504.26 717.55 509.61 717.55 2 L
N
503.71 724.32 504.15 723.56 504.26 722.47 503.71 720.5 4 L
N
504.7 723.67 505.9 723.78 2 L
N
504.59 721.81 505.79 720.61 2 L
N
508.3 724.43 508.74 723.67 508.85 722.58 508.3 720.61 4 L
N
509.28 723.78 510.48 723.89 2 L
N
509.17 721.92 510.37 720.72 2 L
N
504.91 711.66 505.79 710.78 506.01 709.91 505.68 709.15 504.91 708.38 5 L
N
509.28 711.66 509.5 710.02 509.06 708.71 3 L
N
54 87.5 540 686.02 R
7 X
V
54 672.98 540 686.02 C
189 680.04 540 680.04 2 L
0.25 H
2 Z
0 X
0 K
N
54 680.04 180 680.04 2 L
0 Z
N
0 0 612 792 C
1 9 Q
0 X
0 K
(FIGURE  2) 54 666.98 T
0 F
(Numeric Representation of Number 87,654,321 on the Abacus) 189 666.98 T
54 503.97 540 512.99 C
189 511.48 540 511.48 2 L
0.25 H
2 Z
0 X
0 K
N
54 511.48 180 511.48 2 L
0 Z
N
0 0 612 792 C
1 12 Q
0 X
0 K
(2.0) 163.33 516.99 T
(Addition) 189 516.99 T
2 10 Q
-0.22 (Addition on the abacus involves registering the numbers on the beads in the straight-for-) 189 483.3 P
(ward left-to-right sequence they are written down in. As long as the digits are placed) 189 471.3 T
(correctly) 189 459.3 T
(, and the carry\325) 223.87 459.3 T
(s noted properly) 283.82 459.3 T
(, the answer to the operation immediately pre-) 348.12 459.3 T
(sents itself right on the abacus. There are 4 approaches to performing additions \050or sub-) 189 447.3 T
-0.1 (tractions\051, each applied to particular situations. Each of these techniques is explained in) 189 435.3 P
(tabular form in the sections that follow) 189 423.3 T
2 8 Q
(1) 344.45 427.3 T
2 10 Q
(.) 348.45 423.3 T
1 F
(2.1) 166.11 393.3 T
(Simple Addition) 189 393.3 T
2 F
(When performing the addition 6+2, one would move 1 bead from the upper deck down) 189 369.3 T
(\050value = 5\051 and one bead from the lower deck up \050value = 1\051; this represents 6. Moving) 189 357.3 T
(2 beads from the lower deck \050in the same column\051 up \050value = 1 * 2 beads = 2\051 would) 189 345.3 T
-0.17 (complete the operation. The answer is then obtained by reading resultant bead positions.) 189 333.3 P
54 302.94 540 315.97 C
189 309.99 540 309.99 2 L
0.25 H
2 Z
0 X
0 K
N
54 309.99 180 309.99 2 L
0 Z
N
0 0 612 792 C
54 279.97 540 293.94 C
189 285.94 540 285.94 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
1 9 Q
0 X
0 K
(T) 54 296.94 T
(ABLE  1) 58.83 296.94 T
0 F
(Simple Addition) 189 296.94 T
0 8 Q
(Original Number) 202.8 274.64 T
(Formula) 312.68 274.64 T
(Bead-Operations Involved) 396 274.64 T
(lower-deck) 411.35 264.64 T
( upper-deck) 488.78 264.64 T
(0,1,2,3,4,5,6,7,8) 198 254.64 T
(plus 1) 313.33 254.64 T
(+1) 427.44 254.64 T
(0,1,2,5,6 or 7) 198 244.64 T
(plus 2) 313.33 244.64 T
(+2) 427.44 244.64 T
(0,1,5 or 6) 198 234.64 T
(plus 3) 313.33 234.64 T
(+3) 427.44 234.64 T
(0 or 3) 198 224.64 T
(plus 4) 313.33 224.64 T
(+4) 427.44 224.64 T
(0,1,2,3 or 4) 198 214.64 T
(plus 5) 313.33 214.64 T
(+1) 508.44 214.64 T
(0,1,2, or 3) 198 204.64 T
(plus 6) 313.33 204.64 T
(+1) 427.44 204.64 T
(+1) 508.44 204.64 T
(0,1 or 2) 198 194.64 T
(plus 7) 313.33 194.64 T
(+2) 427.44 194.64 T
(+1) 508.44 194.64 T
(0 or 1) 198 184.64 T
(plus 8) 313.33 184.64 T
(+3) 427.44 184.64 T
(+1) 508.44 184.64 T
(0) 198 174.64 T
(plus 9) 313.33 174.64 T
(+4) 427.44 174.64 T
(+1) 508.44 174.64 T
54 153.5 540 169.56 C
189 161.49 333 161.49 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
2 8 Q
0 X
0 K
-0.28 (1.  The tables are used as follows: given) 189 148.17 P
3 F
-0.28 (Original Number) 316.57 148.17 P
2 F
-0.28 (, to add/subtract) 371.6 148.17 P
3 F
-0.28 (Formula) 424.04 148.17 P
2 F
-0.28 (, perform) 452.03 148.17 P
3 F
-0.28 (Bead-Operations) 483.21 148.17 P
(Involved) 189 139.17 T
2 F
(.) 216.53 139.17 T
-0.03 ( For example, to add 2 and 7, move down) 189 130.17 P
3 F
-0.03 (Original Number) 323.76 130.17 P
2 F
-0.03 ( column to a row containing 2 and down) 379.03 130.17 P
3 F
-0.03 (Formula) 510.04 130.17 P
2 F
-0.13 (to a row containing 7 \050the 7th row\051 then move across to read the) 189 121.17 P
3 F
-0.13 (Operation Involved) 394.13 121.17 P
2 F
-0.13 (: move 2 bead counters in) 456.4 121.17 P
(the lower-deck and 1 bead counter in the upper-deck to represent the sum; the answer then presents itself on) 189 112.17 T
(the abacus. The \322+\323 symbol, in the) 189 103.17 T
3 F
(Operations Involved) 302.1 103.17 T
2 F
( column represents moving the bead in question,) 367.61 103.17 T
3 F
(towards) 189 94.17 T
2 F
( the rail; the \322-\323 symbol represents moving the bead in question) 214.77 94.17 T
3 F
(away) 419.52 94.17 T
2 F
( from the rail.) 436.4 94.17 T
54 87.5 540 686.02 C
273 547.99 540 663.98 C
273 547.99 540 663.98 R
7 X
0 K
V
4 14 Q
0 X
(8 7 6 5 4 3 2 1) 297 552.98 T
278.88 565.88 424.88 654.88 R
3 X
V
0.5 H
2 Z
0 X
N
281.52 627.49 422.23 650.87 R
7 X
V
0 X
N
282.05 571.24 421.7 621.89 R
7 X
V
0 X
N
90 450 4.5 3.42 288.51 645.58 G
90 450 4.5 3.42 288.51 645.58 A
90 450 4.5 3.42 288.51 637.49 G
90 450 4.5 3.42 288.51 637.49 A
90 450 4.5 3.42 288.51 600.24 G
90 450 4.5 3.42 288.51 600.24 A
90 450 4.5 3.42 288.51 608.95 G
90 450 4.5 3.42 288.51 608.95 A
90 450 4.5 3.42 288.51 591.7 G
90 450 4.5 3.42 288.51 591.7 A
90 450 4.5 3.42 288.51 583.44 G
90 450 4.5 3.42 288.51 583.44 A
90 450 4.5 3.42 288.89 575.34 G
90 450 4.5 3.42 288.89 575.34 A
288.62 651.32 288.62 571.03 2 L
V
N
90 450 4.5 3.42 384.56 645.5 G
