Abstract:
A teaching aid computer, requiring no electrical or spring power, designed for elementary and middle school students, that when properly set and manipulated, provides real number solutions to equations involving multiple degree, as well as single degree unknowns. Parallel, interconnecting balancing beams are marked so as to provide adjustable positions for weights, whose resultant torques represent equation constants and coefficients of each degree of the unknown quantity. Further, by positioning and clamping movable axes beams, students can see and feel the results of adding or subtracting a few, or a series of numbers. Also, sliding beams, as well as sliding weights captively located in channels of each beam, with no required extra loose weights, allow simple decimal settings and answers to equations involving multiplication, and division. Being non-electronic, the student can understand the simple, visable, teeter-tooter-like workability of the repeated mechanical advantages and torque transfers from beam to beam, prior to beam and equation balancing, with this computer; and so gain confidence in rapidly discovering an answer, and also a range of non-answers, to a third degree equation, for example.

Description:
CROSS REFERENCE APPLICATION 
     This is a continuation in part of Ser. No. 07/722,952, filed Jun. 28, 1991, now abandoned. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention is categorized as an educational aid for students of mathematics, The construction and operation of this teaching aid can help elementary and middle grade students understand equations involving addition, subtraction, multiplication, division, and multiple as well as single degree unknowns. 
     2. Description and Relationship of Prior Art 
     U.S. Pat. No. 3,788,541 also covers a mechanical computer. However the goal of U.S. Pat. No. 3,788,541 is to provide accuracies similar to those achieved by a very large slide rule. By utilizing reels of tape, instead of rule length, U.S. Pat. No. 3,788,541&#39;s approach remains compact in size. The goals, construction, and utilization of the computer as described in this application are completely different from referenced U.S. Pat. No. 3,788,541. 
     U.S. Pat. Nos. 3,949,491, 3,928,923, 4,106,220, 4,713,009, 4,731,022, and European Patent EP - 240-574-A, and British Patent 1,407,899, all involve single balancing beam teaching aid apparatus to help young students with solutions to equations involving only single degree unknowns, such as 2+X=7; 3N=18; 2X+(-X)+3=2(-X)+15; and 27/3=R. U.S. Pat. No. 3,000,114 involves multiple balancing beams, but with no facility for changing a lever arm length on either side of any beam. It follows that U.S. Pat. No. 3,000,114 does not indicate the ability to solve any multiple degree equations. 
     Equations having multiple degree unknowns have very important and widespread use throughout science and industry, Examples are included later that involve: 
     (a) a 4th degree equation for finding the radius of a rod having a given twisting torque T, shear modulus E, and length L--as used extensively in the automotive industry; 
     (b) a 3rd degree equation for finding the radius of a sphere--a soft ball--when knowing the ball&#39;s volume; 
     (c) a 2nd degree equation for finding the direct current in an automobile light bulb when knowing its resistance and its wattage rating. 
     A separate list includes some common multiple degree equations. 
     Studies have shown that young students are capable of learning concepts involved with equations having multiple degree unknows. However, there is no manipulating type of mechanical computer teaching aid that will help in understanding and solving a multiple degree equation such as -2X 4  -5X 3  +10X 2  -4X+16=0. Later in this application, an example of the simple set up and solution of such a 4th degree equation is illustrated. This invention&#39;s teaching aid will reinforce the students&#39; understanding and memory of the equation solution process, in addition to discovering problem solutions. 
     Since all of the referenced patents involve only a single balance beam, it will be revealed by a comparison to representative patent &#39;491, FIGS. 4, 6, and 8, versus this application&#39;s FIGS. 1 and 2, that in order for &#39;491 to arrive at a total weight, shown as 44, at a specific beam lever arm, shown as -6 in FIG. 6, or -8 in FIG. 8, several component weights must first be added mentally, then stacked correctly, before a desired confirmation, as in FIG. 8, that 8×3=24. With this application&#39;s approach, by comparison, it is easier, faster, and has less likelihood of error, to slide a 1 unit weight to a +4 position on a second beam, than to stack four 1 unit weights at a number 1 position on a single beam. 
     The stacking of weights, involved with the referenced patents&#39; single beams, could raise the grouped weights&#39; center of gravity too high for a proper, narrow band, accurate, beam balancing movement. 
     All of the referenced patents are limited in that whenever their single beam is used for solutions to multiplication or division problems, at least one stacked weight, or substitution weight is required, along with its disadvantages. 
     Only this application&#39;s multiple interconnecting beam approach overcomes the stated disadvantages or limitations with stacking or substituting weights: (a) increased required time; (b), higher weight center of gravity, and (c), likelihood of error. 
     Further, here, by using a group of interconnecting balancing beams, with each beam having at least a 1 unit and 10 unit weight, along with decimal marked positions for settings; the decimal system can be taught and used, 
