US2421745A - Computer for solving bombing problems - Google Patents

Description

June l0, 1947. G. B. DANTZIG I COMPUTER FOR SOLVING BOMBING PROBLEMS 'Filed sept. 1e, 1942 6 sheets-Sheet -1 June 10, 1947. G. B. DANTZIG COMPUTER FOR SOLVING BOMBING PROBLEMS 6 Sheets-Sheet 2 June 10, 1947.

G. B. DANTZIG COMPUTER FOR SOLVING BOMBING PROBLEMS Filed Sept. 16, 1942 6 Sheets-Sheet 3 June l0, 1947. G. B. DANTZIG COMPUTER FOR SOLVING BOMBING PROBLEMS Filed Sept. 16, 1942 6 Sheets-Sheet 5 l l lllllllllL lllll |Il|| l ITPIIL I June 10, 1947. G. B. DANTzIG 2,421,745 COMPUTER FOR SOLVING BOMBING PROBLEMS v I Filed sept. 1e, 1942 6 Sheets-Sheet 6 zosoo Patented June 10, 1947 UNITED STATES PATENT OFFICE COMPUTER FOR SOLVING BOMBING PROBLEMS (Granted under the act of March 3, 1883, as amended April 30, 1928; 370 0. G. 757) 20 Claims.

The invention described herein may be manufactured and used by or for the Government for governmental purposes, without the payment to me of any royalty thereon.

The invention relates generally to a computer for solving mathematical problems and specifically to a bombardier slip stick or slide-computer for evaluating, by mechanical means, bombing probability formulae for calculating the number of bombs to be dropped under given conditions on a target. It is herein described with reference to its use for solving aviation bombardment problems.

In bombing operations conducted by aircraft, economy of force indicates the dropping of the smallest practicable number of bombs required to provide a certain percentage degree of assurance of securing one hit or a given number of hits on the target. The number of bombs required may, for some conditions, be determined from existing tables, charts and curves of bombing probability values calculated in accordance with the mathematical law of errors. These bombardment probability tables, curves, and charts are available to the bombardment commander (bombardier) through published texts on bombing probabilities but the calculated values cover only a narrow scope of conditions and the practical use thereof for the solution of bomb requirements is limited and involves denite and time-consuming procedure.

The primary object of the invention is t provide a more convenient and rapid means for the solution of bomb requirements. To that end, the data is combined with and arranged on a slide instrument to provide a mechanical bombing p robability calculator or computer which (l) is readily operable to give a direct reading of the answer to a given problem and (2) has a greater range `of possible values than the tables found in published texts. Two species of the invention are disclosed, the preferred one of which is based on a simplifying assumption that the probable error in range is equal to the probable error in deiiection, both of which are expressed in terms of the more convenient circular probable error, as hereinafter explained.

Other objects and advantages of the computer will appear from the following detailed description of the construction and operation of the same, considered with reference to the accompanying drawing of the device, wherein:

Figure 1 is a fragmentary top plan view of one species of the invention;

Figure 2 is a vertical cross section on line 2-2 of Figure 1;

Figures 3 to 5 inclusive are detail views of substantially different sections of the data-bearing tape common to both illustrated species of the invention and collectively showing the arrangement and grouping thereon of the various curves and scales;

Figure 6 is a detail view of a table of probable errors for use in connection with illustrated solutions;

Figure '7 is a view similar to Figure l but showing another and preferred species of the invention;

Figure 8 is a vertical cross section on line S- of Figure 7;

Figures 9 to ll inclusive are detail views of elements of the particular species of the invention'shown more completely in Figure 7, and

Figure 12 is a detail view of a guard-bearing end of one of the tape-supporting cylinders.

The values of bombing probability calculations, together with the formulas for computing probable errors, as applied to the present invention, are herein dened and explained by Way of preface to the detailed description of the instrument.

The accuracy of individual bombing units for different altitudes is determined from tables of average errors representative of actual results of dropping a large number of bombs. Each bomb which does not fall on the center of the target has an error in range or in deflection-or in both. Range error is defined as the distance from the center of the target to the point of impact, measured in feet parallel to the direction of approach to the target. Deilection error is defined as the distance from the center of the target to the point of impact, measured in feet perpendicular to the direction of approach. The values of range and deflection errors may be combined in the form of a 'circular error which is the radial distance from the center of the target to the point of impact of the bomb. Average errors in range (Rea), cleilection (Dea), or radial distance (Cea) are equal to the arithmetic sum of the error in range, deflection, or radial distance, respectively divided by the number of bombs dropped. In determining average errors, the actual errors in feet is desired without regard to overs and shorts or rights and lefts For example assume a total number of six bombs to be dropped from a given altitude. The point of impact of each bomb is carefully plotted and the distance in feet from the point of impact to the center of the target in the direction of approach (range error) and perpendicular to the direction of approach (deflection error) is recorded. Assume, also, that the recorded arithmetic sum of the range errors of the individual bomb impact points totals 5&0 feet. Then, this sum divided by 6 (number of bombs dropped) equals feet which represents the range arithmetic mean or average error (Rea). In similar manner, the arithmetic mean or average deflection error (Dea) and the arithmetic mean or average circular error (Cea) are determined. From recorded average errors for different altitudes, tables of probable errors for those altitudes are prepared. Since various bombing units have different accuracies and since the accuracy of bombing is less at higher altitudes, the computed probable error may differ from one unit to the next. These tables give the general precision at Various altitudes in terms of Range probable error (Rep) and Deflection probable error (Dep) and Circular probable error (Cep). A specimen table of probable errors is shown in Figure 6. In reading the table, the indicated Cep accuracy for a given altitude is interpreted in terms of the radial dimensions of the circular area or target upon which will fall half of the number of bombs dropped. For example, the Cep accuracy of 173 feet at 10,000 feet altitude means that on the average half the number of bombs dropped at that altitude will fall in a circular target of 173 feet radius and half will miss the target.