90 450 4.5 3.42 384.56 645.5 A
90 450 4.5 3.42 384.56 637.41 G
90 450 4.5 3.42 384.56 637.41 A
90 450 4.5 3.42 384.56 600.15 G
90 450 4.5 3.42 384.56 600.15 A
90 450 4.5 3.42 384.56 608.86 G
90 450 4.5 3.42 384.56 608.86 A
90 450 4.5 3.42 384.56 591.62 G
90 450 4.5 3.42 384.56 591.62 A
90 450 4.5 3.42 384.56 583.35 G
90 450 4.5 3.42 384.56 583.35 A
90 450 4.5 3.42 384.94 575.26 G
90 450 4.5 3.42 384.94 575.26 A
384.67 651.23 384.67 570.94 2 L
V
N
90 450 4.5 3.42 394.84 645.41 G
90 450 4.5 3.42 394.84 645.41 A
90 450 4.5 3.42 394.84 637.32 G
90 450 4.5 3.42 394.84 637.32 A
90 450 4.5 3.42 394.84 600.06 G
90 450 4.5 3.42 394.84 600.06 A
90 450 4.5 3.42 394.84 608.78 G
90 450 4.5 3.42 394.84 608.78 A
90 450 4.5 3.42 394.84 591.53 G
90 450 4.5 3.42 394.84 591.53 A
90 450 4.5 3.42 394.84 583.26 G
90 450 4.5 3.42 394.84 583.26 A
90 450 4.5 3.42 395.21 575.17 G
90 450 4.5 3.42 395.21 575.17 A
394.95 651.14 394.95 570.85 2 L
V
N
90 450 4.5 3.42 404.96 645.5 G
90 450 4.5 3.42 404.96 645.5 A
90 450 4.5 3.42 404.96 637.41 G
90 450 4.5 3.42 404.96 637.41 A
90 450 4.5 3.42 404.96 600.15 G
90 450 4.5 3.42 404.96 600.15 A
90 450 4.5 3.42 404.96 608.86 G
90 450 4.5 3.42 404.96 608.86 A
90 450 4.5 3.42 404.96 591.62 G
90 450 4.5 3.42 404.96 591.62 A
90 450 4.5 3.42 404.96 583.35 G
90 450 4.5 3.42 404.96 583.35 A
90 450 4.5 3.42 405.34 575.26 G
90 450 4.5 3.42 405.34 575.26 A
405.08 651.23 405.08 570.94 2 L
V
N
90 450 4.5 3.42 415.62 645.41 G
90 450 4.5 3.42 415.62 645.41 A
90 450 4.5 3.42 415.62 637.32 G
90 450 4.5 3.42 415.62 637.32 A
90 450 4.5 3.42 415.62 600.06 G
90 450 4.5 3.42 415.62 600.06 A
90 450 4.5 3.42 415.62 608.78 G
90 450 4.5 3.42 415.62 608.78 A
90 450 4.5 3.42 415.62 591.53 G
90 450 4.5 3.42 415.62 591.53 A
90 450 4.5 3.42 415.62 583.26 G
90 450 4.5 3.42 415.62 583.26 A
90 450 4.5 3.42 415.99 575.17 G
90 450 4.5 3.42 415.99 575.17 A
415.73 651.14 415.73 570.85 2 L
V
N
90 450 4.5 3.42 373.37 645.32 G
90 450 4.5 3.42 373.37 645.32 A
90 450 4.5 3.42 373.37 637.23 G
90 450 4.5 3.42 373.37 637.23 A
373.49 651.05 373.49 570.77 2 L
V
N
299.81 651.5 299.81 571.21 2 L
V
N
310.08 649.5 310.08 571.12 2 L
V
N
320.21 651.5 320.21 571.21 2 L
V
N
90 450 4.5 3.42 330.75 645.67 G
90 450 4.5 3.42 330.75 645.67 A
90 450 4.5 3.42 330.75 631.58 G
90 450 4.5 3.42 330.75 631.58 A
90 450 4.5 3.42 330.75 600.33 G
90 450 4.5 3.42 330.75 600.33 A
90 450 4.5 3.42 330.75 609.04 G
90 450 4.5 3.42 330.75 609.04 A
90 450 4.5 3.42 330.75 591.79 G
90 450 4.5 3.42 330.75 591.79 A
90 450 4.5 3.42 330.75 583.53 G
90 450 4.5 3.42 330.75 583.53 A
90 450 4.5 3.42 331.13 575.43 G
90 450 4.5 3.42 331.13 575.43 A
330.87 651.41 330.87 571.12 2 L
V
N
287.6 622.2 417.21 627.26 R
3 X
V
0 X
90 450 4.5 3.42 373.37 600.47 G
90 450 4.5 3.42 373.37 600.47 A
90 450 4.5 3.42 373.37 617.69 G
90 450 4.5 3.42 373.37 617.69 A
90 450 4.5 3.42 373.37 591.94 G
90 450 4.5 3.42 373.37 591.94 A
90 450 4.5 3.42 373.37 583.67 G
90 450 4.5 3.42 373.37 583.67 A
90 450 4.5 3.42 373.75 575.58 G
90 450 4.5 3.42 373.75 575.58 A
90 450 4.5 3.42 363.1 575.17 G
90 450 4.5 3.42 363.1 575.17 A
90 450 4.5 3.42 342.32 645.41 G
90 450 4.5 3.42 342.32 645.41 A
90 450 4.5 3.42 342.32 637.32 G
90 450 4.5 3.42 342.32 637.32 A
90 450 4.5 3.42 341.82 609.06 G
90 450 4.5 3.42 341.82 609.06 A
90 450 4.5 3.42 341.82 617.78 G
90 450 4.5 3.42 341.82 617.78 A
90 450 4.5 3.42 341.82 600.53 G
90 450 4.5 3.42 341.82 600.53 A
90 450 4.5 3.42 341.82 592.26 G
90 450 4.5 3.42 341.82 592.26 A
90 450 4.5 3.42 342.69 575.17 G
90 450 4.5 3.42 342.69 575.17 A
342.43 651.14 342.43 570.85 2 L
V
N
90 450 4.5 3.42 352.59 645.32 G
90 450 4.5 3.42 352.59 645.32 A
90 450 4.5 3.42 352.59 637.23 G
90 450 4.5 3.42 352.59 637.23 A
90 450 4.5 3.42 352.59 608.97 G
90 450 4.5 3.42 352.59 608.97 A
90 450 4.5 3.42 352.59 617.69 G
90 450 4.5 3.42 352.59 617.69 A
90 450 4.5 3.42 352.59 600.44 G
90 450 4.5 3.42 352.59 600.44 A
90 450 4.5 3.42 352.59 583.17 G
90 450 4.5 3.42 352.59 583.17 A
90 450 4.5 3.42 352.97 575.08 G
90 450 4.5 3.42 352.97 575.08 A
352.71 651.05 352.71 570.77 2 L
V
N
90 450 4.5 3.42 362.72 645.41 G
90 450 4.5 3.42 362.72 645.41 A
90 450 4.5 3.42 362.72 637.32 G
90 450 4.5 3.42 362.72 637.32 A
90 450 4.5 3.42 363.22 609.06 G
90 450 4.5 3.42 363.22 609.06 A
90 450 4.5 3.42 363.22 617.78 G
90 450 4.5 3.42 363.22 617.78 A
90 450 4.5 3.42 362.72 591.53 G
90 450 4.5 3.42 362.72 591.53 A
90 450 4.5 3.42 362.72 583.26 G
90 450 4.5 3.42 362.72 583.26 A
362.83 651.14 362.83 570.85 2 L
V
N
90 450 4.5 3.42 319.75 645.67 G
90 450 4.5 3.42 319.75 645.67 A
90 450 4.5 3.42 319.75 631.58 G
90 450 4.5 3.42 319.75 631.58 A
90 450 4.5 3.42 309.75 645.67 G
90 450 4.5 3.42 309.75 645.67 A
90 450 4.5 3.42 309.75 631.58 G
90 450 4.5 3.42 309.75 631.58 A
90 450 4.5 3.42 299.75 645.67 G
90 450 4.5 3.42 299.75 645.67 A
90 450 4.5 3.42 299.75 631.58 G
90 450 4.5 3.42 299.75 631.58 A
90 450 4.5 3.42 310.6 575.17 G
90 450 4.5 3.42 310.6 575.17 A
90 450 4.5 3.42 300.09 608.97 G
90 450 4.5 3.42 300.09 608.97 A
90 450 4.5 3.42 300.09 617.69 G
90 450 4.5 3.42 300.09 617.69 A
90 450 4.5 3.42 300.09 600.44 G
90 450 4.5 3.42 300.09 600.44 A
90 450 4.5 3.42 300.09 583.17 G
90 450 4.5 3.42 300.09 583.17 A
90 450 4.5 3.42 300.47 575.08 G
90 450 4.5 3.42 300.47 575.08 A
90 450 4.5 3.42 310.72 609.06 G
90 450 4.5 3.42 310.72 609.06 A
90 450 4.5 3.42 310.72 617.78 G