     A large part of a student&#39;s attraction to, and confidence in, a teaching aid computer stems from an understanding of how it works, Today&#39;s popular teaching aid computers have complex electronic circuitry that is seldom understood by the user. Therefore, the student most often must proceed on blind faith the computer&#39;s capability. If an error occurs, the student can only check his programming, not the way that program is processed through the computer&#39;s circuitry. This application&#39;s teaching aid computer has exposed workability that is easily understood and therefore bolsters confidence and attraction in its use. 
     Later, FIGS. 3 through 7, are directed to an understanding of the computer&#39;s workability as well as its use in problem solution. 
     SUMMARY OF THE INVENTION 
     Objects of this invention are to provide: 
     (a) a mechanical, analog type, teaching aid computer--free of any required electrical, or mechanical spring, power source; 
     (b) a computer that requires a sense of manipulative touch, as well as a sense of balance--motor nerve inputs to student&#39;s brain that bolster understanding and memory; 
     (c) a computer that allows solutions to multiple degree, as well as single degree, equations; 
     (d) a computer that allows a student to use one or more sliding weights, and/or placable torques or forces provided by one or more adjacent moving axis beams, to achieve solution to equations involving addition, subtraction, division, and single and multiple degree unknowns; 
     (e) a computer with more than two connectable beams and with multiple weight channels, that may be needed accomplishing solutions to addition and subtraction problems involving many numbers--by locking connected moveable beam mounting strips in positions so that all beam weight channels can be used; 
     (f) a computer that allows a student to observe a range of non-answers, as well as one or more answers, to single and multiple degree equations; 
     (g) a computer whose workability can be easily seen and understood. The basic concept of balancing torques is gained by a child early on. A heavy child automatically sits closer to the fulcrum of a teeter-toter to offset the torque of a lighter child that sits further back from the fulcrum. Thus, a student may be attracted to the use of this teaching aid because of his or her confidence in understanding the computer&#39;s basic functions. 
     (h) a computer composed of a group of interconnecting balancing beams that only requires understandable settings and manipulative balancing. Such activity should challenge and intrigue young students with their sense of accomplishment having an ability to solve otherwise too-complex problems--the type of multiple degree problems that are involved with equations similar to ones that one of their relatives or acquaintances may utilize at work. Thus, the young student may be viewed by elders with astonishment and accompaning praise. What better motivation would there be to encourage our country&#39;s hoped for, forthcoming generation of scientists and engineers. 
     In achieving the above objects, this invention utilizes a cross combing effect of two sets of parallel, equally spaced, interconnected, alternately stationary and moveable axes mounted, balancing beams--to seek real number solutions to single or multiple degree equations that have been arranged to have both sides equal to zero. 
     Each beam has a torque transfer shaft on its left side, at its numbered 1 position. Marks are made and numbered at an equal spacing of positions from 1 to 10 starting from its centered pivoting axis toward each outside. Each of those marked spaces are further subdivided and marked into 10 equal spaces, with marked sub numbers 1 to 9 arranged in succession from each inboard to outboard position. Alternate beams have stationary and movable axes. Each beam has a designated positive and negative half of its length on either side of its 0 marked pivoting axes. The stationary axes beams have common positive sides, which are at opposite ends to the common positive sides of the movable axes beams. Each beam has a depth centered, linear groove along its length, on its right side. Each beam&#39;s torque transfer shaft&#39;s end mounted rotatable block shaped bearing mates and slides snugly within its adjacent beam&#39;s groove. 
     The movable axis beams&#39; supports are fastened, and move in unison, so that the distance marked space between each torque transfer shaft and the pivoting axis of the groove engaged beam, is always the same from beam to beam. Therefore, that common distance is utilized as the unknown in a multiple degree equation which has each side set to equal 0. Setting an equation whose both sides equal zero, allows a state of balance of beams when an equation solution is indicated. 
     Each beam includes a 1 unit and a 10 unit weight that can be placed and held in any position along the beam&#39;s length. Beams from left to right can be designated a,b,c,d and e for simple identification. The weights on beam &#34;a&#34; are placed to produce torque that represents the + or - constant in an equation. Weights on beam &#34;b&#34; are placed to produce torque that represents the + or - coefficient of the first degree of the unknown. Beam &#34;c&#34; weights are placed to produce torque that represents the coefficient of the 2nd degree of the unknown. Beam &#34;d&#34; weights are placed to produce torque that represents the coefficient of the 3rd degree of the unknown. Beam &#34;e&#34; weights are placed to produce torque that represents the coefficient of the 4th degree of the unknown. Each additional added beam allows solution of an equation with one higher degree of the unknown. 
     After the constants and coefficients of the equation are set as above, the movable axes beams, being fastened together, are moved in unison from an extreme left to right hand end of the computer while noting any indicated position at which the beams tip back and forth. The positioning of the movable beams is then adjusted until all beams balance and the torque transfer shafts point to the answer. 
    
