# Expressions derived from proportion C1/D1=C2/D2

For any slide position for any two cursor positions numbers on scales C and D under the cursor always satisfy the equation C1/D1=C2/D2.

Leaving only one variable on the left side of the equation and using relations between scales many formulas can be calculated with only one slide movement. They are really 'simple' algorithms since they do not include tricks like reverting slide, transfering values from scale to scale, matching pair of values with a cursor or hand calculations. You still have to deal with digit counts, though.

The following table displays such formulas for slide rule with [K A[B CI C]D] scales (this notation means that B, CI and C scales are located on the slide). Interesting that reversing slide does not give any additional formulas of this kind.

Records are ordered by powers of multiplicands. Each line shows scales used. The first scale in the list is the scale where the final result is found. (Cases with result on stator scales are more useful since this result can be used in further calculations.)

Short notation is used to describe these algorithms because of their enormous number.

Formula Scales Algorithm
r=u3v3w3 [K C D CI] wCI->vD; uC=>K(r)
r=u3v3w3/2 [K C CI A] vCI->wA; uC=>K(r)
[K D CI B] vCI->uD; wB=>K(r)
r=u3v3w [K C CI] vCI->wK; uC=>K(r)
r=u3v3w-3/2 [K C D B] wB->vD; uC=>K(r)
r=u3v3w-3 [K C D] wC->vD; uC=>K(r)
[K D CI] vCI->uD; wCI=>K(r)
r=u3v3/2w3/2 [K CI B A] uCI->wA; vB=>K(r)
r=u3v3/2w [K CI B] uCI->wK; vB=>K(r)
r=u3v3/2w-3/2 [K C A B] wB->vA; uC=>K(r)
[K D B] wB->uD; vB=>K(r)
r=u3v3/2w-3 [K C A] wC->vA; uC=>K(r)
[K CI A] uCI->vA; wCI=>K(r)
[K D B C] wC->uD; vB=>K(r)
r=u3vw-3/2 [K C B] wB->vK; uC=>K(r)
r=u3vw-3 [K C] wC->vK; uC=>K(r)
[K CI] uCI->vK; wCI=>K(r)
r=u3v-3/2w-3 [K D B CI] vB->uD; wCI=>K(r)
r=u3v-3w-3 [K D CI C] wC->uD; vCI=>K(r)
r=u2v2w2 [A C D CI] wCI->vD; uC=>A(r)
r=u2v2w [A C CI] vCI->wA; uC=>A(r)
[A D CI B] vCI->uD; wB=>A(r)
r=u2v2w2/3 [A C CI K] vCI->wK; uC=>A(r)
r=u2v2w-2/3 [B D C K] vC->wK; uD=>B(r)
r=u2v2w-1 [B D C A] vC->wA; uD=>B(r)
[A C D B] wB->vD; uC=>A(r)
r=u2v2w-2 [B D C] vC->wD; uD=>B(r)
[A C D] wC->vD; uC=>A(r)
[A D CI] vCI->uD; wCI=>A(r)