The formulas for computing probable error are:

(l) Range probable error (Rep):0.845 Rea (2) Deflection probable error (Dep) :0.845 Dea (3) Circular probabl e error (Cep) :0.939 Cea (4) Cep- 11.746 \/Rep Dep (Where Rep :Dep)

Probable error is dened as that error or deviation which is as likely to be exceeded as not. In other words, a probable error of an indicated value, for example- 72 feet, does not mean that 72 feet is the most probable value of the error, either in range or deflection, but simply that there is a 50-50 chance or probability that the error of 72 feet may or may not be exceeded. The probable errors (Rep-Dep-Cep) for a given altitude are used in connection with the target dimensions to determine the vulnerability factor for that target and the given altitude. The formulas for computing the vulnerability factors are as follows:

(5) Range vulnerability factor Rep (probable error) (6) Deflection vulnerability factor DDT (allowable error) Dep (probable error) in which RDT represents the length or range dimension of the target and DDT represents the width or deflection dimension of the target. The distance from the center of a given target to the edge thereof represents the largest allowable error that can be made in bombing the target and still obtain a hit thereon. The allowable error applies to both range and deflection and is equal to one-half the dimensions of the target in range or deection.

The vulnerability factors, sometimes referred to as probability factors, are used to determine the single shot probability (SSP) for the given altitude, probable error and target dimensions. Each vulnerability factor has a calculated percentage value representing the probability of a hit with a single bomb dropped on a target of the dimensions used .in determining the vulnerability factor and under conditions for which the average error, measuring the efficiency and eectiveness of the particular bombing team or unit, is applicable. For example, if RVFzl, it means that there is a 50% probability of a hit. A hit by a single shot requires the concurrence of two events, i. e., a hit in range and a hit in deflection. Hence, when using rectangular errors (range and deflection errors) separate probabilities must be (nvr) (DVF) found for a hit with respect to both range (RSSP) and defiection (DSSP). The product of the range single shot probability (RSSP) and the deflection single shot probability (DSSP) gives the probability of both events occurring with a single bomb and, hence, the combined single shot probability (SSP) of a hit on the target. The values of the single shot probabilities have been calculated and are readily determined from tables and curves of vulnerability factors and probabilities published in well known ordnance texts. Example of such tables are shown on page 471, Chapter 15, of Ordnance and Gunnery by Earl M. McFarland. The single shot probability (SSP) is the basis upon which calculations may be made to determine the number of bombs required to provide a certain percentage degree of assurance of securing a given number of hits on the target. The results of some bombing probability calculations covering different percentage values of single shot probability and assurances of suceess for one or more hits are shown in tables and charts in standard text books on ordnance and the principles and methods governing the plotting thereof are well understood. The curves and tables described in connection with the instant invention are in accordance with these Well known principles but are of wider scope and cover a greater combination of events than the published charts and tables.

As shown in Figures 1 to 6 inclusive, illustrating one form of the invention, the instrument is cased within a receptacle or box I having a hinged cover 2 and cover-latching means 3 as Well as a handle 4 for conveniently carrying the case. Upon the inner surface of the hinged cover is suitably affixed a sheet 5 preferably containing a table 6 of probable errors, such as shown in Figure 6. The table may be accompanied by printed instructions for the use of the instrument. Within and xedly secured to the receptacle I and extending upwardly and slightly above the same is the body of the instrument which includes a framework I upon the top of which is mounted and supported a fiat face plate 8. The face plate is attached by screws 9 or other suitable fasteners to the framework and together with the latter, completely covers the interior opening of the receptacle, The depth of the frame work and the receptacle is greater at the back or hinge-bearing end I0 of the case than at the front or handle-bearing end Il. Hence, the face plate and the mutually contacting edges I2 of the box and cover are disposed in parallel planes slightly inclined to the horizontal when the device is resting on its bottom or base I3, as depicted in Figure 2 of the accompanying drawings. This inclination of the face plate facilitates the viewing of data and the manipulation of the movable parts associated therewith, as hereinafter described, and provides also compactness and reduction in the overall dimensions of the case.

The face plate is provided with a long narrow window I4 extending diagonally across its central portion and a pair of relatively smaller windows I5 and I6 to the right of the diagonal window and elongated in a direction parallel to the side edges of the face plate, as shown in Figure 1. Adjacent the diagonal window is a legend II and an arrow mark I8 indicating that the answer as to the number of bombs to be dropped for a given condition will be found from data exposed through the window. Similarly, appropriate legends I 9 and 20 adjacent the smaller windows I5 and I6 indicate that other inderdependent data, such 'as the` number of bomb hits required for a successful raid and the percentage chance' of a successful raid are tol be viewed therethrough. The larger of the two small windows is provided with a hairline-or an index mark 2 I. A series of guide curves 22 of equiprobability extend from the lower'longitudinal edge of the diagonal window to` points ina line adjacent and parallel to the -top edge of a guide rail 23 fastened, as at 24, to

the'face plate 8 inwardly of and in parallelism with the bottom edge of the plate. Provided on the'face plate in spaced relation to the lower edge of the guide rail 23 andk xed in parallelism therewith is an altitude error scale 25 in which `the values of deflection probable errors (Dep) in `eet'corresponding to a given range of altitudes For convenience, however, the scale reads directly in width dimensions of target which are twice the allowable 'error in deflection. This scale is shown r`at29 and is designated hereinafter as the target deflection scale.