90 450 4.5 3.42 310.72 617.78 A
90 450 4.5 3.42 310.22 591.53 G
90 450 4.5 3.42 310.22 591.53 A
90 450 4.5 3.42 310.22 583.26 G
90 450 4.5 3.42 310.22 583.26 A
90 450 4.5 3.42 320.37 599.97 G
90 450 4.5 3.42 320.37 599.97 A
90 450 4.5 3.42 320.37 617.19 G
90 450 4.5 3.42 320.37 617.19 A
90 450 4.5 3.42 320.37 591.44 G
90 450 4.5 3.42 320.37 591.44 A
90 450 4.5 3.42 320.37 583.17 G
90 450 4.5 3.42 320.37 583.17 A
90 450 4.5 3.42 320.75 575.08 G
90 450 4.5 3.42 320.75 575.08 A
2 10 Q
(1) 442.57 614.47 T
380.84 616.85 441.27 616.85 2 L
4 X
N
335.98 586.86 346.84 622 R
3 X
N
370.27 609.57 451.99 609.57 2 L
4 X
N
337.41 629.57 434.13 629.57 2 L
N
0 X
(5) 436.29 627.18 T
(2) 454.29 607.76 T
357.56 603.14 369.13 621.14 R
3 X
N
347.55 595.14 462.7 595.14 2 L
4 X
N
0 X
(4) 467.29 593.33 T
54 87.5 540 686.02 C
0 0 612 792 C
FMENDPAGE
%%EndPage: "7" 8
%%Page: "8" 8
612 792 0 FMBEGINPAGE
72 750.02 558 750.02 2 L
7 X
0 K
V
2 H
0 Z
0 X
N
72 730.01 558 745.2 R
7 X
V
1 9 Q
0 X
(Addition) 207 739.2 T
72 726.98 558 726.98 2 L
7 X
V
0.25 H
0 X
N
72 74.02 558 74.02 2 L
7 X
V
2 Z
0 X
N
72 55.01 558 66.96 R
7 X
V
0 X
(The Abacus) 207 60.96 T
1 10 Q
(8) 527.85 60.96 T
1 9 Q
( of) 533.41 60.96 T
1 10 Q
(12) 546.89 60.96 T
498.25 742.67 502.85 742.67 2 L
1 H
N
500.55 742.55 500.55 735.65 2 L
N
498.25 735.78 502.85 735.78 2 L
N
498.25 739.48 502.85 739.48 2 L
N
509.74 746.12 509.74 731.18 2 L
N
506.8 742.16 512.68 742.16 2 L
N
505.53 738.21 513.7 738.21 2 L
N
507.44 735.78 505.14 732.33 2 L
N
511.65 735.4 513.19 732.33 2 L
N
505.35 743.7 M
 505.67 742.81 505.67 742.81 505.75 742.49 D
 505.83 742.17 505.83 742.17 505.87 741.85 D
 505.91 741.53 505.91 741.53 505.35 740.49 D
N
522.12 733.79 528.11 741.74 R
N
517.84 731.84 533 731.84 2 L
N
522.12 737.1 528.11 737.1 2 L
N
522.12 739.3 528.11 739.3 2 L
N
521.51 746.88 521.99 746.02 522.12 744.8 521.51 742.6 4 L
N
522.61 746.14 523.95 746.27 2 L
N
522.48 744.07 523.83 742.72 2 L
N
526.64 747 527.13 746.14 527.25 744.92 526.64 742.72 4 L
N
527.74 746.27 529.09 746.39 2 L
N
527.62 744.19 528.96 742.84 2 L
N
522.85 732.69 523.83 731.72 524.07 730.74 523.71 729.88 522.85 729.03 5 L
N
527.74 732.69 527.98 730.86 527.5 729.39 3 L
N
517.15 710.58 532 710.58 2 L
N
519.31 710.58 518.59 713.6 529.12 713.6 528.11 710.58 4 L
N
522.63 713.6 522.49 710.72 2 L
N
525.37 713.32 525.22 710.58 2 L
N
516.72 720.23 523.64 720.67 2 L
N
520.03 722.83 521.62 721.1 2 L
N
521.04 719.66 522.05 717.93 2 L
N
518.16 715.77 518.88 716.78 519.17 718.79 518.88 723.69 522.34 724.56 522.2 715.91 6 L
N
519.03 724.27 520.18 726 2 L
N
525.22 719.95 526.38 720.81 526.23 723.69 527.53 723.98 528.97 723.84 529.12 721.39 528.97 720.23
 530.27 719.8 531.13 720.67 9 L
N
524.65 714.9 526.81 715.77 528.83 717.35 529.4 718.79 525.22 718.79 527.1 716.34 528.4 715.19
 531.13 715.19 8 L
N
504.26 712.64 509.61 719.74 R
N
500.44 710.89 513.98 710.89 2 L
N
504.26 715.59 509.61 715.59 2 L
N
504.26 717.55 509.61 717.55 2 L
N
503.71 724.32 504.15 723.56 504.26 722.47 503.71 720.5 4 L
N
504.7 723.67 505.9 723.78 2 L
N
504.59 721.81 505.79 720.61 2 L
N
508.3 724.43 508.74 723.67 508.85 722.58 508.3 720.61 4 L
N
509.28 723.78 510.48 723.89 2 L
N
509.17 721.92 510.37 720.72 2 L
N
504.91 711.66 505.79 710.78 506.01 709.91 505.68 709.15 504.91 708.38 5 L
N
509.28 711.66 509.5 710.02 509.06 708.71 3 L
N
72 228.98 558 686.02 R
7 X
V
0 X
(2.2) 184.11 679.35 T
(Combined Adding-up And T) 207 679.35 T
(aking Off) 338.96 679.35 T
2 F
-0.11 (When the original number registered on a rod is smaller than 5, but will become greater) 207 655.35 P
-0.14 (than 5 after the addition, one bead from the upper) 207 643.35 P
-0.14 (-deck is moved down \050added on to the) 403.7 643.35 P
(beam\051 and one or more beads from the lower deck removed from the beam.) 207 631.35 T
72 600.98 558 614.02 C
207 608.04 558 608.04 2 L
0.25 H
2 Z
0 X
0 K
N
72 608.04 198 608.04 2 L
0 Z
N
0 0 612 792 C
72 578.02 558 591.98 C
207 583.99 558 583.99 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
1 9 Q
0 X
0 K
(T) 72 594.98 T
(ABLE  2) 76.83 594.98 T
0 F
(Combined Adding-up And T) 207 594.98 T
(aking Of) 316.9 594.98 T
(f) 350.21 594.98 T
0 8 Q
(Original Number) 220.8 572.68 T
(Formula) 330.68 572.68 T
(Bead-Operations Involved) 414 572.68 T
(lower-deck) 429.35 562.68 T
( upper-deck) 506.78 562.68 T
(4) 207 552.68 T
(plus 5 \050+5 - 4\051) 317.23 552.68 T
(- 4) 445.33 552.68 T
(+1) 526.44 552.68 T
(4 or 3) 207 542.68 T
(plus 6 \050+5 - 3\051) 317.23 542.68 T
(- 3) 445.33 542.68 T
(+1) 526.44 542.68 T
(4,3, or 2) 207 532.68 T
(plus 7 \050+5 - 2\051) 317.23 532.68 T
(- 2) 445.33 532.68 T
(+1) 526.44 532.68 T
(4,3,2 or 1) 207 522.68 T
(plus 8 \050+5 -1\051) 318.34 522.68 T
(- 1) 445.33 522.68 T
(+1) 526.44 522.68 T
1 10 Q
(2.3) 184.11 493.35 T
(Combined T) 207 493.35 T
(aking-off And Place Advancement) 264.01 493.35 T
2 F
(When a sum greater than 10 occurs on a certain rod, beads are removed from either or) 207 469.35 T
-0.1 (both the upper and lower decks and 1 bead is added to the rod directly to the left. Exam-) 207 457.35 P
(ple: When adding 9 \05010-1\051 to 8, one bead from the lower deck is removed \050-1\051 and one) 207 445.35 T
(bead from the lower deck on the row directly to the left is added \050+10\051.) 207 433.35 T
72 402.98 558 416.02 C
207 410.04 558 410.04 2 L
0.25 H
2 Z