r=u2vw [A CI B] uCI->wA; vB=>A(r)
r=u2vw2/3 [A CI B K] uCI->wK; vB=>A(r)
r=u2vw-2/3 [B C A K] uC->wK; vA=>B(r)
[B D K] vB->wK; uD=>B(r)
r=u2vw-1 [B C A] uC->wA; vA=>B(r)
[B D A] vB->wA; uD=>B(r)
[A C B] wB->vA; uC=>A(r)
[A D B] wB->uD; vB=>A(r)
r=u2vw-2 [B C A D] uC->wD; vA=>B(r)
[B D] vB->wD; uD=>B(r)
[A C] wC->vA; uC=>A(r)
[A CI] uCI->vA; wCI=>A(r)
[A D B C] wC->uD; vB=>A(r)
r=u2v2/3w-2/3 [B C K] uC->wK; vK=>B(r)
r=u2v2/3w-1 [B C K A] uC->wA; vK=>B(r)
[A C K B] wB->vK; uC=>A(r)
r=u2v2/3w-2 [B C K D] uC->wD; vK=>B(r)
[A C K] wC->vK; uC=>A(r)
[A CI K] uCI->vK; wCI=>A(r)
r=u2v-2/3w-2 [B D K CI] wCI->vK; uD=>B(r)
r=u2v-1w-2 [B D A CI] wCI->vA; uD=>B(r)
[A D B CI] vB->uD; wCI=>A(r)
r=u2v-2w-2 [B D CI] vCI->wD; uD=>B(r)
[A D CI C] wC->uD; vCI=>A(r)
r=u3/2v3/2w-3/2 [K B A] wB->vA; uB=>K(r)
r=u3/2v3/2w-3 [K B A C] wC->vA; uB=>K(r)
r=u3/2vw-3/2 [K B] wB->vK; uB=>K(r)
r=u3/2vw-3 [K B C] wC->vK; uB=>K(r)
r=u3/2v-3/2w-3 [K A B CI] vB->uA; wCI=>K(r)
r=u3/2v-3w-3 [K A CI C] wC->uA; vCI=>K(r)
r=uvw [D C CI] wCI->vD; uC=>D(r)
r=uvw1/2 [D C CI A] vCI->wA; uC=>D(r)
[D CI B] vCI->uD; wB=>D(r)
r=uvw1/3 [D C CI K] vCI->wK; uC=>D(r)
r=uvw-1/3 [C D K] vC->wK; uD=>C(r)
[CI D K] uCI->vD; wK=>CI(r)
r=uvw-1/2 [C D A] vC->wA; uD=>C(r)
[CI D A] uCI->vD; wA=>CI(r)
[D C B] wB->vD; uC=>D(r)
r=uvw-2/3 [B A K] vB->wK; uA=>B(r)
r=uvw-1 [C D] vC->wD; uD=>C(r)
[CI D] uCI->vD; wD=>CI(r)
[B A] vB->wA; uA=>B(r)
[A B] wB->vA; uB=>A(r)
[D C] wC->vD; uC=>D(r)
[D CI] vCI->uD; wCI=>D(r)
r=uvw-2 [B A D] vB->wD; uA=>B(r)
[A B C] wC->vA; uB=>A(r)
r=uv2/3w-2/3 [B K] uB->wK; vK=>B(r)
r=uv2/3w-1 [B K A] uB->wA; vK=>B(r)
[A B K] wB->vK; uB=>A(r)
r=uv2/3w-2 [B K D] uB->wD; vK=>B(r)
[A B K C] wC->vK; uB=>A(r)
r=uv1/2w1/2 [D CI B A] uCI->wA; vB=>D(r)
r=uv1/2w1/3 [D CI B K] uCI->wK; vB=>D(r)
r=uv1/2w-1/3 [C A K] uC->wK; vA=>C(r)
[C D B K] vB->wK; uD=>C(r)
[CI A K] uCI->vA; wK=>CI(r)
r=uv1/2w-1/2 [C A] uC->wA; vA=>C(r)
[C D B A] vB->wA; uD=>C(r)
[CI A] uCI->vA; wA=>CI(r)
[D C A B] wB->vA; uC=>D(r)
[D B] wB->uD; vB=>D(r)
r=uv1/2w-1 [C A D] uC->wD; vA=>C(r)
[C D B] vB->wD; uD=>C(r)
[CI A D] uCI->vA; wD=>CI(r)
[D C A] wC->vA; uC=>D(r)
[D CI A] uCI->vA; wCI=>D(r)
[D B C] wC->uD; vB=>D(r)
r=uv1/3w-1/3 [C K] uC->wK; vK=>C(r)
[CI K] uCI->vK; wK=>CI(r)