    
     BRIEF DESCRIPTION OF DRAWINGS 
     FIG. 1 shows a proposed complete teaching aid computer assembly with four beams as needed to solve a third degree equation. 
     FIG. 2 illustrates a closeup of either a moving or stationary mounting strap and axis, beam (17 or 15). 
     FIGS. 3, 4, 5, 6, and 7 schematically use an open circle to represent the location of a ten unit weight and a smaller solid circle to represent the location of a one unit weight, for each beam, when illustrating a problem solution method. 
     FIG. 3 is a schematic drawing representation of a method of fixing the moving strap base and axis beams with a locking pin (33). 
     FIG. 4 is a schematic drawing that illustrates the placement of weights that can be used when solving a typical fourth degree equation for determining the radius of a torsional rod. 
     FIG. 5 is a schematic drawing that illustrates the placement of weights used in solving the fourth degree equation indicated. 
     FIG. 6 is a schematic drawing of the placement of weights used in solving a third degree equation when seeking the diameter of a sphere when the volume is known. 
     FIG. 7 is a schematic drawing of the required placement of weights for solution of a second degree equation for finding the electrical current when a light bulb&#39;s wattage and resistance are known. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The following is a list of items as numbered in FIGS. 1 through 7, with an accompanying brief description. 
     
         ______________________________________ItemNo.  Description______________________________________ 1   Rectangular base of the teaching aid computer 2   Left hand stationary mounting strip 3   Right hand stationary mounting strip 4   Left hand, linearly moveable, mounting strip 5   Right hand, linearly moveable, mounting stripThe listed items 2, 3, 4, and 5, mounting strips are all stiff,rectangular, parallel, and have the same, width, and depth. Thetwo stationary strips 2, and 3 are fixed to base 1, have thesame length, and have their ends in alignment. The two movingstrips 4 and 5 have the same length, and have their endsmaintained in alignment. The stationary strips prevent anylateral movement of the moveable strips, and guide theirlongitudinal movement. 6   A crossbar fixed to the same, right, ends of both strips4 and 5. A similar bar can be fixed to their left ends. 7   A knob fixed to crossbar 6 which when moved, causes bothmoveable strips 4 and 5 to always move linearly, in unison. 8   A typical yoke shaped beam support, shown with a typicalmounting on strip 5, and having two parallel uprights. 9   A typical cutout allowing protrusion of the right handupright of a typical yoke support 8.10   A typical left hand axle shaft.11   A typical right hand axle shaft.12   A typical top inserted pin in an upright, used toengage, and prevent lateral movement of, shaft 10. Similarpins prevent lateral movement of later described shafts.Typical beam insertable antifriction bearings, unshown,surrounding the ends of shafts 10 and 11 that are not pined13   Right hand, narrow width, stationary strip of the samelength and depth as stationary strips 2 and 3. These threestationary strips are utilized as guides for the parallelmovement of the movable strips.The above items 8 through 13 all are typical components usedto pivotally mount the following balancing beams on theirpictured strip bases.14   Left hand (stationary axes) balancing beam that hastypical inserted antifriction bearings, unshown, that matewith shafts 10 and 11 which are pinned to a yoke, like 8,which, in turn, is mounted on strip 2. Thereby, beam 14 ispivotally mounted on stationary strip 2.15   Right hand, stationary axes, balancing beam pivotallymounted on stationary strip 2.16   Left hand, movable axes, balancing beam pivotallymounted on movable strip 4.17   Right hand, movable axes, balancing beam pivotallymounted on movable strip 5.18   Evenly marked and numbered spaces, from the 0, centeredposition adjacent to the pivoting axes, to a numbered 10position near each outside end of each beam. Unshown onFIGS. 