Each of the scalesV 25 and 29, as laid off, covers a range from twenty-live (25) to ve thousand (5000) feet and they are exactly the same except that scale 29 is laid off from left to right whereas scale 25 is laid off from right to left. Suitably engaged with the slide 26 and adjustable longitudinally thereof is a runner 3U marked fwidth. 'Fastened to the face plate in parallelism with the guide rail 23 but appropriately spaced below the latter is another guide rail 3l on which is mounted an adjustable slide 32. Slide 32 is provided with an allowable error scale '33, corresponding to the scale 29, but which reads `directly in range dimensions of the target. The

xed altitude error scale 34, to which the slide 32 is set by means of the arrow 35, corresponding to the scale`25 but is offset relative toy the latter and "reads in values of range probable errors (Rep). Adjustable on slide 32 is a runner 3B marked length Runner 30 is adapted to be set slides, the right hand edge of runner 30 and the left hand edge of runner 36 (Figure l) should be aligned with or over the scale graduation corresponding to the given target dimension. Extending 'from the upper right hand corner of runner 30 and xed rigidly thereto is a long narrow arm 31. A similar arm 38 extends from the upperleft hand corner of the runner 36. These arms extend at right angles of forty-live (45) and 'l one hundred and thirty-five (135) degrees respectively, as measured in an anti-clockwise direction from the upper longitudinal edges of the respective slides. They cross each other at right angles and serve as coordinates establishing at their intersection a reference point 39 over the portion of the face plate on which are located the guide curves 22;` the position of the point relative to the face plate being determined by the setting of the runners and changing as the spacing between the In any rerunners-is varied. The arms 31 and-38, as well "as'the top'panels of thefrunners- 30 and 36, prefare not obscured.

`rEach arrangement of altitude error scale, slide,

' and runner corresponds essentially to an ordinary logarithmic slide rule with a reciprocal scale to facilitate division; vthe two divisions (Formulas 5 and 6) being `carried-out 0n the separate slide rules. When the slides are setto given Rep and Dep errors and the runners are set to given target dimensions, the positions of 'the arms relative to the face plate determine the values of the vrange and deflection vulnerability factors which may be translated directly into percentage values representing the probabilities of hitting the target in range and deflection. Arm 31 may 'be considered, therefore, as representing the deyilection single shot probability (DSSP) and arm '3B as representing th'e range single shot probability (RSSP). For every setting of the slides and runners, the `arms intersect and determine a point on the face plate to which isassigned a single percentage value which is the product of the separate` probability values represented by thel arms. "This single-percentage value represents the'probability of hitting the target with a single shot (SSP). Points on the face plate vof vequi-'probability, that is, having the same SSP Vor cylinders located within the framework and vincluding a main cylinderM-at the back of the framework, a pair of'vertically spaced auxiliary cylinders i2 and 43 at the front of the framework, and a tape-tautening cylinder. immediately in front of the main cylinder. The main and auxiliary cylinders are journaled in side members of the framework l but the tape-tautcylinder is supported by and journaled betw en a `pair of upstanding members and 46. These support members have their lower ends respectively pivoted, as at 4l to the bottom of the case for movement about a commonv axis extending transversely to the direction of travel of the tape and are resiliently connected at their respective upper ends, by springs 48 to the sides of the framework. The springs are tensioned to exert a pull on the support members constantly tending -to displace the tape-tautening cylinder de in the direction and with a magnitude required to take up any slack in the tape. The tape is held tightly to th'e main cylinder by lock ro at? on shaft and is moved when the main cylinder 4| is turned by means of the hand knob 5S en the extended right end of the cylinder stub shaft 52', Displacement of the tape laterally of the supporting cylinders is prevented, without binding, by cone-shaped guards 53 mounted on the cylinders adjacent the edges of th'e tapa-as shown in Figures 2 and 12.

The data carried'by the tape 40 comprises a plurality of curves and 'scales vbased'on bombing probability calculations. These are arranged in separate groups or charts covering diiierent given conditions throughout a range of broad or limited scope, as may be found expedient. In the instant case, four groups or charts of such curves and scales, are provided on the tape and are marked D, C, B and A, as shown collectively in Figures 3, 4 and 5. These charts correspond to representative chance levels for success of 99.5, 95, 85, and 70 percent, respectively. Each chart comprises a set of curves in which each curve indicates a. number of bombs; a non-uniform scale in which each graduation indicates a number of bomb hits; and a series of similar markings indicating the particular percentage degree of assurance of success upon which the curves and scales of the particular chart are based. Chart A comprises the set of bomb-indicating curves 54, the bomb hit scale 55, and the percentage markings 56. Chart B comprises the bomb-indicating curves 51, bomb-hit scale 58, and percentage markings 59. Chart C comprises the bomb-indicating curves E0, bomb-hit scale 6| and percentage markings 62. Chart D comprises the bomb-indicating curves 63, bomb-hit scale 64, and percentage markings 65. These charts are arranged one behind another on and in the length wise direction of the tape in the natural sequence of their respective indicated percentage values. A set of curves 65 for evaluating the guide curves 22, and obtaining thereby the diierent values of single shot probability, is located on the tape between the iirst and last charts A and D and is coordinated with the marking SSP on the tape, shown in Figure 3. The curves 54, 5l, 60, 63, and 6E extend diagonally across the tape in the opposite sense to the diagonal window I4 of the face plate 8 and are positioned on the tape to be exposed through the window I4 at a given setting of the tape. The bomb-hit scales 55, 58, Si, and G, and the SSP marking, are arranged in a. column lengthwise of the tape and to the right of the curves in position to be exposed through the window l5. The percentage markings 5S, 59, 62, and 65 are similarly arranged on the tape for exposure through the face plate window i6.

The curves and scales oi the respective charts A, B, C and D cover a comparatively large number of bombs and bomb-hits, ranging in the ior mer from one to at least twenty thousand bombs (7i-20h89) and in the latter from one to one hundred bomb hits (1.-180). The values of the single shot probabilities represented by the curves 55 range from one-half percent to ninety-nine one-half percent. The curves (bombs) of each chart are distinguishably marked to designate the number of bombs individually represented thereby and are coordinated with the corresponding non-uniform scale (for the bomb hits required) of the chart. This coordination is accomplished in the process of laying out the various charts as will be clear from the following explanation of the manner of laying out chart A. The tape is moved relative to the face plate and its position thereto for the solution of bomb requirements is determined by means of a scale, such as shown at 55, on the tape and the hairline indicator 2i on the face plate: the scale can be non-uniform and arbitrary. The surface of the tape is viewed through the diagonal window at the upper end of the guide curves 22, For every combination of guide curve (or SSP) and position of the tape (or graduation of scale 55 there corresponds a point on the surface of the tape which is directly at or below the upper end of the guide curve. To this point is assigned a numerical value which is a function of the SSP as represented by the guide Curve, and the number of bomb hits as represented by the scale graduation determining the position of the tape; this numerical value represents the number oi bombs to be dropped to obtain the desired number of hits in accordance with the percentage degree of success on Which the scale is based. On the surface of the tape, a curve 54 is empirically drawn Where the numerical values are constant. By selection of several fixed values of number of bombs required, the various curves 54 are drawn. Each curve is marked at spaced points along its length with a value indicative of the number of bombs represented by the curve so that it can be evaluated.