0 X
0 K
N
72 410.04 198 410.04 2 L
0 Z
N
0 0 612 792 C
72 380.02 558 393.98 C
207 385.99 558 385.99 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
1 9 Q
0 X
0 K
(T) 72 396.98 T
(ABLE  3) 76.83 396.98 T
0 F
(Combined Adding-up And Place Advancement) 207 396.98 T
0 8 Q
(Original Number) 220.8 374.68 T
(Formula) 301.39 374.68 T
(Bead-Operations Involved) 386.86 374.68 T
(lower-deck) 361.65 364.68 T
( upper-deck) 409.21 364.68 T
(lower-deck, previous rod) 460.77 364.68 T
(9) 216 354.68 T
(plus 1 \050-9 +10\051) 289.11 354.68 T
(- 4) 373.33 354.68 T
(-1) 428.45 354.68 T
(+1) 481.44 354.68 T
(8 or 9) 216 344.68 T
(plus 2 \050-8 +10\051) 289.11 344.68 T
(- 3) 373.33 344.68 T
(- 1) 427.33 344.68 T
(+1) 481.44 344.68 T
(7,8 or 6) 216 334.68 T
(plus 3 \050-7 +10\051) 289.11 334.68 T
(- 2) 373.33 334.68 T
(- 1) 427.33 334.68 T
(+1) 481.44 334.68 T
(6,7,8 or 9) 216 324.68 T
(plus 4 \050-6 +10\051) 289.11 324.68 T
(- 1) 373.33 324.68 T
(- 1) 427.33 324.68 T
(+1) 481.44 324.68 T
(5,6,7,8 or 9) 216 314.68 T
(plus 5 \050-5 +10\051) 289.11 314.68 T
(- 1) 427.33 314.68 T
(+1) 481.44 314.68 T
(4or 9) 216 304.68 T
(plus 6 \050-4 +10\051) 289.11 304.68 T
(- 4) 373.33 304.68 T
(+1) 481.44 304.68 T
(3,4,8 or 9) 216 294.68 T
(plus 7 \050-3 +10\051) 289.11 294.68 T
(- 3) 373.33 294.68 T
(+1) 481.44 294.68 T
(2,3,4,7,8 or 9) 216 284.68 T
(plus 8 \050-2 +10\051) 289.11 284.68 T
(- 2) 373.33 284.68 T
(+1) 481.44 284.68 T
(1,2,3,4,6,7,8,9) 216 274.68 T
(plus 9 \050-1 +10\051) 289.11 274.68 T
(- 1) 373.33 274.68 T
(+1) 481.44 274.68 T
FMENDPAGE
%%EndPage: "8" 9
%%Page: "9" 9
612 792 0 FMBEGINPAGE
54 750.02 540 750.02 2 L
7 X
0 K
V
2 H
0 Z
0 X
N
54 730.01 540 745.2 R
7 X
V
1 9 Q
0 X
(Subtraction) 189 739.2 T
54 726.98 540 726.98 2 L
7 X
V
0.25 H
0 X
N
54 74.02 540 74.02 2 L
7 X
V
2 Z
0 X
N
54 55.01 540 66.96 R
7 X
V
1 10 Q
0 X
(9) 54 60.96 T
1 9 Q
( of) 59.56 60.96 T
1 10 Q
( 12) 70.54 60.96 T
1 9 Q
(The Abacus) 189 60.96 T
498.25 742.67 502.85 742.67 2 L
1 H
N
500.55 742.55 500.55 735.65 2 L
N
498.25 735.78 502.85 735.78 2 L
N
498.25 739.48 502.85 739.48 2 L
N
509.74 746.12 509.74 731.18 2 L
N
506.8 742.16 512.68 742.16 2 L
N
505.53 738.21 513.7 738.21 2 L
N
507.44 735.78 505.14 732.33 2 L
N
511.65 735.4 513.19 732.33 2 L
N
505.35 743.7 M
 505.67 742.81 505.67 742.81 505.75 742.49 D
 505.83 742.17 505.83 742.17 505.87 741.85 D
 505.91 741.53 505.91 741.53 505.35 740.49 D
N
522.12 733.79 528.11 741.74 R
N
517.84 731.84 533 731.84 2 L
N
522.12 737.1 528.11 737.1 2 L
N
522.12 739.3 528.11 739.3 2 L
N
521.51 746.88 521.99 746.02 522.12 744.8 521.51 742.6 4 L
N
522.61 746.14 523.95 746.27 2 L
N
522.48 744.07 523.83 742.72 2 L
N
526.64 747 527.13 746.14 527.25 744.92 526.64 742.72 4 L
N
527.74 746.27 529.09 746.39 2 L
N
527.62 744.19 528.96 742.84 2 L
N
522.85 732.69 523.83 731.72 524.07 730.74 523.71 729.88 522.85 729.03 5 L
N
527.74 732.69 527.98 730.86 527.5 729.39 3 L
N
517.15 710.58 532 710.58 2 L
N
519.31 710.58 518.59 713.6 529.12 713.6 528.11 710.58 4 L
N
522.63 713.6 522.49 710.72 2 L
N
525.37 713.32 525.22 710.58 2 L
N
516.72 720.23 523.64 720.67 2 L
N
520.03 722.83 521.62 721.1 2 L
N
521.04 719.66 522.05 717.93 2 L
N
518.16 715.77 518.88 716.78 519.17 718.79 518.88 723.69 522.34 724.56 522.2 715.91 6 L
N
519.03 724.27 520.18 726 2 L
N
525.22 719.95 526.38 720.81 526.23 723.69 527.53 723.98 528.97 723.84 529.12 721.39 528.97 720.23
 530.27 719.8 531.13 720.67 9 L
N
524.65 714.9 526.81 715.77 528.83 717.35 529.4 718.79 525.22 718.79 527.1 716.34 528.4 715.19
 531.13 715.19 8 L
N
504.26 712.64 509.61 719.74 R
N
500.44 710.89 513.98 710.89 2 L
N
504.26 715.59 509.61 715.59 2 L
N
504.26 717.55 509.61 717.55 2 L
N
503.71 724.32 504.15 723.56 504.26 722.47 503.71 720.5 4 L
N
504.7 723.67 505.9 723.78 2 L
N
504.59 721.81 505.79 720.61 2 L
N
508.3 724.43 508.74 723.67 508.85 722.58 508.3 720.61 4 L
N
509.28 723.78 510.48 723.89 2 L
N
509.17 721.92 510.37 720.72 2 L
N
504.91 711.66 505.79 710.78 506.01 709.91 505.68 709.15 504.91 708.38 5 L
N
509.28 711.66 509.5 710.02 509.06 708.71 3 L
N
54 90.98 540 686.02 R
7 X
V
1 10 Q
0 X
(2.4) 166.11 679.35 T
(Combined Adding-up, T) 189 679.35 T
(aking-off And Place Advancement) 301.53 679.35 T
2 F
(There are 4 cases when beads are added to the lower) 189 655.35 T
(-deck, removed from the upper) 398.61 655.35 T
(-) 522.47 655.35 T
(deck and one bead added to the adjacent rod.) 189 643.35 T
-0.27 (Example: When adding 7 to 6 \050+1-5+10\051, one bead is added to the lower) 189 619.35 P
-0.27 (-deck, one bead) 475.3 619.35 P
(removed from the upper) 189 607.35 T
(-deck and one bead is added to the left rod \050lower) 285.67 607.35 T
(-deck\051.) 483.64 607.35 T
54 576.98 540 590.02 C
189 584.04 540 584.04 2 L
0.25 H
2 Z
0 X
0 K
N
54 584.04 180 584.04 2 L
0 Z
N
0 0 612 792 C
54 554.02 540 567.98 C
189 559.99 540 559.99 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
1 9 Q
0 X
0 K
(T) 54 570.98 T
(ABLE  4) 58.83 570.98 T
0 F
(Combined Adding-up And T) 189 570.98 T
(aking Of) 298.9 570.98 T
(f and Place Advancement) 332.21 570.98 T
0 8 Q
(Original Number) 202.8 548.68 T
(Formula) 283.39 548.68 T
(Bead-Operations Involved) 368.86 548.68 T
(lower-deck) 343.65 538.68 T
( upper-deck) 391.21 538.68 T
(lower-deck, previous rod) 442.77 538.68 T
(5,6,7 or 8) 198 528.68 T
(plus 6 \050+1-5+10\051) 267.67 528.68 T