1 and 2, because of anticipated crowding, are the 10equal spaces with sub marks, between each of the mainnumbered positions. Typical marks, and sub marks, areshown later, in FIG. 4. Each beam is to be similarlymarked with main, and sub marks.19   A capital letter P, and (unshown) purple dye, indicationof the assigned positive ends of stationary axes beam2 and 3.20   A capital letter P, and (unshown) purple dye indicationof the assigned positive ends of movable axes beam 4 and 5.(opposite the ends of the stationary axes beams)21   A capital letter N, and (unshown) nut brown indication ofthe assigned negative ends of stationary axes beams 2 and 3.22   A capital letter N, and (unshown) nut brown indication ofthe assigned negative ends of movable axes beams 4 and 5.23   A separate torque transfer shaft fixed perpendicularlyto the middle of the left side, at the numbered 1, positiveend, position of stationary axes beams 14 and 15.24   A separate torque transfer shaft fixed perpendicularlyto the middle of the left side, at the numbered 1, positiveend, position of movable axes beams 16 and 17.25   A separate groove along the length, at the middle of theright side, of the stationary axes beams 14 and 15.26   A separate groove along the length, at the middle of theright side of the movable axes beams 16 and 17.27   A typical 1 unit weight with its centered positionplacement mark.28   A typical 10 unit weight with its centered positionplacement mark.29   A typical shaft along the length of each beam on which atypical 1 unit weight 27 can slide with sufficient snugness tohold the weight in its placed position.30   A typical shaft along the length of each beam on which atypical 10 unit weight 28 can slide with sufficient snugnessto hold the weight in its placed position.31   A block, fixed to strip 13, that confines strip 5 tohorizontal, not vertical, movement. A similar blockconfines the left side of strip 4 to horizontal movementonly.32   typical balance weights applied during each beam&#39;sassembly, that cause each beam to remain in any set angularposition, when the supporting axes are in a preassemblyhorizontal position, and the 1 unit and 10 unit weights arekept in their centered, 0 position.32A  Shown with dashed lines as hidden is a short counter-balancing shaft weight fixed to the far side of typicalfixed axis beam 15 or typical moveable axis beam 17; shownat the same depth, and symetrically placed opposite pin 23with respect to colinear axes 10 and 11 as indicated inFIG. 2; shaft weight 32A counterbalances pin 23 and itsbearing 38. Pin 23 and shaft 32A are as pictured formoveable axis beams 16 and 17, but are positioned inreverse for stationary axis beams 14 and 15. Pin 23 (aswith pin 24) is always located on the &#34;P&#34; (positive) endof a beam.33   A pin that can be fully inserted horizontally throughend strip 31 into the normally movable strip 5 when torquetransfer shafts 23 of movable axes beams 16 and 17 pointto the positive end numbered 1 positions. In this lockedposition the 1 unit and 10 unit weights can representnumbers in problems involving addition and subtractiononly.34   Block, similar to 31, fixed on strip 2 and usedto hold down the left out side of moveble strip 4. Thatblock 34 is shown in its position in the later FIG.______________________________________3 
    