The spacing of the graduations oi the scale 55 governs the formation of the curves 54 and in the process of laying out the chart the spacing of the scale graduations is empirically varied until the curves 54 of the tape are approximately straight lines parallel to each other and at right angle to the long lower edge oi the diagonal window i4 so as to achieve ease in reading the answer. When the tape moves relative to the face plate, the positions of the curves 54 along the lower edge of the diagonal Window change and each change in position of the curves 54 gives rise to different evaluations of the guide curves E?. in terms of the number of bombs to be dropped for given conditions. When the tape is moved relatively to the face plate for evaluation of the guide curves 22 in terms of single shot probability (SSP), its position is determined by means of the SSP marking on the tape and the hairline indicator 2l on the face plate. Evaluation of the guide curves 22 in terms of single shot probability is given by the curves B6 on the tape. These curves 6G are drawn and coordinated with the SSP marking so as to be exposed at the diagonal Window i4 substantially as straight line continuations of the guide curves 22 when the SSP marking is opposite the hairline indicator The value of SSP represented by each oi the curves 66 is indicated at one end of the curve.

As an illustrative example of a problem confronting a bombardment commander, let it be assumed that the commander of a bombing unit is given an order to neutralize a certain objective-a factory, for instance. His problem is to determine the number of bombs to be dropped on this target to attain a reasonable chance of destroying or neutralizing the same Without dropping more than a necessary number of bombs. In order to do this, he must know, among other things, the size of the target; the accuracy of his bombing unit; the minimum altitude of bombing that may be considered in operating against this particular target (often very high because of enemy antiaircrait fire), and the number of actual hits necessary to reduce the target. Since there is, also, an element of chance that every aimed bomb Will miss the target and that the mission will be unsuccessful and will not cariy out the required number of hits, he must deter mine what percentage chance of success of ac complishing the mission is a reasonable chance in View of the urgency of the situation or other tactical considerations. The highest chance ol success is set usually at 99.5% (i. e. only 5 raids out of 1000 Will fail) since absolute certainty of success would require an impossible number (theoretically infinite) of bombs to be dropped.

ceptable on and justifying a bombing 'mission again'st'targets which, though important, offer no immediate menace and the destruction 'of which is not urgent, The target dimensions,

sizef of bombs, required number of bomb fhits,

andother specicinformationof his targets required by the bombardment commanderare ohmy tained from an oobjectivefolder which is a le or ,booklet of information concerning the targets andl collected by` observation aviation andthe various intelligence agencies; suchy information being compiled and arranged rior the use 'ofthe commander inplanning a bombing mission. The decisions as to the minimum altitudefrorn which to ybomb and the'` percentage degree of success to .be attained are tactical and are made in accordance with the judgment of the commander. The accuracy of his bombing unit for the conditio'nsgivenfare obtained kfrom table ,6. The number ofbombsto be dropped ordinarily is computed directlydrom or by interpolationwof bombardment tablesA orcurves. Such computa-v tion is avoided by the use ofthe present invention in solving the problemas willrbe apparent from the succeeding explanation yof, its ,use and operation.

Forr some conditionsof bombing it may be desirable td know the number. of, individually sighted and releasedr bombs which' `shouldmbe dropped onv a `given target to give therequiredv degree of assurance of ,at least,v one. hit. In such a case, it is necessary first to determine the single snot probability which is applicable tothe givenlactorsw of the problem. 'For example, assumey the values., ofV the given factors to be: target dimensions-e500 :feet bylDO l-feet; altitude-15,00()l feet g required percentage degree of obtainir'igl yat least y one r hit-70%. It n is required i. to. Afind (a) the single shot probability'v (SSP) and (b) the number ofrbombs to bedropped.. The particular setting oiy the u Ainstrument which solves this problemi.; ism., shown in Figure 1 and is obtained yby, the following procedure; The runner 30 is set over |00 on thedeilection dimension scale, 29vvand runner` 35 isset over A500 on the range dimension l scale 33 the longer of lthe two given rtarget dimensions being taken `as the length. From the table 6, the probable errors in range (Rep) and inrdeflection (Dep). corresponding toV the given altitude of 15,000 feetare found to be 130 feet and, 168y feet respectively. The slides 2G and` 32, withthe runners 3U and 36 Set thereon, as just described, are positioned so that the index arrowZB. is centered over I68.on the Dep scale 25 andthe index arrow 35 is centered over 130 on Vthe Rep scale 34. The knob is turned to. movey the tape until the marking .SSP (singleshot probability) on the tape is centered beneaththe hairline4 2i of the window l5.. As a result of this movement of the tape and thev arrangement ofthe scales and curves thereon, the curves `|6.for the different values of single shot probability (SSP) are exposed through the diagonal window I4 but no data'will show in the percentage window t6.

From this setting of the instrument, the percentage chance of a single bomb hitting the target is found by noting the position of` the reference point 39' relative tothe guide curves v22, particularly with reference to its distance or spacing fromV the nearestguide' curve within the,

and the v'alue'of this product (whichgives theL percentage chance of hitting the target with a single bomb, abbreviated SSP) is obtained from curves B by followingthe guide curve 22near est to the reference point; The answer is found.

to be a single shotfprobability (SSP). of 0.13 percent. Having thus. determined the percentage chance orprobability. ofr hittingthe target with a singlebomb'in 'one trial, the same is. used as a basis for. further determining the. number of trials necessary (bombs'dropped) to secure a desired percentv chance, from` 1%.l to 100%, 0f. hitting the target with at least one bomb. In current practice', tables based fon'the desiredy de gree of certaintymay be entered with the known or calculated value of` single shot probabilityto secure opposite it .the required number of bombs to be' dropped toy attain the result. sought, vWith the herein described instrument, however, it is required only to turnthe knob 5l untilthe'number 1 of scale 55 on. chart A of theftapeis cen-` tered in window. I5 withthe required percentage number (70%) showing in window I6. The move* ment of the tape occasioned .by .the turning 4of the knob results in the displacement .of the SSP. curves B6 from thediagonal window andzthe substitution therefor of'the .bomb curves 54 of chart A. The number of bombstobe `dropped with` the requiredpercentage degree of chance oiy getting at least'onev hit, based on the single shot probabilityof 0.1'3 percent,'would be read orf` the bomb curves inthe .diagonal'window by iollowing the procedure'previously explained and described'in connection with reading the SSP.