(+1) 355.44 528.68 T
(-1) 410.45 528.68 T
(+1) 463.44 528.68 T
(5,6 or 7) 198 518.68 T
(plus 2 \050+2-5+10\051) 267.67 518.68 T
(+2) 355.44 518.68 T
(- 1) 409.33 518.68 T
(+1) 463.44 518.68 T
(5 or 6) 198 508.68 T
(plus 3 \050+3-5+10\051) 267.67 508.68 T
(+3) 355.44 508.68 T
(- 1) 409.33 508.68 T
(+1) 463.44 508.68 T
(5) 198 498.68 T
(plus 4 \050+4-5+10\051) 267.67 498.68 T
(+4) 355.44 498.68 T
(- 1) 409.33 498.68 T
(+1) 463.44 498.68 T
54 425.99 540 435.02 C
189 433.5 540 433.5 2 L
0.25 H
2 Z
0 X
0 K
N
54 433.5 180 433.5 2 L
0 Z
N
0 0 612 792 C
1 12 Q
0 X
0 K
(3.0) 163.33 439.02 T
( Subtraction) 189 439.02 T
2 10 Q
(Subtraction is performed by \336rst registering the minuend and then subtracting, starting) 189 405.33 T
-0.17 (from the left, by removing beads form either or both the lower or upper decks. The \336nal) 189 393.33 P
(bead-positions represent the answer) 189 381.33 T
(.) 331.39 381.33 T
1 F
(3.1) 166.11 351.33 T
(Simple T) 189 351.33 T
(aking-off) 229.92 351.33 T
2 F
(This is achieved by simply taking of) 189 327.33 T
(f one or more beads from the lower deck, or some-) 334.29 327.33 T
(times both. Example: When subtracting 7 \050represented by -5-2= -7\051 from 9, remove 1) 189 315.33 T
(bead from the upper) 189 303.33 T
(-deck \050-5\051 and 2 beads from the lower deck \050-2\051. The remaining 2) 269.57 303.33 T
(beads represent the result.) 189 291.33 T
54 260.96 540 273.99 C
189 268.02 540 268.02 2 L
0.25 H
2 Z
0 X
0 K
N
54 268.02 180 268.02 2 L
0 Z
N
0 0 612 792 C
54 237.99 540 251.96 C
189 243.97 540 243.97 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
1 9 Q
0 X
0 K
(T) 54 254.96 T
(ABLE  5) 58.83 254.96 T
0 F
(Simple T) 189 254.96 T
(aking-of) 223.46 254.96 T
(f) 255.26 254.96 T
0 8 Q
(Original Number) 202.8 232.66 T
(Formula) 312.68 232.66 T
(Bead-Operations Involved) 396 232.66 T
(lower-deck) 411.35 222.66 T
( upper-deck) 488.78 222.66 T
(1,2,3,4,5,6,7,8) 198 212.66 T
(minus 1) 310 212.66 T
(-1) 428.45 212.66 T
(2,3,4,7,8 or 9) 198 202.66 T
(minus 2) 310 202.66 T
(-2) 428.45 202.66 T
(3,4,8 or 9) 198 192.66 T
(minus 3) 310 192.66 T
(-3) 428.45 192.66 T
(4 or 9) 198 182.66 T
(minus 4) 310 182.66 T
(-4) 428.45 182.66 T
(5,6,7,8 or 9) 198 172.66 T
(minus 5) 310 172.66 T
(-1) 509.45 172.66 T
(6,7,8 or 9) 198 162.66 T
(minus 6) 310 162.66 T
(-1) 428.45 162.66 T
(-1) 509.45 162.66 T
(7,8 or 9) 198 152.66 T
(minus 7) 310 152.66 T
(-2) 428.45 152.66 T
(-1) 509.45 152.66 T
(8 or 9) 198 142.66 T
(minus 8) 310 142.66 T
(-3) 428.45 142.66 T
(-1) 509.45 142.66 T
(9) 198 132.66 T
(minus 9) 310 132.66 T
(-4) 428.45 132.66 T
(-1) 509.45 132.66 T
FMENDPAGE
%%EndPage: "9" 10
%%Page: "10" 10
612 792 0 FMBEGINPAGE
72 750.02 558 750.02 2 L
7 X
0 K
V
2 H
0 Z
0 X
N
72 730.01 558 745.2 R
7 X
V
1 9 Q
0 X
(Subtraction) 207 739.2 T
72 726.98 558 726.98 2 L
7 X
V
0.25 H
0 X
N
72 74.02 558 74.02 2 L
7 X
V
2 Z
0 X
N
72 55.01 558 66.96 R
7 X
V
0 X
(The Abacus) 207 60.96 T
1 10 Q
(10) 522.29 60.96 T
1 9 Q
( of) 533.41 60.96 T
1 10 Q
(12) 546.89 60.96 T
498.25 742.67 502.85 742.67 2 L
1 H
N
500.55 742.55 500.55 735.65 2 L
N
498.25 735.78 502.85 735.78 2 L
N
498.25 739.48 502.85 739.48 2 L
N
509.74 746.12 509.74 731.18 2 L
N
506.8 742.16 512.68 742.16 2 L
N
505.53 738.21 513.7 738.21 2 L
N
507.44 735.78 505.14 732.33 2 L
N
511.65 735.4 513.19 732.33 2 L
N
505.35 743.7 M
 505.67 742.81 505.67 742.81 505.75 742.49 D
 505.83 742.17 505.83 742.17 505.87 741.85 D
 505.91 741.53 505.91 741.53 505.35 740.49 D
N
522.12 733.79 528.11 741.74 R
N
517.84 731.84 533 731.84 2 L
N
522.12 737.1 528.11 737.1 2 L
N
522.12 739.3 528.11 739.3 2 L
N
521.51 746.88 521.99 746.02 522.12 744.8 521.51 742.6 4 L
N
522.61 746.14 523.95 746.27 2 L
N
522.48 744.07 523.83 742.72 2 L
N
526.64 747 527.13 746.14 527.25 744.92 526.64 742.72 4 L
N
527.74 746.27 529.09 746.39 2 L
N
527.62 744.19 528.96 742.84 2 L
N
522.85 732.69 523.83 731.72 524.07 730.74 523.71 729.88 522.85 729.03 5 L
N
527.74 732.69 527.98 730.86 527.5 729.39 3 L
N
517.15 710.58 532 710.58 2 L
N
519.31 710.58 518.59 713.6 529.12 713.6 528.11 710.58 4 L
N
522.63 713.6 522.49 710.72 2 L
N
525.37 713.32 525.22 710.58 2 L
N
516.72 720.23 523.64 720.67 2 L
N
520.03 722.83 521.62 721.1 2 L
N
521.04 719.66 522.05 717.93 2 L
N
518.16 715.77 518.88 716.78 519.17 718.79 518.88 723.69 522.34 724.56 522.2 715.91 6 L
N
519.03 724.27 520.18 726 2 L
N
525.22 719.95 526.38 720.81 526.23 723.69 527.53 723.98 528.97 723.84 529.12 721.39 528.97 720.23
 530.27 719.8 531.13 720.67 9 L
N
524.65 714.9 526.81 715.77 528.83 717.35 529.4 718.79 525.22 718.79 527.1 716.34 528.4 715.19
 531.13 715.19 8 L
N
504.26 712.64 509.61 719.74 R
N
500.44 710.89 513.98 710.89 2 L
N
504.26 715.59 509.61 715.59 2 L
N
504.26 717.55 509.61 717.55 2 L
N
503.71 724.32 504.15 723.56 504.26 722.47 503.71 720.5 4 L
N
504.7 723.67 505.9 723.78 2 L
N
504.59 721.81 505.79 720.61 2 L
N
508.3 724.43 508.74 723.67 508.85 722.58 508.3 720.61 4 L
N
509.28 723.78 510.48 723.89 2 L
N
509.17 721.92 510.37 720.72 2 L
N
504.91 711.66 505.79 710.78 506.01 709.91 505.68 709.15 504.91 708.38 5 L
N
509.28 711.66 509.5 710.02 509.06 708.71 3 L
N
72 164.98 558 686.02 R
7 X
V
0 X
(3.2) 184.11 679.35 T
(Combined Adding-up And T) 207 679.35 T
(aking Off) 338.96 679.35 T
2 F
(When the number of beads in the lower deck is less than the subtracter \050the number) 207 655.35 T