r=uv1/3w-1/2 [C K A] uC->wA; vK=>C(r)
[CI K A] uCI->vK; wA=>CI(r)
[D C K B] wB->vK; uC=>D(r)
r=uv1/3w-1 [C K D] uC->wD; vK=>C(r)
[CI K D] uCI->vK; wD=>CI(r)
[D C K] wC->vK; uC=>D(r)
[D CI K] uCI->vK; wCI=>D(r)
r=uv-1/3w-1/2 [CI D K B] wB->uD; vK=>CI(r)
r=uv-1/3w-1 [C D K CI] wCI->vK; uD=>C(r)
[CI D K C] wC->uD; vK=>CI(r)
r=uv-1/2w-1/2 [CI D A B] wB->uD; vA=>CI(r)
r=uv-1/2w-1 [C D A CI] wCI->vA; uD=>C(r)
[CI D A C] wC->uD; vA=>CI(r)
[CI D B] vB->uD; wD=>CI(r)
[D B CI] vB->uD; wCI=>D(r)
r=uv-2/3w-2 [B A K CI] wCI->vK; uA=>B(r)
r=uv-1w-1 [C D CI] vCI->wD; uD=>C(r)
[CI D C] wC->uD; vD=>CI(r)
[D CI C] wC->uD; vCI=>D(r)
r=uv-1w-2 [B A CI] wCI->vA; uA=>B(r)
[A B CI] vB->uA; wCI=>A(r)
r=uv-3/2w-3 [K B CI] vB->uK; wCI=>K(r)
r=uv-2w-2 [B A CI D] vCI->wD; uA=>B(r)
[A CI C] wC->uA; vCI=>A(r)
r=uv-3w-3 [K CI C] wC->uK; vCI=>K(r)
r=u2/3v-2/3w-2 [B K CI] wCI->vK; uK=>B(r)
r=u2/3v-1w-2 [B K A CI] wCI->vA; uK=>B(r)
[A K B CI] vB->uK; wCI=>A(r)
r=u2/3v-2w-2 [B K CI D] vCI->wD; uK=>B(r)
[A K CI C] wC->uK; vCI=>A(r)
r=u1/2v1/2w-1/3 [C A B K] vB->wK; uA=>C(r)
r=u1/2v1/2w-1/2 [C A B] vB->wA; uA=>C(r)
[D B A] wB->vA; uB=>D(r)
r=u1/2v1/2w-1 [C A B D] vB->wD; uA=>C(r)
[D B A C] wC->vA; uB=>D(r)
r=u1/2v1/3w-1/3 [C B K] uB->wK; vK=>C(r)
r=u1/2v1/3w-1/2 [C B K A] uB->wA; vK=>C(r)
[D B K] wB->vK; uB=>D(r)
r=u1/2v1/3w-1 [C B K D] uB->wD; vK=>C(r)
[D B K C] wC->vK; uB=>D(r)
r=u1/2v-1/3w-1/2 [CI A K B] wB->uA; vK=>CI(r)
r=u1/2v-1/3w-1 [C A K CI] wCI->vK; uA=>C(r)
[CI A K C] wC->uA; vK=>CI(r)
r=u1/2v-1/2w-1/2 [CI A B] wB->uA; vA=>CI(r)
r=u1/2v-1/2w-1 [C A CI] wCI->vA; uA=>C(r)
[CI A C] wC->uA; vA=>CI(r)
[CI A B D] vB->uA; wD=>CI(r)
[D A B CI] vB->uA; wCI=>D(r)
r=u1/2v-1w-1 [C A CI D] vCI->wD; uA=>C(r)
[CI A D C] wC->uA; vD=>CI(r)
[D A CI C] wC->uA; vCI=>D(r)
r=u1/3v-1/3w-1/2 [CI K B] wB->uK; vK=>CI(r)
r=u1/3v-1/3w-1 [C K CI] wCI->vK; uK=>C(r)
[CI K C] wC->uK; vK=>CI(r)
r=u1/3v-1/2w-1/2 [CI K A B] wB->uK; vA=>CI(r)
r=u1/3v-1/2w-1 [C K A CI] wCI->vA; uK=>C(r)
[CI K A C] wC->uK; vA=>CI(r)
[CI K B D] vB->uK; wD=>CI(r)
[D K B CI] vB->uK; wCI=>D(r)
r=u1/3v-1w-1 [C K CI D] vCI->wD; uK=>C(r)
[CI K D C] wC->uK; vD=>CI(r)
[D K CI C] wC->uK; vCI=>D(r)

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