The values of Rep and Dep are theoretically needed in planning attacks against most targets, especially rectangular'o'nes, `as long asthe range and deflection errors are unequal. `However,` determination of single shot probabilityris quicker andl easier using Cep values and such use is jus# tied when range and deflection errors are substantially equal. Now, if the values'of rRep and Dep are expressed in terms of circularprobable error (Cep) in'accordance with the rFormula 4, then Rep and Dep caribe represented by a single value and the'evaluationof the formulae by mechanical means is simplied. Reverting to lormula` 4l andfassumingmRep fand Dep to' be equal, itis obvious thatRepequals Cep'divided by 1.74s. Aisoitliatinep equais'cep divided by l.7l. v The constant 1.746 is a conversion factor derivedwas an integral part vof relationship between range, deflection, andlcircular'prob'able error by mathematical calculations. lllence,`l'or mulas 5 and y6 maybe restated in the following forms:

Dep Cep# 1,746

Itis an empirical fact that the value obtained, by dividing the Cepfor a given altitude by. M746,

lies between the true` values of Rep ,and Dep for' the same altitudeand ythat this average value can be used to-rep'resent both the range probable error 'and the deflectionV probable. error. The range probable error, as represented by this average" value, will be larger than the true Ren but the represented deflection probable error, on the other hand, will be less than the true Dep. The two discrepancies therefore compensate each other and, as a result, the value of single shot probability determined by using this average value is substantially the same as the value of single shot probability determined by using the true values of Rep and Dep. This may be shown in connection with the problem illustrated in Figure l. As therein depicted, the Dep slide 26 is set to 168 on the Dep scale 25 and the Rep slide 33 is set to 130 on the Rep scale 34. As shown in table E, the corresponding Cep is 266. If the value of the single shot probability were to be determined by using only the Cep, the settings of the Dep slide and the Rep slide on their respective scales would have to be identical, that is, have a common value (Dep=Rep). This common value is obtained by dividing 266 by 1.746 (Formulas 7 and 8) and is found to be 152. This common or average value (152) is larger than 130 (Rep) but less than 168 (Dep). If the range and deflection slides of the instrument were set to 152 instead of to 130 and 168 respectively, the guide curve 22, on which the reference point 39 would fall in this setting, would be the same curve upon which it was positioned in the setting shown in Figure 1. As seen, this curve has an SSP value of 0.13 percent.

Formulas 7 and 8 are preferred over Formulas and 6 because in the former the denominators are equal and are expressed in terms of circular probable error. Also, the numerator of Formula 7 is greater than that of Formula 8. Hence, the two divisions can be carried out simultaneously on one ordinary reciprocal slide rule with a single setting for the denominators and the use of two runners or indicators on one slide to give separate readings for the numerators. It is upon the empirical fact herein explained, and the advantages flowing therefrom, that the preferred form of the invention, shown in Figures 7 to 11, inclusive, is based as hereinbefore stated. Since Formulas 7 and 8 require only values of Cep and not Rep or Dep values, a single table converting altitude into Cep values is all that is needed when using the instrument, in its preferred form, to determine the range and deflection vulnerability factors. A specimen table of altitude versus circular probable error values, intended for use with the instrument, is shown in Figure 1l. Only a few representative values of circular probable error corresponding to given altitudes are shown but it is to be understood that all blank spaces are to be lled in for convenient reference by the bombardment commander; the inserted values depending of course on the rated accuracy or bombing ability of his unit for the indicated altitudes.

As is apparent from a comparison of Figures 1 and 7 of the drawings, both forms of the instrument are substantially alike in respect to their various parts and general constructional features; the preferred form being distinctive in respect to the number and arrangement of slide rule parts. Referring to Figure 9, the face plate S is provided with a single scale 6l in which the Rep and Dep for a given range of altitudes are represented, in feet, logarithmically and by average values (Cep-1.746 Formulas 7 and 8). For convenience the sca-le reads directly in terms cf Cep errors corresponding to the average values- This permits the instrument to be set directly to the Cep values given in the table 63,

shown in Figure 1l. Scale 6l is, therefore, desig! nated the altitude error or Cep scale and it constitutes the fixed part of a slide rule; the movable part 0f which, as shown in Figure 10, is a slide 69 dove-tailed with and slidably adjustable on the guide rail 10, which is fastened to the face plate inwardly of and in parallelism with the Cep scale El. An extension 'll on the slide normally overlies the Cep scale and is provided with a window 12 having a hairline 13 for guidance in setting the slide to a given Cep value on the scale 6l. The slide is provided on its upper surface with a scale 75, in feet, of allowable errors in range and deiiection expressed logarithmically. For convenience, scale reads directly in dimensions of the target which are twice the allowable error in range or deflection and is thereinafter designated the target dimension scale. The Cep scale El and the target dimension scale 15 are exactly the same except that the former is laid off from right to left and the latter from left to right. Each scale covers a range from twenty-five (25) to five thousand (5000) feet. Suitably engaged with the slide 69 for independent adjustment longitudinally thereof are two runners 16 and l'l marked width and length respectively. The former is adapted to be set relative to the target dimension scale 'I5 in accordance with the given width dimension of a target and the latter in accordance with the given length dimension; the inner vertical edge of each runner being used as the setting-guide 0r index. These runners are similar in construction to the runners 30 and 36 of Figure l and in like manner the arms 3T and 38 cross each other and serves as coordinates establishing at their intersection a reference point 39 over the portion of the face plate bearing the guide curves 22. The arrangement of scale El, slide 69, and runners 'l5 and l1 corresponds essentially to an ordinary slide rule with a reciprocal scale to facilita-te division and the operation thereof is the same as in the form of instrument rst described: a chief feature of distinction being that in the preferred form of, instrument the two divisions (Formulas 7 and 8) are carried out simultaneously on the same slide, made possible because of the equality of denominators.