     The ends of cross bar 6 are sufficiently recessed to avoid interference with hold down blocks 31 and 34. 
     SOME CONSTRUCTION SPECIFICATIONS 
     (Item Descriptions Resume Later) 
     (a) the movable axes beams all have the same axes centerline, 
     (b) the stationary axes beams all have the same axes centerline, 
     (c) the torque transfer shafts of the movable axes beams would have the same centerline when all beams are balanced, 
     (d) the torque transfer shafts of the stationary axes beams would have the same centerline when all beams are balanced. More beams can be added to allow solution of equations involving 4th degree and higher unknowns. Five beams are shown in FIGS. 3 through 5, and in model that was used to prove the workability of this computer, 
     (e) the center lines of all beam axes, and torque transfer shafts are always parallel, 
     (When all of the beams are positioned parallel to base 1, a common plane parallel to base 1 would include: all beams&#39; axes, all beams&#39; torque transfer shafts, all beams&#39; weights&#39; centers of gravity, and the centerlines of the right side grooves on all beams.) 
     (f) the top surfaces of all beams would be in a second, separate common plane, parallel to the base, 
     (g) the bottom surfaces of all beams would be in a third, separate, common plane parallel to the base, 
     (h) all beams have the same length, width, and thickness, 
     (i) stationary axes beams have the same ends designated as positive; the same ends designated as negative; and the same torque transfer shaft locations, 
     (j) movable axes beams have the same ends designated as positives; the same ends designated as negative; and the same torque transfer shaft locations, 
     (k) the moving axes beams have designated positive and negative half beam positions that are opposite to the stationary axes beam positions, 
     (l) adjacent beams alternate between being the stationary, and the moving axes type. 
     As the moving axes beams&#39; torque transfer shaft pointings are moved from positive numbered then to negative numbered sides of the adjacent stationary axes beams, real negative number solutions to equation unknowns can be revealed when the beams reach a balanced condition. 
     
         ______________________________________Further Item Descriptions______________________________________35   Index mark on stationary strip 13.36   Index mark on movable strip 5 that lines up with mark 35when the movable axes beams&#39; torque transfer shafts pointto positive 1 numbered locations on the stationary axesbeams, at which position locking pin 33 can be inserted.The following items apply to FIG. 2.34   A block, previously mentioned, similar to block 31, thatrestrains the left side of the combined, movable, item 4 and5, to linear, horizontal, not vertical, movement.37   Bore in movable strip 5 and stationary strip 13 intowhich pin 33 has a snug fit when marks 35 and 36 arealigned..38   Typical block shaped bearing that pivots on the extendedend of all torque transfer shafts and slides snugly in themating grooves like 25 or 26.The following items apply to FIG. 3.The schematic representation, as described below for items 10,11, 23, 24, 27, and 28 apply to all FIGS. 3 through 7.10   axles are represented schematically by a set of two,and  crossed lines, with a small 0 to the left of center, as each11   beam&#39;s pivoting axes.23   torque transfer shafts are shown as a solid lineand  projecting from the left, positive numbered 1 position from24   both the stationary and moving axes type beams, to engagein the groove of the adjacent beam to the left.27   the 1 unit weight is represented schematically by a smallcircle with a darkened center.28   the 10 unit weight is represented by a larger circlethan 27, with an open center.39   The sum of the listed numbers as added.40   This balance beam teaching aid requires that allproblems involving addition, subtraction, multiplication,division, and single and multiple degree equations, utilize anequation such as this equaling 0. When the beams (includingthe left hand stationary beam on which all torques aresummed) are in balance, there is zero unbalanced CW orCCW torque.41   Manipulative action needed to achieve equation 40solution.42   This is the teaching aid answer to the addition of theseven positive and negative numbers listed, with the 1 unitand 10 unit weights associated with each beam positionedin the correct places to represent the numbers to be added,and with all beams shown in their balanced positions. Thisanswer is also confirmed as checked by the numberssummed at item 39.The following item numbers occur in FIG. 4.Note that the schematic representations, as decribed above, forthe 1 unit and 10 unit weights, are shown to the right of eachscale that represents one of the five separate beams, aslabeled.43   This is an equation that indicates the twisting torqueon a rod having shear modulus E, length L, and radius R.The automotive industry, for example, has used torque rodsto spring up trunk and hood lids, as well as give springsupport to front axles of cars.44   If the required spring torque is known, and the radiusof the rod is desired, this equation applies.45   To find the radius using this computer, this equationapplies when beams are in balance.46   The radius answer that becomes evident with theconstants as inserted in 44 above and with the beams, aspictured, in their balanced state.In further detail: (a) a 1 unit weight is placed in the +1position on the added, stationary beam 15A - to representthe coefficient 1 for X.sup.4. Since there are no X.sup.3, X.sup.2,or X quantities in the equation 45, no coefficients need tobe set on beams 17, 15, or 16. The constant -10.49 is seton beam 14 with the 10 unit weight set at -1, and the 1 unitweight set at -.49 . The movable axes beams 16 and 17 arethen moved in unison by knob 7 through tie bar 6 until thebeams rock back and forth on each side of the positionwhen all torque transfer shaft point to +1.8. Then finalbalance of all beams occurs at +1.8. The most detaileddescription of the workability of this computer is given forthe 4th degree equation associated with FIG. 5.The following items apply to FIG. 5.47   A typical 4th degree equation that requires, as does FIG. 4,the addition of stationary beam 15A for solution.48   The computer&#39;s answer to equation 47.The following shorthand will be employed for reviewing thesettings for FIGS. 5 through 7.B15A = beam 15A; W1 = weight 1 unit; N2 = negative side 2W10 = weight 10 units; P0, or N0 = 0, occurs at the pivotingaxes.So, the settings for FIG. 5, equation 47 are: B15A,W1N2,W10P0; B17,W1N5,W10P0; B15W1P0,W10P1; B16,W1N4, W10P0;B14,W1P0,W10P1.6______________________________________ 
    