-0.25 (being subtracted\051, one or more beads are added in the lower deck and 1 bead is removed) 207 643.35 P
(from the upper) 207 631.35 T
(-deck.) 266.2 631.35 T
(Example: When subtracting 4 \050+1-5 = -4\051 from 7 \050represented by 1 bead in the upper) 207 607.35 T
(-) 545.87 607.35 T
-0.18 (deck and 2 beads in the lower deck \050less than 4, the subtracter\051, one bead is added to the) 207 595.35 P
-0.02 (lower deck \050+1\051 and 1 bead is removed from the upper) 207 583.35 P
-0.02 (-deck \050-5\051 leaving 3 beads, repre-) 424.88 583.35 P
(senting the result.) 207 571.35 T
72 540.98 558 554.02 C
207 548.04 558 548.04 2 L
0.25 H
2 Z
0 X
0 K
N
72 548.04 198 548.04 2 L
0 Z
N
0 0 612 792 C
72 518.02 558 531.98 C
207 523.99 558 523.99 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
1 9 Q
0 X
0 K
(T) 72 534.98 T
(ABLE  6) 76.83 534.98 T
0 F
(Combined Adding-up And T) 207 534.98 T
(aking Of) 316.9 534.98 T
(f) 350.21 534.98 T
0 8 Q
(Original Number) 220.8 512.68 T
(Formula) 330.68 512.68 T
(Bead-Operations Involved) 414 512.68 T
(lower-deck) 429.35 502.68 T
( upper-deck) 506.78 502.68 T
(5) 207 492.68 T
(minus1 \050-5 + 4\051) 315.01 492.68 T
(+ 4) 444.33 492.68 T
(-1) 527.45 492.68 T
(5 or 6) 207 482.68 T
(minus 2 \050-5 + 3\051) 313.89 482.68 T
(+ 3) 444.33 482.68 T
(-1) 527.45 482.68 T
(5,6 or 7) 207 472.68 T
(minus 3 \050-5 + 2\051) 313.89 472.68 T
(+ 2) 444.33 472.68 T
(-1) 527.45 472.68 T
(5,6,7 or 8) 207 462.68 T
(minus 4 \050-5 + 1\051) 313.89 462.68 T
(+ 1) 444.33 462.68 T
(-1) 527.45 462.68 T
1 10 Q
(3.3) 184.11 409.35 T
(T) 207 409.35 T
(aking-off From A Rod Of Higher Order And Adding-up) 212.37 409.35 T
2 F
(When a number on a speci\336c rod is smaller than the subtrahend \0504 is the subtrahend) 207 393.35 T
(when performing 13 - 4; note that in the ones column, the 3 is less than the 4\051 one bead) 207 381.35 T
-0.16 (for the order of tens and one bead from the lower) 207 369.35 P
-0.16 (-deck has to be) 401.75 369.35 P
3 F
-0.16 (taken off,) 463.85 369.35 P
2 F
-0.16 (and one bead) 503.24 369.35 P
(from the upper) 207 357.35 T
(-deck is counted.) 266.2 357.35 T
72 326.98 558 340.02 C
207 334.04 558 334.04 2 L
0.25 H
2 Z
0 X
0 K
N
72 334.04 198 334.04 2 L
0 Z
N
0 0 612 792 C
72 304.02 558 317.98 C
207 309.99 558 309.99 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
1 9 Q
0 X
0 K
(T) 72 320.98 T
(ABLE  7) 76.83 320.98 T
0 F
(T) 207 320.98 T
(aking-of) 211.49 320.98 T
(f From A Rod Of Higher Order And Adding-up) 243.3 320.98 T
0 8 Q
(Original Number) 220.8 298.68 T
(Formula) 301.39 298.68 T
(Bead-Operations Involved) 386.86 298.68 T
(lower-deck, right 1 rod) 361.65 288.68 T
( lower-deck) 442.97 288.68 T
(upper-deck) 507.86 288.68 T
(0) 216 278.68 T
(minus 1 \050+9 -10\051) 285.78 278.68 T
(-1) 392.45 278.68 T
(+ 4) 462.33 278.68 T
(+ 1) 516.33 278.68 T
(0 or 1) 216 268.68 T
(minus 2 \050+8 -10\051) 285.78 268.68 T
(- 1) 391.33 268.68 T
(+ 3) 462.33 268.68 T
(+ 1) 516.33 268.68 T
(0,1 or 2) 216 258.68 T
(minus 3 \050+7 -10\051) 285.78 258.68 T
(- 1) 391.33 258.68 T
(+ 2) 462.33 258.68 T
(+ 1) 516.33 258.68 T
(0,1,2 or 9) 216 248.68 T
(minus 4 \050+6 -10\051) 285.78 248.68 T
(- 1) 391.33 248.68 T
(+ 1) 462.33 248.68 T
(+ 1) 516.33 248.68 T
(0,1,2,3 or 4) 216 238.68 T
(minus 5 \050+5 -10\051) 285.78 238.68 T
(- 1) 391.33 238.68 T
(+ 1) 516.33 238.68 T
(0 or 5) 216 228.68 T
(minus 6 \050+4 -10\051) 285.78 228.68 T
(- 1) 391.33 228.68 T
(+ 4) 462.33 228.68 T
(0,5 or 6) 216 218.68 T
(minus 7 \050+3 -10\051) 285.78 218.68 T
(- 1) 391.33 218.68 T
(+ 3) 462.33 218.68 T
(0,1,2,5,6 or 7) 216 208.68 T
(minus 8 \050+2 -10\051) 285.78 208.68 T
(- 1) 391.33 208.68 T
(+ 2) 462.33 208.68 T
(0,1,2,3,5,6,7,8) 216 198.68 T
(minus 9 \050+1 -10\051) 285.78 198.68 T
(- 1) 391.33 198.68 T
(+ 1) 462.33 198.68 T
FMENDPAGE
%%EndPage: "10" 11
%%Page: "11" 11
612 792 0 FMBEGINPAGE
54 750.02 540 750.02 2 L
7 X
0 K
V
2 H
0 Z
0 X
N
54 730.01 540 745.2 R
7 X
V
1 9 Q
0 X
(Acknowledgments) 189 739.2 T
54 726.98 540 726.98 2 L
7 X
V
0.25 H
0 X
N
54 74.02 540 74.02 2 L
7 X
V
2 Z
0 X
N
54 55.01 540 66.96 R
7 X
V
1 10 Q
0 X
(11) 54 60.96 T
1 9 Q
( of) 65.11 60.96 T
1 10 Q
( 12) 76.1 60.96 T
1 9 Q
(The Abacus) 189 60.96 T
498.25 742.67 502.85 742.67 2 L
1 H
N
500.55 742.55 500.55 735.65 2 L
N
498.25 735.78 502.85 735.78 2 L
N
498.25 739.48 502.85 739.48 2 L
N
509.74 746.12 509.74 731.18 2 L
N
506.8 742.16 512.68 742.16 2 L
N
505.53 738.21 513.7 738.21 2 L
N
507.44 735.78 505.14 732.33 2 L
N
511.65 735.4 513.19 732.33 2 L
N
505.35 743.7 M
 505.67 742.81 505.67 742.81 505.75 742.49 D
 505.83 742.17 505.83 742.17 505.87 741.85 D
 505.91 741.53 505.91 741.53 505.35 740.49 D
N
522.12 733.79 528.11 741.74 R
N
517.84 731.84 533 731.84 2 L
N
522.12 737.1 528.11 737.1 2 L
N
522.12 739.3 528.11 739.3 2 L
N
521.51 746.88 521.99 746.02 522.12 744.8 521.51 742.6 4 L
N
522.61 746.14 523.95 746.27 2 L
N
522.48 744.07 523.83 742.72 2 L
N
526.64 747 527.13 746.14 527.25 744.92 526.64 742.72 4 L
N
527.74 746.27 529.09 746.39 2 L
N
527.62 744.19 528.96 742.84 2 L
N
522.85 732.69 523.83 731.72 524.07 730.74 523.71 729.88 522.85 729.03 5 L
N
527.74 732.69 527.98 730.86 527.5 729.39 3 L
N
517.15 710.58 532 710.58 2 L
N
519.31 710.58 518.59 713.6 529.12 713.6 528.11 710.58 4 L
N
522.63 713.6 522.49 710.72 2 L
N