For example, assume the values of the given factors of a problem to be solved are as follows: target dimensions- 1000 feet by 200 feet; altitude 20,300 feet; number of bomb hits required- 40; chance of successful raid- 70%. Itis required to find the number of bombs to be dropped. The setting of the instrument which solves the problem is shown in Figure 7 and is obtained in the following manner. Runners 'l and 'Vl are set over 200 and 1000 respectively on the target dlmension scale 75; the longer dimension being taken as the length. Slide 69 is then positioned so that the hairline 13 of the Window '.2 is centered over 400 on the Cep scale 6l; this number representing the Cep corresponding to the given altitude as taken from table 68. Tape 40 is then moved relative to the face plate 8 by turning the knob 5l until the number 40 is centered under the hairline 2l' of window I5 with the number 70% showing at window i6. The number of bombs to be dropped is found by following the guide curve 22 nearest to the reference point 39' as in previous examples. The answer, 215 bombs, is read off the set of curves 54 (chart A), appearing in the diagonal window I4', by interpolation. When the guide curves 22 are to be evaluated in terms of single short probability, the knob 5I' is turned until the marking SSP on the 1.3,. tape is centered `beneath the hairline 2| oi window` |57, at whichtime the curves 6 5 will be exposed through. the Ldiagonal window.

From the foregoing description,y the many advantages of the instrument, are apparent. The device is compact and durablewithout involving complicated structure andthe calculations there included cover a broad scope of conditions, The numbers set in are allvisible and easily checked and only a few seconds are required to work out a given problem. The instrument is advantageous not only for everyday use in the field. but also as a valuable adjunct to training and as a gauge of proiiciency. The use oi the instrument, by visibly presenting results asthe measure of the bombing accuracy of a particular bombing team or unit, emphasizes and drives home the importance of improving the accuracy so that a small bombing force can do the work of a large force.

It is obvious, also, that the invention is not restricted to the exact construction and use hereinbefore set forth but is adapted for general use as a computer. It is understood, therefore, that omissions, substitutions and alterations in the form and details of the instrument may be made within the scope of the appended claims and without departing from the spirit of the invention.

Having thus described the invention, I claim:

1. A computer comprising a pair of superposed relatively movable members, one cf said membershaving thereon an index mark and also a row of value-representing characters, the latter extending transversely With respect to the direction of relative movement of the members,l and the other of said members having two separate groups of value-indicating characters arranged thereon so that during relative movement of the members the characters of one group align successively with the said index mark while the characters of the other group register with the said row of value-representing characters, the characters of the said one group being arranged in a single column extending in parallelism with the direction of relative movement cf the members and the characters of the said other group being arranged in several approximately straight lines parallel to each other and obliquely disposed with respect to the direction of relative movement of the members, each line including only characters of a constant value which is a function of the values of a given value-representing.character on the index-carrying member and a selected valueindicating character of the column on the other member when the said selected character aligned with the said index mark.

2. A computer having in combination a pair of superposed relatively movable members, one member having a series of Value-indicating characters and a set of evaluated curves and the other member having a fixed mark and a narrow elongated window extending transversely with respect to the direction of relative movement of the said members, said other member having markings representing a scale of numerical values along an edge of the said window, the said characters and curves of the one member being arranged relative to the said mark and window so that during relative movement of the members characters successively register with the said mark while at the same time the curves travel past 'the Window as approximately straight lines parallel to but obliquely disposed with respect to the ydirection of relative movement ofk theV members `whereby the point of intersection of each line 14r with-the Window scalev markings. is continuously displaced along the said markings, each line repf resenting a constant value which is a function of a given numerical value of the particular window scale marking intersected thereby and a. given value of the value-indicating character in registry with the said mark in a predetermined relative position of the saidmembers.

3'. A computer comprising a member having a set of curves, aslide rule mounted on the member and comprising two scale-bearing unitsl disposed for relative sliding movement against yeach other inl parallelism with and adjacent the axis of abscissas of the curves, one of said scale-bearing units 'being fixed against movementl in paralellism with the said axis andthe other scale-bearing unit being free to be so moved, a pair ofy runners on the said other scale-bearing unit, and an index member carried by each runner and extending angularly therefrom and over the curves and across the index member of the other runner to establish at their intersection over the curves a reference point.

4. Al computer comprising a body having a scale and a series ofr evaluated curves, a slider movable relative to thebody and carrying a scale, and a pair of runners on. the body and slidably and separately settable to given positionsv relative tothe scale on the slider, each runner having an arm extending angularly therefrom, and overthe said curves, said arms crossing each other to establish at their intersectionA over the curves a reference point, the evaluation of which is dependent upon and4 varies withits position over the curves as determined by the relative settings of the scales andrunners.

5. A computer comprising a body having a scale and aseries of curves, a slider. movable relative to the body and. carrying a scale, a pair of runners on the body and slidably and separately settable to given positions relative to the scale on the slider, each runner having an arm extending angularlyV therefrom and over .the said curves, said arms crossing each other to establish at their intersection over the curves a reference point, the evaluation of which is dependent upon and varies with its position over the curves as determined by the relative settings of the scales and runners, a member mounted on the body and slidablymow able to various predetermined positions relative to the said curves, said member having a plurality of value-indicating characters arranged thereon to give different evaluations of the curves in the various positions of the member, and cooperative indicia on the said body and member for setting the latter to any predetermined. position.

