 4 9(«) 121 9 14 12-0 9-72 121 9-74 For telescope \o. 7<>7, Fig. 4, the exi)erimental j)()ints lie vi-r\ dose lo the curve: the aKreenient is ahnost jK-rfect. The observations at niagniticatioii Maqnific3f:on F?-to* 01 02 03 04 05 06 07 08 C9 10 I.I FiQ. 4— Expcrinicnts with tplosropo \o. "97. 1.3 1.4 1.5 1.6 16 fartorn Km.ti-r than 1-65 hav mit Ik-o., pl.,iu-.l Urau^H. the fixure woul.l exten.I Ux. far to th.- riKht ; the risoluti.,,, Uuim for all ilusc ohK-nations is very m ..rly or,.? sn that they all lie <,n the extensi.m of the curve parallel to the axis of ahsciHsae. These ol,Her%-ations have also laeii omit ted fron. the other figurcH for the same reason. Mignrfication Ftcioi S^ 0? 3 0« 0> "« 0? 08 09 '''' ^ — i» — ^~Tf — 08 — 05 — iis — r Ti ra i3 — w — ii — iV Flu. 5- K«piTiiiicnt» with (I'luM'tipc No. 2|. but ,h^;*'-''T'l^' ^'"n-^'* ^l''- ^' """ ^K'-^'^''"'-'"^ '« «°«1 "t high magnifications but there is a slight falling off at medium magnifications. The falling off at medium magn.ncations is a little more marked with tele- scope No. 13167. Fig. 5. but the agreement is good at high and low magnifica- tions. " ^ = Mignification Ftetor °' <" 03 04 -ol 06 07 08 09 10 li TS ti M k k" Fio. 6— Kxperimcnts with tcltwop.^ Xo. 1316" On the whole, the agreement is about as gocnl as could be expected in obsemng with objectives and oculars which were not corrected to work together Experiments with an objective of poor quality are given below: they are plotted in Fig. 7. 17 KESOLirnoN KXI'KKI.MKNTS WITM TKI.KStOI't No. 1074. A-J80mm. l>-4i 5mm. Magnification. Ki'Milutiiin . . , 11 9 18 I 17 21 M 41 8 21 S At the low maKnifiratitm of 1 1 4 (exit j.upil -.?.66 mm) which is little m. o than oiuMiuarttT of the ol.jecliveV ajR-rture in millinu-tris, tho resolution of t*« ttlescojK' is as k'mkI as that of a iKrfecl insiriimeni. It is limited hv what ib- JO \i_ itji' 'l^■^■ iRii •IIQ Kui. 7- KxpprinicntN «iil. ii.l.«opi. x„. 1074. eye is able to see throujjh the ocular, which is less than is resoK ed l>> »h= The luality of the objective does nm come into play. The defective iKrformance c.f the objective is i)roukht 1 1; .i the .nagnifications of 41S and 92-6 (Exit pu,)ils=l()4 and 0-47 ,„fni. F ' obj.-ctive were a Rood one, the resolution should be 21 and 27 4 while ii ^ K 2l5and21-8. The eye could see through the (Kular more deiuii ihanisresctS d by the objective. On the figure, the resolution factor of the objective, 050. i- npfescH i-d b» the line A B parallel to the axis of abscissae and intersecting ''^> curve -h eye's resolution factor at A . The curve for the resolution fa. of ih t»t should therefore be the line OA B. The observed data aj; fairU >- ' *h=. this conclusion but the sharp angle at A is rounded off. Tht ffcct of t» K^l quality of the objective is felt before th. limit of its resolution is reach. Resolution of a Perfect Objective. The maximum resolution factor of telescopes 797, 21, and 1J167 1<. l |jy observation is 0-63; it could not be more because that is the limit for th eye. What the ouservatioii shows is that the factor for the objectives is no less than 0-63, but it may be greater. The test fixed only a minimum limit for the objective's factor; the actual value cannot be determined by visual observation. In order to elucidate what the actual factoi may be, four consecutive sets of the dot transparency near the limit of resolution were photographed on a bromide dry plate. It Thf mide plate, alwut 56 and 45 mirronii respectively ~ /? (actinic) -0-706 In other woriU, if 0-567 is the resolution for visual rays, it is 0-706 for actinic ra}'.4. The sets photographed were Nos. 0-656, 0-689, 0-723, and 0-75Q. The first two only should have l>een resolved. The photograph, enlarged three and a half times, ii'cpnxlucetllKlow iFig. 8). The upiK'r pjirt of the figure is the geometrical imajjo of the dots and the lower part the actual phot«.graph. Structure is quin distinct in Nos. 0()56 and 0-689; it is fairly plain in No. 0-723 but is nearly obliterated in No. 0-759. In No.O.flM No.O.«W No. 0.723 No. 0.7M Geometriral Imace. ^% Actual Photograph. Fio, 8— Ti-Bnspaj-en>-y dot* and their photograph near the limit o( resolution the visual tests, an image like this would be read as 0-75; it corresponds for visual rays to a resolution of 0-60 and a resolution factor of 0-67. The objective apiiears to resolve 6 per cent more than can be seen by the eye. However, iiiriher experiments are required before a definite conclusion can be reached. 1» General ConitderatkNU. An King ai« the diamiti-c «if the exit ])upil iH not \vm than 2 5 mm. the refwiution ot an ordinary tele«-o|H! of fair nualWy should not Ik- Icsh than the theoretical value, Ltmer value** are howe r foumi in prismatic l.in hall a minute is only 2-8; the power of the telescojMs specitied is therefore more than ample for the measurement of angles. Tile si/e of the telescope of a le\el instrumeiil de(H>nds uiM)n the graduation of the rod and the disLmce at wliicli it is to l)e read. The necessary resolution is gi\C'n liy t'cpiation ( 1 ).