525.37 713.32 525.22 710.58 2 L
N
516.72 720.23 523.64 720.67 2 L
N
520.03 722.83 521.62 721.1 2 L
N
521.04 719.66 522.05 717.93 2 L
N
518.16 715.77 518.88 716.78 519.17 718.79 518.88 723.69 522.34 724.56 522.2 715.91 6 L
N
519.03 724.27 520.18 726 2 L
N
525.22 719.95 526.38 720.81 526.23 723.69 527.53 723.98 528.97 723.84 529.12 721.39 528.97 720.23
 530.27 719.8 531.13 720.67 9 L
N
524.65 714.9 526.81 715.77 528.83 717.35 529.4 718.79 525.22 718.79 527.1 716.34 528.4 715.19
 531.13 715.19 8 L
N
504.26 712.64 509.61 719.74 R
N
500.44 710.89 513.98 710.89 2 L
N
504.26 715.59 509.61 715.59 2 L
N
504.26 717.55 509.61 717.55 2 L
N
503.71 724.32 504.15 723.56 504.26 722.47 503.71 720.5 4 L
N
504.7 723.67 505.9 723.78 2 L
N
504.59 721.81 505.79 720.61 2 L
N
508.3 724.43 508.74 723.67 508.85 722.58 508.3 720.61 4 L
N
509.28 723.78 510.48 723.89 2 L
N
509.17 721.92 510.37 720.72 2 L
N
504.91 711.66 505.79 710.78 506.01 709.91 505.68 709.15 504.91 708.38 5 L
N
509.28 711.66 509.5 710.02 509.06 708.71 3 L
N
54 87.98 540 686.02 R
7 X
V
1 10 Q
0 X
(3.4) 166.11 679.35 T
(Combined T) 189 679.35 T
(aking-off From A Rod Of Higher Order) 246.01 679.35 T
(, Adding-up in the) 425.93 679.35 T
(Upper-deck and T) 189 667.35 T
(aking-off in the Lower-Deck) 272.67 667.35 T
2 F
(This technique is called for when a number on a speci\336c rod is smaller than the sup-) 189 651.35 T
(posed subtrahend \050I have no idea what this means\051, but only in such cases as exempli-) 189 639.35 T
(\336ed by 12 - 6.) 189 627.35 T
54 596.98 540 610.02 C
189 604.04 540 604.04 2 L
0.25 H
2 Z
0 X
0 K
N
54 604.04 180 604.04 2 L
0 Z
N
0 0 612 792 C
54 574.02 540 587.98 C
189 579.99 540 579.99 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
1 9 Q
0 X
0 K
(T) 54 590.98 T
(ABLE  8) 58.83 590.98 T
0 F
(Combined T) 189 590.98 T
(aking-of) 236.95 590.98 T
(f...) 268.75 590.98 T
0 8 Q
(Original Number) 202.8 568.68 T
(Formula) 283.39 568.68 T
(Bead-Operations Involved) 368.86 568.68 T
(lower-deck, 1 rod right) 343.65 558.68 T
( lower-deck) 436.55 558.68 T
(upper-deck) 490.53 558.68 T
(1,2,3 or 4) 198 548.68 T
(minus 6 \050-1+5-10\051) 265.34 548.68 T
(-1) 374.45 548.68 T
(-1) 446.45 548.68 T
(+1) 499.44 548.68 T
(2,3 or 4) 198 538.68 T
(minus 7 \050-2+5-10\051) 265.34 538.68 T
(-1) 374.45 538.68 T
(- 2) 445.33 538.68 T
(+1) 499.44 538.68 T
(3 or 4) 198 528.68 T
(minus 8 \050-3+5-10\051) 265.34 528.68 T
(-1) 374.45 528.68 T
(- 3) 445.33 528.68 T
(+1) 499.44 528.68 T
(4) 198 518.68 T
(minus9 \050-4+5-10\051) 266.45 518.68 T
(-1) 374.45 518.68 T
(- 4) 445.33 518.68 T
(+1) 499.44 518.68 T
2 10 Q
(If you\325ve actually read this user) 189 495.35 T
(\325) 316.21 495.35 T
(s guide up to this point, hoping to learn how to use the) 318.99 495.35 T
(abacus to its maximum potential, I should tell you that you\325d be better of) 189 483.35 T
(f using a hand-) 480.31 483.35 T
(held \050or pop-up\051 digital calculator) 189 471.35 T
(. \050I hear you asking: Why did I bother going through) 323.91 471.35 T
-0.11 (all this if I was going to expound this heresy at the conclusion? Essentially) 189 459.35 P
-0.11 (, the exercise) 485.37 459.35 P
(was designed to be didactic; from my programming stand-point, it was an interesting) 189 447.35 T
(program to attempt and hopefully) 189 435.35 T
(, the user would learn a thing or two. Unquestionably) 322.71 435.35 T
(,) 534.97 435.35 T
(though, it was unmitigated fun!\051. T) 189 423.35 T
(echnically) 328.49 423.35 T
(, the abacus) 368.92 423.35 T
3 F
(is) 418.32 423.35 T
2 F
( a hand-held digital calcula-) 424.99 423.35 T
(tor) 189 411.35 T
(, but that the user must perform some sort of manipulations before the solution is) 199.7 411.35 T
(arrived at.) 189 399.35 T
54 349.99 540 359.02 C
189 357.5 540 357.5 2 L
0.25 H
2 Z
0 X
0 K
N
54 357.5 180 357.5 2 L
0 Z
N
0 0 612 792 C
1 12 Q
0 X
0 K
(4.0) 163.33 363.02 T
(Acknowledgments) 189 363.02 T
2 10 Q
-0.22 (A great debt of gratitude goes to Agustine Lee, instructor at the R) 189 329.33 P
-0.22 (yerson Electrical Engi-) 448.29 329.33 P
-0.14 (neering Department, who supplied a real, live abacus without which) 189 317.33 P
3 F
-0.14 (xabacus) 462.41 317.33 P
2 F
-0.14 ( would not) 495.16 317.33 P
(be possible, for supplying invaluable documentation, that was shamelessly plagiarized) 189 305.33 T
(into the documentation you are now reading, for the Chinese characters, and for testing) 189 293.33 T
3 F
(xabacus) 189 281.33 T
2 F
( and providing helpful comments on improving it....) 221.75 281.33 T
(...thanks also to Nick Colonello, systems administrator and all-round technical-support) 189 257.33 T
(person for beta testing....) 189 245.33 T
(...and to Eva Dudova, who has expertise in unmercifully crashing applications \050has a) 189 221.33 T
(future as a beta-tester\051, and is cute too...) 189 209.33 T
-0.17 (... and \336nally) 189 185.33 P
-0.17 (, thanks to those who have written X-applications, from whose code I have) 240.48 185.33 P
(learned the art of X.) 189 173.33 T
(This document was typeset using FrameMaker 2.1 \050and then we got version 3.0 which) 189 149.33 T
(would have really come-in handy for making all those tables\051 running on a) 189 137.33 T
(SP) 189 125.33 T
(ARCstation 1.) 199.19 125.33 T
FMENDPAGE
%%EndPage: "11" 12
%%Page: "12" 12
612 792 0 FMBEGINPAGE
72 750.02 558 750.02 2 L