6. A computer comprising a set of curves, a pair of logarithmic scales arranged in parallelism with the axis of abseissas of the curves for sliding movement against each other, one of the scales being fixed relatively to the curves and having values which are reciprocals of the values` on the other scale and said other scale being movable relatively to the curves, a pair of slidable runners on the movable scale, separate arms xed respecat;j 1,745

associated with the said curves and graduated to provide a scale of values in terms of circular probable errors extending in parallelism with the axis of abscissas of the curves, a movable scalebearing member slidable longitudinally of the fixed scale-bearing member and graduated to provide a scale of allowable error values, the values oi one scale being the reciprocals of the values of the other scale, means on the slidable scale-bearing member lor setting it to a given graduation of the fixed scale-bearing member, a p-air of runners on the slidable scale-bearing member, a vulnerability factor-representing element carried by each runner and extending angularly therefrom over the said curves in crossed relation with the corresponding element of the other runner, said elements serving as coordinates establishing at their intersection a reference point having a value of single shot probability dependent upon its position over the curves, and means associated with e ch curve for indicating the value represented thereby.

8. A computer comprising a support, a chart mounted in the support for sliding movement relative thereto and having value lines extending diagonally of its direction of movement, a. slide rule mounted on the support and extending transversely of the direction of the movement of the chart, said slide rule having a stationary scale fixed to the support and a movable scale reading against and slidable longitudinally of the xed scale, runners on the movable scale, separate index members respectively carried by the runners and extending angularly over the value lines relatively crossed rela-tion to serve as coordinates establishing at their intersection a reference point having a determinable value dependent upon its position with respect to the value lines.

9. A computer comprising a member having a series of points and marked to provide a scale remote from the said points and fixed relatively thereto, said member also having guide curves extending rom the points and between the points and the fixed scale, a scale bearing slide movable lengthwise of the xed scale, a pair of runners on the slide and having setting guides to be selectively aligned with the graduations of the slide scale, an index member carried by each runner and extending angularly therefrom over the guide curves and across the corresponding member of the companion runner to establish at their intersection a reference point having a value dependent upon its position with respect to the series of points as determined by its position to the nearest guide curve, and means associated With the said member and operatively positioned in relation to the said points for indicating the value represented by each point.

l0. A bombing probability computer comprising a member having a set of curves, means for indicating the values of the curves in terms of single shot probability values, a set of indicia on said member and representing a scale of probable error values, a body slidable on said member and along the scale of probable error values, said body carrying a scale of allowable error values, the values of both of said scales being expressed logarithmically and the values of one of the scales being the reciprocals of the values of the other scale, index means on the slidable body for registry with the values of the probable error scale, a pair of runners carried by the slidable body and slidably settable thereon with respect to the values of the allowable error scale, a vulnerability factor-representing member fixedly secured to each runner and extending angularly over the curves in crossed relation with the corresponding member of the other runner to establish at their intersection a reference point having a value ci single shot probability determinable from its position over the curves either directly or by interpolation between curves.

il. A bombing probability computer comprising a member' having a set of equi-probability curves and a set of indicia representing a scale of probable error values, the probable error scale being remote from the said curves, a movable body on said member adjacent to and slidable along the probable error scale, said body carrying a scale of allowable error values, the values oi both of said scales being expressed logarithrnicaliy and the values or" one scale being the reciprocals oi the values of the other scale, a pair oi runners carried by said movable body and settable thereon with respect to the values of the allowable error scale, a member carried by each runner and extending angularly over the curves crossed relation with the corresponding member oi the other runner to establish at their intersection a reference point having a determinable value dependent upon its position relative to the said curves and upon the values assigned to the curves, and a movable member adjacent the said curves and having separate sets of indicated values to be alternately positioned with respect t0 the curves for ssigning different values thereto.

l2. A computer comprising a set of evaluated curves, a scale-bearing member disposed over the cu Yes and having a pair of logarithmic scales extending in parallelism with the axis of abscissas of the curves and slidably movable against each other, said scale-bearing member and the said set ci curves being relatively movable in a direction parallel to the axis of ordinates of the curves but one oi said scales being fixed against movement relative to the curves in a direction parallel to the axis of abscissas and having values which are the reciprocals of the values on the other scale, a pair of slidable runners on the said other scale, separate arms fixed respectively to the runners and extending over the curves, said arms crossing each other and serving as coordinates establishing at their intersection a reference point having a value dependent upon its position over the curves, and means associated with each curve for indicating the value represented thereby.

13. A computer comprising a face plate having a diagonal edge and a set of indicia representing a iixed scale remote from and opposite the diagonal edge, said face plate also having a series of guide curves extending from the said diagonal edge to points adjacent and parallel with the nxed scale, a member' movable past the said diagonal edge and carrying a set of value curves arranged thereon to appear at the said diagonal edge approximately as straight line continuations of the guide curves parallel to each other and to the said diagonal edge, a movable scale-bearing member operatively associated with the face plate and slidably against the fixed scale, a pair of runners on the movable scale-bearing member, a member carried by each runner and extending angularly therefrom and over the said guide curves in crossed relation with the corresponding member of the other runner', said runnercarried members establishing at their intersection a reference point having a value dependent upon its position relative to the value curves as determined by its distance from the nearest guide L17 curve, and means associated with each value -curve for indicating the value Irepresented thereby.

14. A bombing probability computer comprising a member having a set of bomb requirement lcurves based von a constant percentage degree of assurance of obtaining certain numbers of hitsthroughout a given range rand for different values of single shot probability, a `slide rule mounted in parallelism with the axis of abscissas of the curves and movable relative to and over -the curves in parallelism with axis of ordinates ofthe curves, said rule including a unit fixed against movement laterally of the 'curves and graduated to provide la scale of probable error values and a unit Vsli'dably movable against the fixed unit, and graduated to provide a scale of allowable error values, the values on one of the scales being the reciprocals of the 'values on the other scale, a pair of runners on the slidably movable unit of the rule, each runner having an index member extending over 'the curves in crossed relation with the index member of the other runner tc establish at their intersection a reference-point having a Value dependent upon its position relative to the rcurves and means associated Awith each curve `for indicating the value represented thereby.