     Following the above settings, movable axis beams 16 and 17 are moved in unison until the beams rock back and forth on either side of the -4 (or N4) positions indicated by all of the torque transfer shafts . Then beams are balanced at -4. Therefore a real number solution to equation 47 is X=-4. 
     A further view to understand the workability of the computer in solving for the unknown in equation 47, as illustrated in FIG. 5, follows. 
     There is a 4 to 1 lever arm mechanical advantage repeated four times, using beams 17,15,16, and 14, caused by the moving beams&#39; placement of the torque transfer shafts at -4. Therefore the -2 CCW, negative torque placed on beam 15A is multiplied by -4 four times. Thus producing -512 units of CCW, negatives torque on beam 14 around its 0 axis. 
     In a similar manner there is a 4 to 1 mechanical advantage repeated three times, using beams 15, 16, and 14, again caused by the moving beams&#39; placement of the torque transfer shafts at the -4 position. The -5 units of torque caused by the 1 unit weight placed at the -5 positions is multiplied by -4 three times, and thus imposes +320 units of CW, positive torque around beam 14&#39;s 0 axis. 
     There is a 4 to 1 mechanical advantage repeated two times, using beams 16 and 14, again caused by the transfer shafts&#39;  placement at -4. The +10 units of torque caused by the 10 unit weight placed at the +1 position is multiplied by -4 two times and thus imposes +64 units of CW, positive torque around beam 14&#39;s 0 axis. 
     There is an additional effective single 4 to 1 mechanical advantage caused by the beam 16&#39;s transfer shaft&#39;s placement at -4 on beam 14. The -4 units of torque caused by the 1 unit of weight set at the -4 position on beam 16 is thereby multiplied by -4, by the beam 16&#39;s shaft applying a force at the -4 position on beam 14. That action imposes +16 units of CW, positive torque around beam 14&#39;s 0 axis. 
     The +16 setting of the constant of the equation is caused by the placement of a 10 unit weight at the +1.6 position on beam 14. Thereby imposing +16 units of CW torque around beam 14&#39;s 0 axis. 
     The summation of the above torques around beam 14&#39;s 0 axis includes: -512 CCW, +320 CW, +160 CW, +16 CW, +16 CW or a combined -512 CCW and +512 CW torques or a resultant zero torque. Thus, confirming a beam balance condition when all of the torque transfer shafts point to the -4 solution to the equation. 
     
         ______________________________________The following items apply to FIG. 649   The equation for determining the volume of a spherehaving a radius R.50   The equation for determining a sphere&#39;s diameter whenits volume is known.51   Equation 50 rearranged for use with this computer.52   This computer&#39;s indicated answer.The above described shorthand is used for making settings forthe item 51 equation, as follows:B17, W1P1, W10P0; B15, W1P0,W10P0; B16, W1P0, W10P0B14, W1P0, W10N2.7Beams 16 and 17 are then slid in unison within their tracksuntil the beams rock back and forth on opposite sides of P3.Then all beams are balanced at P3 (or +3) a real numbersolution to equations 50, and 51.The following items apply to FIG. 7.53   An equation that indicates the relationship betweenwatts W, direct current I, and resistance R in an electricalcircuit. Such a relationship would be used for example, tocalculate the current drawn by an automobile&#39;s light bulb.54   In this specific equation, the square of the current can bedetermined if the bulbs watts and resistance are known.55   Equation 54 rearranged for use in this computer.56   This computer&#39;s answer in amps of current.______________________________________ 
    