7 X
0 K
V
2 H
0 Z
0 X
N
72 730.01 558 745.2 R
7 X
V
1 9 Q
0 X
(Do those Chinese characters really mean anything?) 207 739.2 T
72 726.98 558 726.98 2 L
7 X
V
0.25 H
0 X
N
72 74.02 558 74.02 2 L
7 X
V
2 Z
0 X
N
72 55.01 558 66.96 R
7 X
V
0 X
(The Abacus) 207 60.96 T
1 10 Q
(12) 522.29 60.96 T
1 9 Q
( of) 533.41 60.96 T
1 10 Q
(12) 546.89 60.96 T
498.25 742.67 502.85 742.67 2 L
1 H
N
500.55 742.55 500.55 735.65 2 L
N
498.25 735.78 502.85 735.78 2 L
N
498.25 739.48 502.85 739.48 2 L
N
509.74 746.12 509.74 731.18 2 L
N
506.8 742.16 512.68 742.16 2 L
N
505.53 738.21 513.7 738.21 2 L
N
507.44 735.78 505.14 732.33 2 L
N
511.65 735.4 513.19 732.33 2 L
N
505.35 743.7 M
 505.67 742.81 505.67 742.81 505.75 742.49 D
 505.83 742.17 505.83 742.17 505.87 741.85 D
 505.91 741.53 505.91 741.53 505.35 740.49 D
N
522.12 733.79 528.11 741.74 R
N
517.84 731.84 533 731.84 2 L
N
522.12 737.1 528.11 737.1 2 L
N
522.12 739.3 528.11 739.3 2 L
N
521.51 746.88 521.99 746.02 522.12 744.8 521.51 742.6 4 L
N
522.61 746.14 523.95 746.27 2 L
N
522.48 744.07 523.83 742.72 2 L
N
526.64 747 527.13 746.14 527.25 744.92 526.64 742.72 4 L
N
527.74 746.27 529.09 746.39 2 L
N
527.62 744.19 528.96 742.84 2 L
N
522.85 732.69 523.83 731.72 524.07 730.74 523.71 729.88 522.85 729.03 5 L
N
527.74 732.69 527.98 730.86 527.5 729.39 3 L
N
517.15 710.58 532 710.58 2 L
N
519.31 710.58 518.59 713.6 529.12 713.6 528.11 710.58 4 L
N
522.63 713.6 522.49 710.72 2 L
N
525.37 713.32 525.22 710.58 2 L
N
516.72 720.23 523.64 720.67 2 L
N
520.03 722.83 521.62 721.1 2 L
N
521.04 719.66 522.05 717.93 2 L
N
518.16 715.77 518.88 716.78 519.17 718.79 518.88 723.69 522.34 724.56 522.2 715.91 6 L
N
519.03 724.27 520.18 726 2 L
N
525.22 719.95 526.38 720.81 526.23 723.69 527.53 723.98 528.97 723.84 529.12 721.39 528.97 720.23
 530.27 719.8 531.13 720.67 9 L
N
524.65 714.9 526.81 715.77 528.83 717.35 529.4 718.79 525.22 718.79 527.1 716.34 528.4 715.19
 531.13 715.19 8 L
N
504.26 712.64 509.61 719.74 R
N
500.44 710.89 513.98 710.89 2 L
N
504.26 715.59 509.61 715.59 2 L
N
504.26 717.55 509.61 717.55 2 L
N
503.71 724.32 504.15 723.56 504.26 722.47 503.71 720.5 4 L
N
504.7 723.67 505.9 723.78 2 L
N
504.59 721.81 505.79 720.61 2 L
N
508.3 724.43 508.74 723.67 508.85 722.58 508.3 720.61 4 L
N
509.28 723.78 510.48 723.89 2 L
N
509.17 721.92 510.37 720.72 2 L
N
504.91 711.66 505.79 710.78 506.01 709.91 505.68 709.15 504.91 708.38 5 L
N
509.28 711.66 509.5 710.02 509.06 708.71 3 L
N
72 87.98 558 686.02 R
7 X
V
1 12 Q
0 X
(5.0) 181.33 678.02 T
(Do those Chinese characters really mean) 207 678.02 T
72 650.99 558 660.02 C
207 658.5 558 658.5 2 L
0.25 H
2 Z
0 X
0 K
N
72 658.5 198 658.5 2 L
0 Z
N
0 0 612 792 C
1 12 Q
0 X
0 K
(anything?) 207 664.02 T
2 10 Q
(A few) 207 634.33 T
2 8 Q
(1) 231.7 638.33 T
2 10 Q
( people have asked me this, so here\325) 235.69 634.33 T
(s the translation:) 379.48 634.33 T
72 99.98 558 116.04 C
207 107.98 351 107.98 2 L
0.25 H
2 Z
0 X
0 K
N
0 0 612 792 C
2 8 Q
0 X
0 K
(1.   A grand total of 1, actually.) 207 94.65 T
72 87.98 558 686.02 C
206.82 438.97 558 630.99 C
215.82 526.6 224.46 526.6 2 L
1 H
2 Z
0 X
0 K
N
220.14 526.36 220.14 513.4 2 L
N
215.82 513.64 224.46 513.64 2 L
N
215.82 520.6 224.46 520.6 2 L
N
237.43 533.08 237.43 504.99 2 L
N
231.9 525.64 242.95 525.64 2 L
N
229.5 518.2 244.87 518.2 2 L
N
233.1 513.64 228.78 507.15 2 L
N
241.03 512.92 243.91 507.15 2 L
N
229.17 528.52 M
 229.77 526.86 229.77 526.86 229.92 526.26 D
 230.07 525.65 230.07 525.65 230.15 525.05 D
 230.23 524.45 230.23 524.45 229.17 522.48 D
N
268.87 510.49 280.13 525.44 R
N
260.82 506.82 289.33 506.82 2 L
N
268.87 516.7 280.13 516.7 2 L
N
268.87 520.84 280.13 520.84 2 L
N
267.72 535.09 268.64 533.49 268.87 531.19 267.72 527.05 4 L
N
269.79 533.72 272.32 533.95 2 L
N
269.56 529.81 272.09 527.28 2 L
N
277.37 535.33 278.29 533.72 278.52 531.42 277.37 527.28 4 L
N
279.44 533.95 281.97 534.18 2 L
N
279.21 530.04 281.74 527.51 2 L
N
270.25 508.43 272.09 506.59 272.55 504.75 271.86 503.14 270.25 501.53 5 L
N
279.44 508.43 279.9 504.98 278.98 502.22 3 L
N
310.73 507.83 338.65 507.83 2 L
N
314.8 507.83 313.45 513.52 333.23 513.52 331.33 507.83 4 L
N
321.03 513.52 320.76 508.1 2 L
N
326.18 512.97 325.91 507.83 2 L
N
309.92 525.98 322.93 526.8 2 L
N
316.15 530.86 319.14 527.61 2 L
N
318.05 524.9 319.95 521.65 2 L
N
312.63 517.58 313.99 519.48 314.53 523.27 313.99 532.49 320.49 534.11 320.22 517.85 6 L
N
314.26 533.57 316.43 536.83 2 L
N
325.91 525.44 328.08 527.07 327.81 532.49 330.25 533.03 332.96 532.76 333.23 528.15 332.96 525.98
 335.4 525.17 337.02 526.8 9 L
N
324.83 515.96 328.89 517.58 332.69 520.56 333.77 523.27 325.91 523.27 329.44 518.67 331.88 516.5
 337.02 516.5 8 L
N
3 10 Q
(The Calculation with) 215.82 579.05 T
(The Abacus) 276.91 606.16 T
2 F
(W) 216.67 475.33 T
(ith respect to the arrangement in the top-left corner of every page, the characters) 225.7 475.33 T
3 F
(Bead) 215.82 543.05 T
(Calculate) 257.95 543.05 T
(Plate) 311.72 543.05 T
(the Abacus) 234.68 570.16 T
249.32 558.99 249.32 495.99 2 L
0.5 H
3 X
N
302.49 585.99 302.49 496.83 2 L
N
251.15 621.99 251.15 594.99 2 L
N
350.82 621.99 350.82 495.99 2 L
N
2 F
0 X
(may be read left-to-right or top-to-bottom.) 219.37 462.18 T
72 87.98 558 686.02 C
0 0 612 792 C
FMENDPAGE
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