r15. A bombing probability computer comprising a member having a set of equi-probability curves and a tset of indicia representing a scale of probable error values xed relative to the said curves, a movable body on said member adjacent to vand slidable against the `iixed scale, said body.,carrying a scale of allowable error values, the values of the xed and movable scales being expressed logarithmically and the values of one scale being the reciprocals of the values of the other scale, a pair vof runners carried by the said body and settable thereon with respect to the values of the allowable error scale, a member carried by each runner and extending angularly over the curves in crossed relation with the corresponding member of the other runner to establish at their intersection a reference point having a determinable value dependent upon its position relative to the said curves and upon the values assigned to the curves,l and a movable member adjacent the said curves and having separate sets of value curves to be alternately positioned with respect to the set of equiprobability curves for assigning different values thereto, the value curves of one set representing different values of single shot probability and value curves of the other set representing the varying number of bombs required to provide a given percentage degree of assurance of securing a given number of hits, and means associated with each value curve for indicating the value represented thereby.

16. A computer comprising a body; two slide rules on the body, each rule comprising a scalebearing unit fixed relatively to the body and a scale-bearing unit slidably movable along the xed scale-bearing unit and a runner mounted on the said slidably movable scale-bearing unit and having an angularly extending arm; said rules being relatively arranged on the body so that the said arms extend crosswise of each other to establish at their intersection a reference point and the said runners being slidably movable along their respective mounting scales to vary the position of the reference point, each position of the reference point having a determinable value; a .series of curves on the said body to be traversed by the said arms, each curve connecting points on the body corresponding to different positions of the reference point which have the same determinable value; a curveevaluating member mounted von the said body and movable in a direction crosswise to the said curves to various predetermined positions relative to the body, said member having a plurality of value-indicating characters arranged thereon to be aligned selectively with the curves by and during the positioning of the curve-evaluating member whereby to give different evaluations of the curves in the various vpredetermined `positions of the curve-evaluating member; and cooperative indicia on the said body and curveevaluating member for setting the latter to any predetermined position.

17. A bombing probability computer comprising a body having a scale of probability errors xed on the body and ascale of allowable errors movable on the body, the latter scale being Aslidably settable along the nxed scaleto a'given value of probable error, a pair of indicators on lthe body slidably and separately settable to given values of allowable errors Iand extending crosswise of each other to establish at theirrintersection a reference point the position of'which'relative to the body varies in accordance with 'the relative settings of the scales and indicators, each indicator representing in terms of probability value the corresponding vulnerability factor which is a function of the given values of probable and allowable errors and each vposition of the reference point having a determinable value which is a function of the probabilities represented by the said indicators, and a `series of curves-on the body to be traversed by the said indicators, each curve connecting 'points on the body corresponding to different positions of the reference point which have the same determinable value.

18. A computer comprising a body having a series of guide curves thereon representing different values of single shot probability, indicatormeans on the body and movable into indicating position relative to the said guide curves, a member associated with the body and movable into various predetermined positions relative to the latter for disposing different selected points on the member in line with the guide curves for every different combination of guide curves and position oi member, each position of the member representing a value in terms of number of bomb hits desired with a given percentage degree of success and each selected point on the member representing a determinable value in terms of number of bombs which is a function of the values represented by a given combination of guide curve and position of member, a series of evaluated curves on the -member and connecting selected points thereon having the same value, said evaluated curves being arranged on the member to be displaced laterally across the guide curves by and during the positioning of the member to give different evaluations ol the guide curves in the various positions of the member, and cooperative means on the body and on the member for determining the setting of the latter to a desired position.

19. A bombing probability computer comprising a body having a series of curves representing different values of SSP, a set of indicia fixed on the body and representing a scale of probability errors, a member slidably settable along the scale of probability errors to a given value of probable error, said member carrying a scale of allow..

able errors, two indicators separately slidable and settable to given values of allowable errors, said indicators extending across the SSP curves and crossing each other at every relative setting of the scales and indicators to chosen values of probable and allowable errors for establishing at their intersection over the SSP curves a datum point, an evaluating member having a series of evaluated curves representing diferent amounts of bombs, said evaluating member being settable into various predetermined positions relative to the body for disposing each of the said evaluated curves in line with a different SSP curve for every position of the evaluating member, and cooperative indicia on the body and evaluating member for determining the setting of the latter to each of its various predetermined positions.

20. A bombing probability computer comprising a member having a plurality of charts thereon, each chart including a set of value curves representing the varying number of bombs required to provide a given percentage degree of assurance of securing a given number of hits-a set of indicia representing a non-uniform scale of bomb hits coordinated with the curvesand a series of indicators of the percentage degree of assurance on which the bomb hit scale and value curves are based, a face plate over the member and movable relative thereto, said face plate having a long window for exposing portions of the value curves diagonally with respect to the direction of relative movement of the member and face plate and having other windows exposing units of the scale and indicator series respectively, a set of indicia, representing a scale of probable error values in which the range and deection probable errors are represented by an average value, said scale-representing indicia being xed to the face plate remote from the long window and reading directly in terms of Cep errors corresponding to the average values, guide curves of equi-probability on the face plate and extending from the probable error scale to the long window, an indexed slide associated with the face plate and slidable longitudinally over the said probable error scale to a position corresponding to a given value of circular probable error, a set of indicia on the slide representing a scale of allowable errors reading in dimensions of target which are twice the allowable error in range and deflection, a pair of runners on the slide and separately settable with respect to the scalerepresenting indicia on the slide to given values of range and deflection dimensions respectively of a target, an arm carried by each runner and extending angularly over the guide curves in crossed relation with the corresponding arm of the other runner, said crossed arms serving as coordinates establishing at their intersection a reference point having a value in number of bombs dependent upon its position relative to the value curves as determined by its distance from the nearest guide curve.

GEORGE B. DANTZIG.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Number Name Date 1,031,852 Johns July 9, 1912 1,200,569 Young Oct. 10, 1916 1,637,222 Hempleman July 26, 1927 1,915,038 Troche June 20, 1933 1,935,021 Engblom et al Nov. 14, 1933 2,295,616 Williamson Sept. 15, 1942

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