     Since there are no equation unknowns higher than the 2nd degree, the weights W1 and W10 would be set to P0 if beams 17, and 15A were employed. So, beams 17, and 15A are not shown here for simplicity. Actually, this computer can always have mope power by utilizing one mope beam than the number representing the exponent of the highest degree used in a specific equation. 
     The following is a review of the required beam settings-- with reference to equation 55: 
     
         B15, W1P1, W10P0, B16W1P0, W10P0, B14. W1N9.6 
    
     Beam 16 is then slid in its track until all beams tip back and forth on opposite sides of a P3.1 (+3.1) position. Then the beams are balanced when the torque transfer shafts point to P 3.1, a real number answer to this useful 2nd degree equation. 
     A simple second degree equation, that also includes a first degree term is 2n 2  -5n-3=0. This equation is solved in the following manner: 
     (a) slide a one unit weight to the number 3 marked position on the negative side of beam a; 
     (b) slide a one unit weight to the number 5 marked position on the negative side of beam b; 
     (c) slide a one unit weight to the number 2 marked position on the positive side of beam c; 
     (d) at a measured place, slide the combined beams b, and d, from a position where the stationary and moving axes are furthest apart, on one side, to the position where the beams are furthest apart on the opposite side; 
     (e) each beam will assume a slope in one direction, then in the opposite direction, on either side of an indicated answer +3, where each beam remains in a balanced condition. Thus, a student can learn a bracketing-in procedure to obtain the answer. 
     (f) the student may then pursue a normal procedure of substituting, a +3 in place of the unknown, to confirm a +3 answer. 
     FIG. 1 is a three dimensional drawing of a teaching aid computer having four beams. Thus, its represented capability is limited to a solution of 3rd degree equations as illustrated later, schematically, in FIG. 6. 
     Mounting strips 2, and 3, fixed to base 1, act as parallel side edge guides for movable beam mounting strips 4 and 5. The supporting yokes 8, and pivoting axles, 10, and 11 for all beams, as covered in the item descriptions above, are shown mounted perpendicularly to the length of each of their mounting strips, allowing all beams to rotate slightly, in parallel vertical planes. 
     The item 18 weight position marks and numbers are shown, with an increased number of divisions, where there is more room, for example in FIG. 6. 
     When each beam is separately assembled, and with the 1 unit weight 27, and the 10 unit weight 28 centered at the 0 position and with torque transfer shaft 23, (or 24) fixed in the numbered 1 assigned positive position (19, or 20) the balancing weights 32 are sized and positioned to permit each beam to remain at any one placed angle. 
     Bar 6 fixed to the ends of both longitudinally moving beam support strips 4 and 5 essentially creates one common moving platform, keeping the moving axes beams 16 and 17 in the same longitudinal position. 
     In a manner better than FIG. 1, FIGS. 3 through 7, with their schematic top views, illustrate mechanically, how an equation&#39;s unknown can be applied an exponent number of times as a repeated torque and summarized along with the effects of the equation&#39;s constants and coefficients as CW and CCW opposing torques applied to beam 14. 
     FIG. 2 shows a blown up portion of a typical moving axes beam 16, or 17--so identified because the torque transfer shaft 23 is at the opposite side to the negative mark 22. 
     Typical small bearing block 38, rotatably mounted on the outside end of torque transfer shaft 23, is sized to slide snugly within the longitudinal groove of the adjacent left hand stationary beam, such as 14, or 15. Block 38 provides a larger, hence more wear immune, bearing surface against the top and bottom surfaces of its mating groove. 
     FIG. 3 schematically illustrates how beams 4 and 5 can be held in their pictured position, with pin 33, when marks 35 and 36 line up. That position occurs when all torque transfer shafts 23, and 24, point to their adjacent beam&#39;s positive, numbered 1, position, In that pinned position a group of positive and negative numbers can be summed, as shown, per item 39. An equation, per item 40 is again utilized so that the beams can be brought to balance. Note that if a CCW torque of -25, 5 units remained on beam 17, (utilized as the torque summing beam in this case)--that degree of unbalance could only be measured by noting that a positive 25.5 units of CW torque (in this case obtained by sliding the 10 unit weight to a positive 2.55 position) caused all beams to balance. 
     Most drawings refer to a four beam arrangment, as in FIG. 1. Four beams compose the preferred embodiment. An indicated alternate #1 is the sometimes required five beam arrangment when higher than 3rd degree equation solutions are desired--as illustrated by the five beam schematic representations per FIGS. 4, and 5. This application&#39;s computer as shown in the FIGS. 1,3, and 6 drawings, item descriptions, and summary covers the preferred embodiment. 
     Another alternate #2, would have each beam incorporate 3 parallel weight channels, one each for a 1, 10, and 100 unit weight. Three weight channels per beam as shown in past drawings, describe a more capable, but more costly alternate arrangement. Those expansive alternates would make the computer more powerful, but also more complex, than is felt necessary. 
     A third altenate arrangement could have the linear connectible device take the form of a linear, horizontal bar, of constant, narrow width and thickness, fixed to the middle of the right, out, side of each beam. From an end view, a channel shaped short lenth of an extrusion, would grip the mating bar, and have a fixed, outward protruding shaft that would be allowed to rotate within a positive number 1 positioned hole in a right hand adjacent beam. This arrangement would seem to have no advantage over the described arrangement herein that includes a torque transfer shaft with a rotatible bearing block end (item 38) that mates and slides within the open groove on the side of left hand adjacent beam. Both arrangements could have a satisfactory amount of flat bearing surfaces to limit wear. 
     A fourth alternate reverses the arrangement of the torque transfer means by having each beam&#39;s torque transfer shaft located on its right side instead of its left side. And, having its linear groove located on its left side instead of its right side. 
     The torques generated by all beams can also be considered as summed on the right hand beam d, since torques are transferred in both directions from beam to beam. The highest, third degree, coefficient of a third degree equation, can, more logically, be set on the left hand beam, in an arrangement similar to the arrangement in an equation. Coefficients of lower degrees of the unknown, in descending order, can then be set on succesive beams in the right hand direction. 
     SOME COMMON MULTIPLE DEGREE EQUATIONS 
     Area of circle of radius r: 
     
         A=πr.sup.2, or A-πr.sup.2 =0 
    
     Surface area of sphere of radius r: 
     
         A=4πr.sup.2, or A-4πr.sup.2 =0 
    
     Volume of sphere of radius r: 
     
         V=4/3πr.sup.3 
    
     Height of a fired projectal after time t, against gravitational pull g: 
     
         H=1/2gt.sup.2 
    
     Centripetal acceleration a, of an object tied at radius r, having a circular velocity v: 
     
         a=v.sup.2 /r 
    
     Kinetic energy (KE) of a body having mass m, and velocity v: 
     
         KE=m v.sup.2 /2 
    
     Twisting torque (T) on a rod having shear modulus E, length l, and radius R: 
     
         T=πθER.sup.4 /2l 
    
     The intensity of radiation on the internal surface of a sphere from a centered point source: 
     
         I=E/4πr.sup.2 
    
     The force action between two poles having strengths of m 1 , and m 2 , in a medium with permeability of u is: 
     
         F=m.sub.1 m.sub.2 /ur.sup.2 
    
     The power in watts in a direct current electrical circuit, with a current level of I, and a circuit resistance of R is: 
     
         W=I.sup.2 R 
    
     The inductance L, of a coil having length l, permeability u, number of turns N, and cross sectional area A, is: 
     
         L=4πN.sup.2 A.u10.sup.-9 /l 
    
     The energy W, in Joules, of a magnetic field having an inductance of L henries, and a current of I amperes is: 
     
         W=LI.sup.2 /2 
    
     The energy W, stored on a capacitor with a capacitance of C farads, and a potential difference of V volts is: 
     
         W=CV.sup.2 /2 
    
     The intensity I, of sound, having a wave propagation of V cm, per sec. frequency f, particle displacement r, and medium density d, is: 
     
         I=2π.sup.2 Vf.sup.2 r.sup.2 d